tag:blogger.com,1999:blog-69953538317331681872018-04-06T11:24:58.709-07:00ALTEREDZINEThe purpose of this blog is to teach Math subjects : Calculus, Pre-calculus, Algebra and Basic Math. It focuses on the learning theories and skills needed in any subject. It educates mostly about the learning skills needed in Math, French, ESL and Spanish and the use of open and web resources for learning. Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.comBlogger55125tag:blogger.com,1999:blog-6995353831733168187.post-34713627065496246002018-03-02T17:56:00.000-08:002018-03-05T06:06:53.953-08:00The formula of the antiderivative of a function simply explained Let's consider the function f(x) = x² +3 its derivative is f '(x) = 2x. The function f '(x) = 2x is the derivative of the function f(x) = x² + 3. The function f(x) = x² +3 is the antiderivative of the function<br />f '(x) = 2x. The antiderivative is the function from which the derivative was calculated.<br /><br />If we consider the function f(x) = 2x it can be considered as the derivative of another function F. This function which is F(x) = x² + 3 is the antiderivative of the function f. Actually, there are several functions whose derivative is f(x) = 2x. Some examples of these functions are F(x) = x²+1, F(x) = x²-3, F(x) = x² + 6. There is a bunch of functions whose derivative is f '(x) = 2x. We can't list them all. For this reason we represent all them by the function F(x) = x²+C.<br /><br /><b>Notation and notion of indefinite integral</b><br /><b><br /></b> The process of finding the antiderivative of a function is called antidifferentiation or integration. If F is the antiderivative of a function f we write ∫f(x)dx = F(x) +C. The left side of this equation is called integral of f(x). The function f(x) is called the integrand. The letter C is the constant of integration. The symbol dx means that the function is integrated with respect to x.<br /><br /><b>Formula to calculate the antiderivative of a single function</b><br /><b><br /></b> Let's retake the function f(x) = 2x. Its derivative is F(x) = x² + C or F(x) = 2/2x² + C. Let's consider another function g(x) = 3x. Its antiderivative can be G(x) = 3/2x²+C. For the function h(x) = x² its antiderivative is H(x) = 1/3x³+C. For the function i(x) =5 x³ its antiderivative is I(x) = 5/4x⁴ + C<br /><br />By observing each of these functions and their antiderivative, two points draw our attention.<br /><br />1) The exponent of x in the antiderivative function is one degree more than the exponent of x in the given function.<br />2) The coefficient of x in the antiderivative function is obtained by dividing the coefficient of x in the given function by the exponent of x in the antiderivative function.<br /><br />Let's come back to the examples above. In the function antiderivative F(x) the exponent of x is 2 and the exponent of x in the function f is 1. The coefficient of x 2/2 in the function F(x) is equal to the coefficient 2 of x in the function f(x) divided by the exponent 2 of x in the antiderivative function.<br /><br /> In the function antiderivative G(x) the exponent of x is 2 and the exponent of x in the function g(x) is 1. The coefficient 3/2 of x in the function G(x) is equal to the coefficient 3 of x in the function f(x) divided by the exponent 2 of x in the function G(x). The same is true for the functions H(x) and I(x).<br /><br />From these observations, we can deduct a rule. In order to find the antiderivative of f((x) = xⁿ add 1 to the exponent and divide the coefficient by the new exponent,<br /><br />The formula that generalizes this procedure is as follows:<br /><br />If f(x) = kxⁿ its derivative is given by F(x) = k/n+1xⁿ⁺¹ + C.<br /><br />That's all for today. To learn more about Calculus visit <a href="http://www.center-for-integral-development.thinkific.com/">Center for Integral Development</a><br /><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/U3FErhwq-gg" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2018/03/notion-of-antiderivative-and-indefinite.htmltag:blogger.com,1999:blog-6995353831733168187.post-11348321696363861112018-02-10T14:28:00.000-08:002018-02-11T06:06:04.564-08:00Formal definition of limitsThe formal definition of limits is mostly skipped in teaching about limits. The intuitive notion is the most taught. The formal definition of limits is interesting and is derived from the intuitive notion.<br /><br />The intuitive definition says that the limit of a function f is L when x approaches a real number "a" if f becomes closer and closer to "L". We write lim f(x) = L when x➡ a .<br /><br /> When x comes closer and closer to a number "a" both to the left and right of that number, x gets values in the neighborhood of a. The variable x in approaching to "a" comes to a certain distance to "a" both from the left and the right. As this distance is very small we call it δ . The variable "x' takes values in the interval ⦐a-ẟ a+ẟ[. The set of values "x" are translated into the equation ᥣx-aᥣ<ε.<br /><br />The independent variable f(x) comes to a certain distance of "L" both from the right and the left. As this distance is very small we call it δ. The independent variable f(x) takes values in the interval ]f(x)-ε f(x)+ε[.<br /><br />When x approaches "a" f(x) approaches "L". In other words you give me an ϵ such that ❙f(x)-a❙<ε I will find you a δ that satisfies the equation ❙x-a❙<δ.<br /><br /><span id="docs-internal-guid-6b64fc77-7c02-de37-c7e9-350a53b0ed9b"><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><img height="514" src="https://lh3.googleusercontent.com/gEw_MYYVHqI2lKaMd5dWgwTNmB4SrALFjFSHtRCM3rKWXQ9X6KkNg1Gmrihu-Xe0F2wm2LKG2hTf1WYh2WNXHxltgSdBqVckkv9Yx6rsEx7gVv23TDAhR8pFTWh34x_QaADWs6S2" style="border: none; transform: rotate(0rad);" width="487" /></span></span><br /><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">The formal definition of limits is stated:</span><br /><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-tYSsgJGfpBU/Wn33WGXBU2I/AAAAAAAAAyg/UIkvp1dc9VY8HVE6xAcLhpR-Iz1MR57xACLcBGAs/s1600/formal%2Bdefinition%2Bof%2Blimits%2B2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="123" data-original-width="637" height="120" src="https://4.bp.blogspot.com/-tYSsgJGfpBU/Wn33WGXBU2I/AAAAAAAAAyg/UIkvp1dc9VY8HVE6xAcLhpR-Iz1MR57xACLcBGAs/s640/formal%2Bdefinition%2Bof%2Blimits%2B2.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><b>Method</b></span><br /><span style="font-family: "arial"; font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"><b><br /></b></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">You give me the ε from the equation ❘f(x)-L❘<ε. I'll transform this equation in ❘x-xindice0❘<δ in order to find δ.</span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c6qfOOv3yrQ/Wn9n-eONm7I/AAAAAAAAAzk/CmU49hVmupIgkHIDbev74GE30tK2N7SFQCLcBGAs/s1600/Example%2B2%2BFormal%2Bdefinition%2Bof%2Blimits.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="242" data-original-width="767" height="200" src="https://2.bp.blogspot.com/-c6qfOOv3yrQ/Wn9n-eONm7I/AAAAAAAAAzk/CmU49hVmupIgkHIDbev74GE30tK2N7SFQCLcBGAs/s640/Example%2B2%2BFormal%2Bdefinition%2Bof%2Blimits.png" width="640" /></a></div><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">However when the inequality ❘f(x)-L❘<ϵ is transformed in a second degree inequality the process becomes more complicated. Follow the process of solving the example below to solve similar examples..</span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-qasUL2m_6EQ/Wn9rJMM_wYI/AAAAAAAAAz8/jD4nQQnQ1ssuQ3n36Pz0qUfBE5wd9JnxwCLcBGAs/s1600/Formal%2Bdefinition%2Bof%2Blimits%2Bexample%2B2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="129" data-original-width="776" height="104" src="https://2.bp.blogspot.com/-qasUL2m_6EQ/Wn9rJMM_wYI/AAAAAAAAAz8/jD4nQQnQ1ssuQ3n36Pz0qUfBE5wd9JnxwCLcBGAs/s640/Formal%2Bdefinition%2Bof%2Blimits%2Bexample%2B2.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-pSeVVgOw3Y8/Wn9r7YTtJZI/AAAAAAAAA0E/OQy4K3BPWxYvMqcc3iw4TYLD37_LBCz1wCLcBGAs/s1600/example%2B2%2Bcontinued.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="335" data-original-width="770" height="273" src="https://4.bp.blogspot.com/-pSeVVgOw3Y8/Wn9r7YTtJZI/AAAAAAAAA0E/OQy4K3BPWxYvMqcc3iw4TYLD37_LBCz1wCLcBGAs/s640/example%2B2%2Bcontinued.png" width="640" /></a></div><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span> <span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><br /><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;"><br /></span></span><span style="font-family: "arial";"><span style="font-size: 14.6667px; white-space: pre-wrap;">Interested in taking online Calculus and tutoring in the same subject visit<a href="http://www.center-for-integral-development.thinkific.com/"> Center for Integral Development</a></span></span><br /><span style="font-family: "arial";"><br /></span><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/ggaGma7r-tk" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2018/02/formal-definition-of-limits.htmltag:blogger.com,1999:blog-6995353831733168187.post-5730376434413855352017-12-01T11:06:00.000-08:002017-12-01T11:06:48.420-08:00Some considerations in the study of CalculusCalculus has been invented a few hundred years ago by Sir Isaac Newton and Leibniz Gottfried to study the motion of planets and moons. After the work of these pioneers, several mathematicians have widened the field of Calculus by developing concepts and methods. The applications of Calculus have been extended to the study of phenomena in the physical, biological and social sciences.<br /><br />Calculus is based on a few simple ideas and these have allowed the development of applications in different fields. The study of Calculus is based on a multi-representational approach to the concepts, methods, and applications represented numerically. analytically and graphically. The interesting element in the study of Calculus is that its core ideas are closed related. For example, the study of limits, derivatives, and integrals form a whole.<br /><br />Calculus is the study of change and this is best modeled by the study of the behavior of functions. Functions have been studied in Pre-calculus, Different combinations of functions such as addition, multiplication, division and composition have been studied. Other properties of functions have been explored. The study of Calculus is more concerned about the behavior of functions closed to certain points. For example, the study of the different values of a function as the dependent variable comes closer and closer to a certain value leads to the notion of limits. The study of the slope of a tangent line to the graph of a function leads to the notion of derivative. The study of the area between curves leads to the study of integrals.<br /><div><br /></div><div>The study of limits helps in the understanding of the derivatives and integrals. The limit of the slope of a secant line to a curve allows to find the slope of a tangent line to this curve. The slope of the tangent line is the derivative of the function. The limit of the sum of the rectangles of the area between two curves leads to a better approximation of the area between these curves. This leads to the notion of integrals.<br /><br />Interested in learning more about Calculus visit this site <a href="http://center-for-integral-development.thinkific.com/collections/mathematical-education-center">Mathematical Education Center </a></div><div><a href="http://center-for-integral-development.thinkific.com/collections/mathematical-education-center">.</a><br /><br /></div><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/kIbyzSNyGYY" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/12/some-considerations-in-study-of-calculus.htmltag:blogger.com,1999:blog-6995353831733168187.post-27570719496095178882017-08-03T11:39:00.000-07:002017-08-03T11:39:04.360-07:00Note to the blog readersHello Readers,<br /><br />I have taken a break since the middle of June in publishing posts about Calculus. These posts are additions to the 2 AP Calculus courses that you can visit at <a href="http://center-for-integral-development.thinkific.com/collections/mathematical-education-center">Mathematical Education Center</a>. I'll be back soon for great content. You can browse the blog to see previous posts in Calculus, math learning, study skills, etc. If you find the content of this blog useful share it to others. If you want to learn more about Calculus you can subscribe to the courses I mentioned earlier. If you are interested in tutoring in Math, French, ESL and Spanish face-to-face and online visit New Direction Services at <a href="http://www.ndes.biz/">www.ndes.biz </a><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/sPtC__KROlQ" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/08/note-to-blog-readers.htmltag:blogger.com,1999:blog-6995353831733168187.post-73365679467369896622017-06-16T20:34:00.001-07:002017-06-17T17:52:55.401-07:00Implicit differentiation<span style="font-size: large;">Implicit differentiation involves differentiating implicit functions. An implicit function is an implicit relation between variables. Differentiating an implicit function leads to differentiate the independent variable with respect to the dependent variable. It's basically finding the derivative using the notation dy/dx.</span><br /><br /><span style="font-size: large;">Two methods can be used:</span><br /><br /><span style="font-size: large;">1) You explicit the function</span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;"><b>Example 1</b></span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;">Find the derivative of 3xy = 2</span><br /><span style="font-size: large;">Let's explicit the function: y = 2/3x</span><br /><span style="font-size: large;">Let's calculate the derivative: dy/dx = d/dx(2/3x)</span><br /><span style="font-size: large;"> = -2(3x)'/(3x)²</span><br /><span style="font-size: large;"> = -6./9x²</span><br /><span style="font-size: large;"> = -2/3x²</span><br /><span style="font-size: large;">2) If expliciting is not possible, you make transformations in order to find the derivative.</span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;">The rules and formulas used to calculate the derivative of different forms of functions apply in the calculations of the derivative of an implicit function.</span><br /><span style="font-size: large;"><br /></span> <span style="font-size: large;">Since an implicit function is a relationship between the independent and the dependent variable the the application of the derivative rules might seem odd. Let's familiarize ourselves with the derivatives of some expressions where the derivative rules are applied.</span><br /><br /><span style="font-size: large;"><b>Example 2</b>. Let's y be a function of x find the derivative of <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 18px;">y³ with </span>respect to x.</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">Let's u = y³. we have two functions: u and y. U is a function of y and y is a function of x. U is a composite function. The chain rule has to be applied in order to find the derivative. The formula to apply here is du/dx = du/dy.dy/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">du/dx = d(y³)/dy.dy/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;"> = 3y²dy/dx.</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;"><b>Example 3</b> Find the derivative of u = 2x²y</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">du/dx = d(2x²y)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">Let's apply the constant rule</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">du/dx = 2d(x²y)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;"> Let's apply the product rule:</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">du/dx = 2[d/dx(x²)y+x² dy/dx]</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;"> = :2(2xy+x²dy/dx)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">du/dx = 4xy+2x²dy/dx </span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;"><b>Example 4</b> Find the derivative of 3y³+x²y = x-3</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">Let's differentiate both sides:</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">d/dx(3y³+x²y) = d/dx(x-3)</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">3y²dy/dx+2xy+x²dy/dx = 1</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">3y²dy/dx+x²dy/dx = 1-2xy</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">(3y²+x²)dy/dx = 1-2xy</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">dy/dx = 1-2xy/3y²+x²</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;"><br /></span></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;"><b>Practice</b>. Find the derivatives of the implicit functions:</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">1) x²+y² = 15</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">2) 3y²-siny = x²</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">3) x²+2xy-y = 2</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;">Interested in learning more about derivatives and Calculus visit <a href="http://center-for-integral-development.thinkific.com/collections/mathematical-education-center">Mathematical Education Center </a></span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: large;"><br /></span></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"> </span><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/aWoSULkBYm0" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/06/implicit-differentiation.htmltag:blogger.com,1999:blog-6995353831733168187.post-15430345638000053602017-06-10T08:04:00.001-07:002017-06-10T08:44:59.186-07:00Derivative of exponential functions<span style="font-size: large;"><b>Derivative of f(x) = </b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-weight: bold;">b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b>x </b></sup></span><br /><span style="font-size: large;"><br /></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: large;">In the expression above b is a positive real number and is called the base of the exponential function.</span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">The formula to calculate the derivative is d/dx[f(x)] = lnb.b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x.</sup></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><br /></sup></span></span><span style="font-size: large;"><span style="font-size: xx-small;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: xx-small;"><b><span style="font-size: xx-small;">Rule: The derivative of an exponential function is equal to the product of the natural logarithm of the base by the function</span>.</b></span></sup></span></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: xx-small;"><b><br /></b></span></sup></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; vertical-align: super;"><b>Example 1</b> calculate the derivative of f(x) = 2</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup></span><br /><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">The given function has the form f(x) = b<sup>x</sup>. By applying the formula d/dx[f(x)] = lnb. b<sup>x</sup> d/dx[f(x)] = ln2.2<sup>x</sup></span></div><span style="font-size: large;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><br /></sup> <b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 13.3333px;">Derivative of </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">f(x) = b</span></b><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b>u</b> </sup></span><br /><span style="font-size: large;"><span style="font-size: x-small;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><br /></sup></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 13.3333px;">Since f is a composite function where u is a function of x the derivative of f is </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 13.3333px;">d/dx [f(x)] = d/du(b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">u</sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 13.3333px;">).du/dx</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">= lnb.</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">u</sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"> .</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">u'</span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b>Rule: The derivative of an exponential function with base b is equal to the product of the natural logarithm of the base by the derivative of u.</b></span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b><br /></b></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b>Example 2</b>. Calculate the derivative of f(x) = 3<sup>2x</sup>. </span></span><br /><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">Let’s apply the formula for the derivative of f(x) = b<sup>u</sup> which is d/dx[f(x)] = lnb.b<sup>u</sup>.u’</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">d/dx[f(x)] = ln3.3<sup>2x</sup>(2<sup>x</sup>)’</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"> = ln3.3<sup>2x</sup>.2</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"> = 2ln3.3<sup>2x</sup></span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"></div><b><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 13.3333px;">Derivative of </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">f(x) = e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup></span></b><br /><span style="font-size: large;"><b><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><br /></sup></b> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">The derivative f(x) = e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x </sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">is a special case of f(x) = b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;"> </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">where b = e</span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">Let's substitute b in the formula d/dx[f(x)] = lnb.</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">b</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup></span><br /><div><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">d/dx[f(x)] = lne.</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup></span></div><span style="font-size: large;">Since lne = 1 d/dx[f(x)] = <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x</sup></span><br /><span style="font-size: large;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><br /></sup> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Rule: The derivative of the function f(x) = e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x </sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">is the function e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">x </sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">itself.</span> </b></span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b><br /></b></span> <b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Derivative of </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">f(x) = e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">u</sup></b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"> </span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Since f is a composite function where u is a function of x its derivative is given by the derivative of a composite function.</span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Then d/dx[f(x)] = </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">d/du(e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">u</sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">).du/dx = e</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">u</sup><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">.u’</span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b style="font-family: "Times New Roman";"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Rule: The derivative</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"> of the composite exponential function with base e is equal to the product of the composite function by the derivative of the function u.</span></b></span></span><br /><span style="font-size: large;"><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span> </span><br /><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"><b>Example 3</b>. Calculate the derivative of f(x) = e3x<sup>2</sup> ( Note this is not e.3x<sup>2</sup> but e with the exponent 3x<sup>2</sup>)</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">Let’s apply the formula for the derivative of f(x) = e<sup>u</sup> which is d/dx[f(x)] = e<sup>u</sup>.u’</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">d/dx[f(x)] = e3x<sup>2</sup>.(3x<sup>2</sup>)’ </span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"> = e3x<sup>2</sup>(6x)</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"> = 6xe3x<sup>2</sup></span></div><span style="font-size: large;"><br /></span> <span style="font-size: large;"><b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span></b> <b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;">Summary</span></b></span><br /><span style="font-size: large;"><b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><br /></span></b> <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><b>The derivative of f(x) = b<sup>x </sup>where b>0 is d/dx(b<sup>x</sup>) - lnb.b<sup>x</sup></b></span></span><br /><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"><b>The derivative of the composite function f(x) = b<sup>u</sup> where u is a function of x is d/dx(b<sup>u</sup>) = lnb.</b><b>b<sup>u</sup></b> <b>u’</b></span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><b><span style="font-size: large;">The derivative of f(x) = e<sup>x</sup> is d/dx(ex) = e<sup>x</sup></span></b></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"><b> The derivative of f(x) = e<sup>u </sup>is d/dx(e<sup>u</sup>) = e<sup>u</sup>.u</b>’</span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;"><br /></span></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><b><span style="font-size: large;">Practice</span></b></div><div style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif; margin: 0em;"><span style="font-size: large;">Calculate the derivative of the following functions: 3x<sup>2</sup></span><br /><div style="margin: 0em;"><span style="font-size: large;">1) f(x) = e<sup>6x</sup> </span></div><div style="margin: 0em;"><span style="font-size: large;">2) f(x) = e3x<sup>2</sup>-4x+3 ( 3x<sup>2</sup>-4x+3 is the exponent )</span></div><div style="margin: 0em;"><span style="font-size: large;">3) f(x) = e<sup>x</sup>-e<sup>-x</sup>/e<sup>x</sup>-e<sup>-x</sup></span></div><div style="margin: 0em;"><sup><span style="font-size: large;"><br /></span></sup></div><div style="margin: 0em;"><sup><span style="font-size: large;">Interested in learning more about Calculus visit this site</span> <a href="https://center-for-integral-development.thinkific.com/collections/mathematical-education-center?q=">Mathematical Education Center</a></sup></div></div><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/rtVkS-dXwJA" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/06/derivative-of-exponential-functions.htmltag:blogger.com,1999:blog-6995353831733168187.post-81067031858872875852017-06-09T08:44:00.002-07:002017-06-09T08:55:24.165-07:00Derivative of logarithmic functionsIn this post I'll show some techniques to remember the formulas for logarithmic functions. I'll do some examples and leave some exercises to practice.<br /><br /><b>Derivative of logarithmic functions </b><br /><b><br /> Derivative of <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log</span><sub style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">b</sub><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">x</span></b><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">d/dx (</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log</span><sub style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">b</sub><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">x) = 1/xlnb</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">To remember this formula let's apply the following technique:</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">1) Multiply the number of which we calculate the logarithm by the natural logarithm of the base. The number here is x and the base is b. Therefore we have xlnb</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">2) Take the inverse of this product. The inverse of the product is 1/xlnx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><b>Derivative of lnx</b></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">d/dx(lnx) = 1/x</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">The derivative of the logarithm of any number is equal to the inverse of this number.</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><b><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">Derivative of </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>b</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u</span></b><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Since </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>b</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u is a composite function its derivative is given by d/dx(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>b</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u) = d/du(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>b</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u).du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">d/dx(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>b</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u) = 1/ulnnb.du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><b>Rule: The derivative of the logarithm of a composite function is equal to its derivative with respect to the new variable (u) multiplied by the derivative of the new variable (u) with respect to x.</b></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><b>Derivative of lnu </b></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><b><br /></b></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Since u<b> </b>is a composite function we have d/dx(lnu) = d/du(lnu).du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = i/u.du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><b>Rule: The derivative of the natural logarithm of a composite function u is equal to the inverse of the function multiplied by its derivative with respect to x</b></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><b><br /></b></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Example 1, Calculate the derivative of y = x³</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">The derivative of y is y" = (x³</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x)'</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Let's apply the product rule:</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Y' = (x³)'(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x) + x³(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x)'</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">The derivative of x³ is obvious. Let's calculate the derivative of </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Let's write u = </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x we have (</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">log</span><sub style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">5</sub><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u)' = d/du(</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">log</span><sub style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">5</sub><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">u),du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = i/uln5.u'</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = 1/2x.ln5.(2x)'</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = 1/2x.ln5.2</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = 1/xln5</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">let's go back to the derivative of y we have:</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">y' = 3x²</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x + x³.1/xlnx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = 3x²</span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;">log<sub>5</sub></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">2x+x²/lnx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Example 2. Calculate the derivative of y = ln(2x²-4x+3)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Let's write u = 2x²-4x+3</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">We have y = lnu</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Then dy/dx = d/dx(lnu)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;">Since lnu is a composite function then dy/dx = d/du(lnu).du/dx</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"> = 1/u(4x-4)</span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;">Substitute u: dy/dx = (1/2x³-4x+3).(4x-4)</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;">dy/dx = 4x-4/2x²-4x+3</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;"> = 4(x-1)/2x³-4x+3</span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;"><br /></span></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;"><b>Practice</b></span></span><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif;"><span style="font-size: 12.5px;"><b><br /></b></span></span>Calculate the derivative of the following functions;<br />1.log₅(2x+5)<br />2. 5/log(x+4)<br />3. ln(sinx)<br /><br />Interested in learning more about Calculus AB visit this site <a href="http://www.center-for-integral-development.thinkific.com/">Center for Integral Development<b> </b></a><br /><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 12.5px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 15px;"><br /></span><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/okSI7K-d7Ag" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/06/derivative-of-logarithmic-and.htmltag:blogger.com,1999:blog-6995353831733168187.post-57821366223314393362017-05-27T15:42:00.001-07:002017-05-27T16:02:02.150-07:00Derivative of Trigonometric functionsIn this post I will state the formulas for the derivative of trigonometric functions. I will also give some techniques to remember them and solve problems. I'll do some examples and give some exercises for practice. The formulas will not be demonstrated here.<br /><br />It's not sufficient to know the formulas for the derivative of trigonometric functions to be able to calculate the derivative of functions containing trigonometric expressions. The calculations of these functions involve being able to apply all the other rules that enable to calculate the derivative of a function.<br /><br /><b>Derivative of the function sine</b><br /><b><br /></b> The derivative of the function sine is equal to the function cosine. If f(x) = sinx f''(x) or d/dx(sinx) = cosx<br /><br /><b>Derivative of the function cosine</b><br /><b><br /></b> The derivative of the function cosine is equal to the opposite of the function sine. If f(x) = cosx f'(x) or d/dx(cosx) = -sinx<br /><br /><b>Derivative of the function tangent</b><br /><b><br /></b> The derivative of the function tangent is equal to the square of the secant function. If f(x) = tanx f'(x) or d/dx(tanx) = sec²x<br /><br /><b>Derivative of the function cotangent</b><br /><b><br /></b> The derivative of the function cotangent is equal to the opposite of the square of the cosecant function. If f(x) = cotx f''(x) or d/dx(cotx) = -csc²x<br /><br /><b>Derivative of the function secant</b><br /><b><br /></b> The derivative of the function secant is equal to the product of the function secant by the function tangent. If f(x) = secx f'(x) or d/dx(secx) = secx.tanx<br /><br /><b>Derivative of the function cosecant</b><br /><b><br /></b> The derivative of the function cosecant is equal to the opposite of the product of the function cosecant by the function cotangent. If f(x) = cosecx f'(x) or d/dx(cosecx) = -cosecx.cotx.<br /><br /><b>Observations that allow to memorize the formulas</b><br /><b><br /></b><b>1) All the derivatives of co-functions have the negative sign.</b> For examples, the derivative of cosx = -sinx, the derivative of cotx = -cosec²x, the derivative of cosecx = -cosecx.tanx<br /><b>2) For the sine and cosine functions the derivative of the first function is equal to the second function </b>The derivative of sinx is cosx.<b> The derivative of the second function is equal to the opposite of the first function. </b>The derivative of cosx is -sinx<br /><b>3) When thinking about the drivative of the tangent and cotangent functions think about the the square of the function secant and cosecant.</b> The derivative of the tangent goes with the square of the secant Example the derivative of tanx = sec²x. <b>The derivative of cotangent goes with the square of the cosecant preceded by the negative sign,</b> Example the derivative of cotx = -csc²x<br />4) For the derivative of the functions secant and cosecant think about multiplying the function secant by the function tangent and the cosecant by cotangent. Example the derivative of secx = secx.tanx. The derivative of coscx = -coscx.cotx <b>In the case of the derivative of the cosecant don't forget to place negative placed before the product. </b><br /><br /><b>Example 1</b><br /><b><br /></b> If f(x) = x²cosx+sinx find f'(x)<br /><br />The derivative of a sum of two functions is equal to the sum of the derivatives of each function.<br />f'(x) = (x²cosx)'+(sinx)'<br /> Applying the product rule to calculate the derivative of x²cosx<br />f'(x) = (x²)'(cosx) + (x²) (cosx)'+ cosx. I apply the formula (uv)' = u'v+uv'<br />f'(x) = 2xcosx + (x²)(-sinx) + cosx<br /> = 2xcosx-x² sinx+cosx<br /> = -x²sinx + 2xcosx + cosx.<br /><b><br /></b><b>Example 2</b><br /><b><br /></b> If f(x) = sin²x find f'(x)<br />Let's write f(x) as f(x) = (sinx)²<br />Let's write sinx = u. Then f(x) = u² and f(u) = u²<br />The function f becomes the function composite f(u)<br />The derivative of the composite function f(u) is f'(u) = f'(u).u'<br />Since f(x) and f(u) are both equivalent we have f(x) = f'(u).u'<br />f'(x) = 2u,u'<br /> = 2sinx.(sinx)' (Substituting u)<br /> = 2sinxcosx.<br /><br /><br /><b>Example 3</b><br /><b><br /></b>Find the derivative of f(x) = sinx-1/sinx+1<br /><br />Applying the quotient rule f'(x) = (sinx-1)'(sinx+1)-(sinx-1)(sinx+1)'/(sinx+1)²<br />Calculating the derivatives: f'(x) = cosx(sinx+1)-(sinx-1)cosx/(sinx-1)²<br />f'(x) = sinxcosx+cosx-sinxcosx+cosx/(sinx-1)²<br />f'(x) - cosx/(sinx-1)²<br /><br /><b>Practice</b><br /><br />1) What are the techniques to memorize the formulas of the derivative of the following functions<br />a) sine and cosine<br />b) tangent and cotangent<br />c) secant and cosecant<br /><br />2) Calculate the derivatives of the following functions:<br />a) f(x) - xsinx+4<br />b) f(x) = xcox-x²tanx-2<br />c) f(x) = cos³x<br />d) f(x) = cosx+sinx/cosx-sinx<br /><br />Interested in knowing more about derivatives visit this site <a href="http://www.center-for-integral-development.thinkific.com/">Center for Integral Development</a><br /><a href="http://www.center-for-integral-development.thinkific.com/"><br /></a><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/daKQpF5bucQ" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/05/derivative-of-trigonometric-functions.htmltag:blogger.com,1999:blog-6995353831733168187.post-10554953713031871692017-05-23T06:27:00.000-07:002017-05-27T10:55:38.985-07:00Derivative of a composite function<br /><b>Derivation of a composite function</b><br /><br />Let's consider a function g. The image by g of any element x of its domain is g(x). Let's consider another function f. The image of g(x) by f is f[g(x)] also written as fog(x). The function fog is called the composed function of g and f.<br /><br />If g is differentiable for any element x and f is differentiable at g(x) fog(x) = f[g(x)] is differentiable at x. The derivative of the function fog is (fog)'(x) = f''[g(x)].g'(x), The demonstration of this formula is not done here.<b></b><br /><br />The derivative of fog is the product of the derivative of fog by the derivative of g.<br /><br />If u is a function of x and f is a function of u then f(u) is a composite function. By applying the rule above the derivative of f(u) or f'(u) is equal to the derivative of f(u) multiplied by the derivative of u. We write [f(u)]' = f''(u).u'. If we introduce the notation (d) of differentiability we can write d/dx[f(u)] =d/du[f(u)].du/dx.<br /><br />In practical applications we have a function f to differentiate with respect to x. We then introduce a function u that is a function of x. Now we have the composite function f(u). The diferentiation or derivative of f with respect to x is equal to the derivative of f with respect to u multiplied by the derivative of u with respect to x . This derivative is called the chain rule. There is a chain of operations to do. First we introduce a new function u. Then we calculate the derivative of f as the the composite function f(u) by applying the formula for the derivative of a composite function.<br /><br />The chain rule holds also the application of the power rule when we work with a complex function with exponents.<br /><br />The power rule applies by introducing the new function u.<br /><br /><b>Example 1 </b><br /><br />Let's calculate the derivative of the function f(x) = (2x+1)²<br /><br />To make the computation of the derivative easy we introduce the function u. Then the function f becomes f(x) = u². The derivative of the function f with respect to x is the derivative of the expression u² with respect to x . We write d/dx[f(x)] = d/dx[u²]<br /><br />By applying the formula for the derivative of a composed function we have d/dx[f(x)] = d/du(u²).du/dx.<br /><br />By calculating d/du(u²) we obtain d/dx[f(x)] = 2u. u'<br /><br />Let's substitute u: d/dx[f(x)] = 2 (2x+1)(2x+1)'<br /><br />By calculating the derivative of 2x+1 we obtain d/dx[f(x)] = 2(2x+1)(2) = 4(2x+1) = 8x+1<br /><br /><b>Example 2</b><br /><b><br /></b> Calulate the derivative of f(x) = (x²+3x+4)²<br /><b><br /></b> Let's write u = x^2+3x+4<br /><b><br /></b> d/dx[f(x)] = d/dx(x²+3x+4)²<br /><b> = </b>d/dx(u²)<br /> = d/du(u²).du/dx (Applying the formula of the derivative of a composite function)<br /> = 2u.u'<br /> = 2(x²+3x+4)(x²+3x+4)' (Substituting u)<br /> = 2(x²+3x+4)(2x+3)<br /> = 2(2x³+6x²+8x+3x²+9x+12)<br /> = 2(2x³+9x²+17x+12)<br /> = 4x³+18x²+34x+24)<br /><br />Interested in learning more about the techniques of calculations for derivatives visit this site and subscribe to the<a href="http://www.center-for-integral-development.thinkific.com/"> Calculus course</a><br /><br /><br /><br /><br /><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/EuxG2pRWGrg" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/05/derivative-of-composed-function.htmltag:blogger.com,1999:blog-6995353831733168187.post-49788849904638605412017-05-12T20:10:00.000-07:002017-05-12T20:10:17.141-07:00Derivative computations<span style="font-size: large;">The formula lim f(x)-f(x+h)/h when x→h that defines the derivative of a function f implies tedious calculations to calculate the derivative of some types of functions and combinations of functions..</span><br /><br /><span style="font-size: large;">Therefore some formulas have been established to determine the derivatives of a combination of functions and some specific types of functions.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The formulas for the constant function and the power functions are called respectively constant rule and power rule. The formulas for the sum, product and quotient of functions are called respectively addition rule, product rule and quotient rule. The derivative of a composition of 2 functions f and g is called the chain rule. It is an extension of the power rule The trigonometric, logarithmic and exponential functions have their specific formula.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The derivative of an implicit function is called implicit differentiation.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">It is essential to memorize the formulas. Otherwise, it would be difficult to calculate the derivatives of these particular functions. Today we are going to limiting ourselves to the learning of the basic formulas: constant, power, sum, product and quotient rule.</span><br /><span style="font-size: large;"><br /></span><b><span style="font-size: large;">Derivative of a constant</span></b><br /><span style="font-size: large;"><b><br /></b> The derivative of the function constant is 0. If f(x) = c the derivative of f(x) is 0. We write: <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;"> </span><span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">f′(x) = 0.</span></span><br /><span style="font-size: large;"><br /></span><b><span style="font-size: large;">The Power rule</span></b><br /><div class="separator" style="clear: both; text-align: center;"><span style="font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><span style="font-size: large;">The derivative of the function power defined by f(x) = <span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">x</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">n</sup> is equal to n multiplied by x to the power of n-1. The formula is .<span style="font-family: "lucida sans unicode" , "lucida grande" , sans-serif; font-size: 18px;">f’(x) = nx</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">n-1</sup></span><br /><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: large;"><br /></span></sup><span style="font-family: lucida sans unicode, lucida grande, sans-serif; font-size: large;"><b>Derivative of the product of a constant by a function</b></span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The derivative of the product of a function by a constant is equal to the product of the constant by the derivative of the function power.</span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">If <span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f(x) = ax</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">n</sup><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"> its derivative is </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f’(x) = ax</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">n-1</sup></span><br /><span style="font-size: large;"><br /></span><span style="font-size: large;"><br /></span><b><span style="font-size: large;">Derivative of the function f(x) = x</span></b><br /><span style="font-size: large;"><br /></span><span style="font-size: large;">The derivative of the function f(x) = x can be calculated using the formula for the derivative of the function power. In order to use this formula we have to write f(x) = x as the function power. We write f(x) = x as <span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f(x) = x</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">1 </sup></span><br /><span style="font-size: large;"><span style="font-family: Lucida Sans Unicode, Lucida Grande, sans-serif;">By applying the formula for the function power we obtain </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f’(x) = x</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">1-1 </sup></span><br /><span style="font-size: large;"><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"> </sup><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f’(x) = x</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">0</sup><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"> ⇒ </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f’(x) = 1</span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"> </span></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: large;"><br /></span></span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;">Derivative of a sum of functions</span></b></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;"><br /></span></b></span><span style="font-size: large;"><span style="font-family: Lucida Sans Unicode, Lucida Grande, sans-serif;">If f. g. h,;;; are differentiable for any value of x of their domain the derivative of the sum of these functions is </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">f’+g’+h’+ ....</span></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: large;"><br /></span></span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;">Derivative of the product of 2 functions</span></b></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;"><br /></span></b></span><span style="font-size: large;"><span style="font-family: Lucida Sans Unicode, Lucida Grande, sans-serif;">If f and g are differentiable for any value x of their domain the derivative of the product f.g is </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">fg’+gf’</span></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><span style="font-size: large;"><br /></span></span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;">Derivative of the quotient of 2 functions</span></b></span><br /><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;"><br /></span></b></span><span style="font-size: large;"><span style="font-family: Lucida Sans Unicode, Lucida Grande, sans-serif;">If f and g are differentiable for any value of their domain the derivative of the quotient f/g is </span><span style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">(f∕g)’ = f’g-gf’∕g</span><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;">2</sup></span><br /><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;"><br /></span></b></sup><span style="font-family: Lucida Sans Unicode, Lucida Grande, sans-serif; font-size: large;">These formulas have to be demonstrated and the learners have to do some exercises to apply them. If anyone is interested in learning more subscribe to these courses via this link <a href="http://www.center-for-integral-development.thinkific.com/">Free Introductory Calculus Course and Complete Calculus Course</a></span><br /><sup style="font-family: "Lucida Sans Unicode", "Lucida Grande", sans-serif;"><b><span style="font-size: large;"><br /></span></b></sup><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /> <img src="http://feeds.feedburner.com/~r/Alteredzine/~4/89u4F9JUvM4" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0North America26.902476886279832 -90.8789062519.717955386279833 -101.20605474999999 34.086998386279831 -80.551757750000007http://alteredzine.blogspot.com/2017/05/derivative-computations.htmltag:blogger.com,1999:blog-6995353831733168187.post-60309725243435071072017-04-25T11:59:00.002-07:002017-05-05T10:09:25.569-07:00Introduction to the notion of derivativeIn studying limit we observe what happens to the values of a function when the values of the independent variable become closer and closer to a certain value. If a function is defined for every value of its domain it is continuous there. Graphically it means that there is no hole, jump or infinite branch. Quantitatively the function has a value for every value of the independent variable that belongs to the domain of the function. In limit and continuity we have been observing some changes in the behavior of a function when the independent variable behaves in a certain way. A function might have a limit when the independent variable becomes closer and closer to a certain value. For other values of the independent variable the same function has no limit. The function is not continuous.<br /><br />The notion of derivative allows us to study more systematically the notion of change in a function. It allows us to study the change at any value of a function. The slope of a function allows us to study the change in this function with respect to the change or the increase of the value of the independent variable. The slope of a line is the rate of change of the independent variable with respect to the change of the dependent variable. Since a non-linear function varies in different ways there is no precise method to define its slope. This leads to the study of the slope of a tangent line to a function. In order to study the change of a function it is important to define the notion of rate of change or slope of a line. the slope of a secant line to a curve or average rate of change or speed and slope at a point of a curve or instantaneous rate of change.<br /><br /><b>Slope of a line</b><br /><br />The notion of slope is familiar to the civil engineers when they build roads. They have to figure out what type of slope they have to give to a road especially when they build it on a hill or in mountains. They have to shape the road in the right slope because if the road is too steep the cars cannot climb it. The slope is calculated by taking the tangent of the angle opposed to the right angle in a right triangle where the hypotenuse is the side that is going to be inclined. The slope is the measure of the inclination.of the hypotenuse. Its measure is calculated by dividing the opposite side to the angle to the adjacent side :<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-pcaq0IVPQDI/WNp56pxnBhI/AAAAAAAAAlM/idcZ1z6Q-SkWByOFWjfUSyBNYr1wqRwWACLcB/s1600/trigonometric%2Bformula%2Bof%2Bthe%2Bslope.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" src="https://4.bp.blogspot.com/-pcaq0IVPQDI/WNp56pxnBhI/AAAAAAAAAlM/idcZ1z6Q-SkWByOFWjfUSyBNYr1wqRwWACLcB/s1600/trigonometric%2Bformula%2Bof%2Bthe%2Bslope.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Em6_Pd5dPWE/WNp4kLCyCZI/AAAAAAAAAlA/_xTvwlpdp-8QOKeKrnwlqErv-ntdP_qrQCLcB/s1600/right%2Btriangle%2B2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="191" src="https://2.bp.blogspot.com/-Em6_Pd5dPWE/WNp4kLCyCZI/AAAAAAAAAlA/_xTvwlpdp-8QOKeKrnwlqErv-ntdP_qrQCLcB/s200/right%2Btriangle%2B2.png" width="200" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>The slope of a function is also called the rate of change of this function. The slope of a line is called the rate of change of this line. It is the rate of the increase of y to the increase of x. It is constant at any part of the graph. It can be positive, negative or equal to zero. The slope of a line is calculated by dividing the difference of the y-ordinates of two points of that line by the difference of the x-ordinates.<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><br /></div> Watch this video to get some understanding of the notion of slope:<br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Iqws-qzyZwc/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Iqws-qzyZwc?feature=player_embedded" width="320"></iframe></div> <br /><b>Slope of a tangent line to a curve</b><br /><b><br /></b><span id="docs-internal-guid-22c56046-a135-ae15-3de9-a2e18207860e"><span style="font-family: "arial"; font-size: 12pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;"><img alt="Graph of the slope of a tangent line.png" height="390" src="https://lh3.googleusercontent.com/fuIymxUtZRNERRstRL6kkfwwBdze6Mu229JIcGVBx34F20P8VibSBoQl_5ofPH5A04LXD_C9EwDIe5JTMkG31NSK9NVilB-a9Yocy5iKEZ29L1o9xDB4dj8mODUO0DIeYvcZqeF_" style="-webkit-transform: rotate(0.00rad); border: none; transform: rotate(0.00rad);" width="499" /></span></span><br />We have a curve (C), a secant line (PQ) and a tangent line L to the curve at the point P. The problem is to find the slope of the tangent line at P. In order to do this we make the point Q become closer and closer to the point P. As the point Q becomes close to the point P the initial secant P occupies different positions. At each position the secant has a different slope, The slope of the tangent line is the limit of the slopes of the different positions of the secant (PQ). In order to come to this conclusion let's calculate the function that allows to find the slope of the secant line (PQ).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-bNYwg5pKhYA/WP-OX5ZxWeI/AAAAAAAAAnA/-3tUtXCakRAiihmVu0djbp8KK7SYhY3GQCLcB/s1600/Definition%2Bof%2Bthe%2Bderivative%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="378" src="https://1.bp.blogspot.com/-bNYwg5pKhYA/WP-OX5ZxWeI/AAAAAAAAAnA/-3tUtXCakRAiihmVu0djbp8KK7SYhY3GQCLcB/s640/Definition%2Bof%2Bthe%2Bderivative%2B2.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/yuEKC93wd_E/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/yuEKC93wd_E?feature=player_embedded" width="320"></iframe></div><br /><div class="separator" style="clear: both; text-align: center;"></div>If you are interested in learning more about these concepts you can subscribe to this free <a href="https://center-for-integral-development.thinkific.com/courses/introduction-to-calculus">Introductory Calculus</a> course to start learning about limits and move on to this complete course <a href="https://center-for-integral-development.thinkific.com/courses/calculus-ab">Calculus AB</a><br /><b><br /></b><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/lgLWEstkMpo" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/04/introduction-to-notion-of-derivative.htmltag:blogger.com,1999:blog-6995353831733168187.post-84058764653996543782017-03-20T19:18:00.000-07:002017-03-20T19:18:03.726-07:00Properties.Methods and Procedures to calculate limits and continuitySometimes we seem lost through the details when studying a subject. However if we get the big picture it becomes easy to continue studying. A math topic is structured in concepts, rules or properties and theorems. This is the theoretical part. Then come the applications. The theories are applied in the applications but the procedures and methods are mastered through practice. Knowing some key theories and procedures can help tremendously in the solutions of problems. In this post I will highlight the properties of limits and continuity, the methods and procedures to solve problems.<br /><br /><b>Properties of limits</b><br /><b><br /></b>The properties of limit show how to calculate the limits of a combination of functions like the sum, the difference, the multiplication and division of functions. It shows also how to calculate the square root of a function.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-5l039EQcwMk/WNAV5fhqN1I/AAAAAAAAAkI/HUrb2Q6-CqQNeGRJl5uRPgafyZrOSnHXQCLcB/s1600/Properties%2Bof%2Blimits.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="338" src="https://1.bp.blogspot.com/-5l039EQcwMk/WNAV5fhqN1I/AAAAAAAAAkI/HUrb2Q6-CqQNeGRJl5uRPgafyZrOSnHXQCLcB/s400/Properties%2Bof%2Blimits.png" width="400" /></a></div><br /><b>Properties of continuous functions</b><br /><b><br /></b><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-NPBNjFsas0k/WNAZrGBFz-I/AAAAAAAAAkc/F8-WnZ__2NsBFLHQgjm5B2M9SVFwkmYowCLcB/s1600/Properties%2Bof%2Bcontinuous%2Bfunctions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="121" src="https://2.bp.blogspot.com/-NPBNjFsas0k/WNAZrGBFz-I/AAAAAAAAAkc/F8-WnZ__2NsBFLHQgjm5B2M9SVFwkmYowCLcB/s400/Properties%2Bof%2Bcontinuous%2Bfunctions.png" width="400" /></a></div><b>Methods for determining limits</b><br /><b><br /></b>There are three methods that can be used to determine a limit. These methods are: graph, table and algebra. The graph method consists in determining a limit from the graph. The table method consists in calculating the limit to the left and to the right by drawing a table for each one-sided limit. The table allows to see the behavior of the values of f(x) as x gets closer and closer to a fixed value. From there we can conclude if the limit to the right or to the left exists. If the limits from both sides exist and are equal then the limit of the function exists at the given value. The algebra method consists by substituting the value of the independent variable in the function.<br /><br /><b>Method for determining if a function is continuous</b> <br /><br />To determine if a function is continuous, we find out if it satisfies the three following conditions:<br />1) It is defined at a specified point "a"<br />2) The limit at the point "a" exists<br />3) The limit of the function at the point "a" is equal to f(a).<br /><br />If you are interested in learning more about these concepts you can subscribe to this free <a href="https://center-for-integral-development.thinkific.com/courses/introduction-to-calculus">Introductory Calculus</a> course or this complete course <a href="https://center-for-integral-development.thinkific.com/courses/calculus-ab">Calculus AB</a><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/sXTUDyJrD7E" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0Boston, MA, USA42.3600825 -71.0588801000000141.984348999999995 -71.704327100000015 42.735816 -70.4134331http://alteredzine.blogspot.com/2017/03/propertiesmethods-and-procedures-to.htmltag:blogger.com,1999:blog-6995353831733168187.post-1627139691731060862017-03-11T10:06:00.000-08:002017-03-11T10:06:04.609-08:00Limits and Continuity vocabularyThese definitions can be best learned by watching some videos and observing the graphs of the functions. If you have learned the previous lessons there shouldn't be any problems mastering them <br /><b><br /></b><b>Limit</b><br /><b><br /></b> If the values of a function f approach a number L as the variable gets closer and closer to a number "a", then L is said to be the limit of the function f at the poin "a".<br /><b><br /></b> <b>Two-sided limit</b><br /><b><br /></b> A two-sided limit is a limit where both the limit to the left and the limit to the right are equal<br /><b><br /></b> <b>One-sided limit</b><br /><b><br /></b> A one-sided limit is a limit taken as independent variable approaches a specific value from one side (from the left or from the right).<br /><b><br /></b> <b>Limit to the left</b><br /><b><br /></b> If the values of a function approach a number L as the independent variable gets closer and closer to a number "a"in the left direction. then the number L is said to be the limit of the function f to the left at the point "a"<br /><b><br /></b> <b>Limit to the right</b><br /><br />If the values of a function approach a number L as the independent variable gets closer and closer to a number in the right direction, then the number L is said to be the limit of the function f to the right at the point "a". <b> </b><br /><b><br /></b> <b>Asymptote</b><br /><br />An asymptote is a straight line to a curve such that as a point moves along an infinite branch of a curve the distance from the point to the line approaches zero as and the slope of the curve at the point approaches the slope of the line <b> </b><br /><b><br /></b> <b>Vertical asymptote</b><br /><br />A vertical asymptote is a vertical line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero<br /><b><br /></b> <b>Horizontal asymptote</b><br /><br />A horizontal asymptote is a horizontal line to a curve such that as a point moves along an infinite branch of the curve the distance from the point to the line approaches zero. <b> </b><br /><b><br /></b> <b>End behavior</b><br /><b><br /></b>This is the behavior of the arm branches or infinite arm branches of a curve. In the case of a curve with a vertical asymptote the arm branch approaches the asymptote more and more.<br /><b><br /></b> <b>Continuity of a function at a point</b><br /><br />A function f is continuous at a point "a" if the limit of the function when x approaches "a" is equal to the value of the function at this point <b> </b><br /><b><br /></b> <b>Continuity of a function on an interval</b><br /><b><br /></b> A function f is continuous on an interval if it is continuous at every point of the interval<br /><b><br /></b> <b>Continuity of a function to the left at a point</b><br /><b><br /></b> A function f is continuous to the left at a point "a" if its limit to the left is equal to the value of the function at this point<br /><b><br /></b> <b>Continuity to the right</b><br /><b><br /></b> A function f is continuous to the right at a point "a" if its limit to the right is equal to the value of the function at this function.<br /><br /><b>Continuous function</b><br /><b><br /></b> A continuous function is a function of which the graph can be drawn without lifting the pencil. Its graph has no hole, jump or asymptote. Algebraically a function f is continuous if for every value of its domain the limit exists.<br /><br /><b>Discontinuous function</b><br /><b><br /></b> A discontinuous function is a function of which the graph has hole, jump or asymptote. Algebraically a discontinuous function is either not defined at a point of its domain, doesn't have a limit at this point or the limit at this point is not equal to the value of the function at this point.<br /><b><br /></b> <b>Removable discontinuity</b><br /><b><br /></b>Graphically a removable discontinuity is a hole in a graph or a point at which the graph is not connected there. The graph can be connected by filling in the single point.<br />Algebraically a removable discontinuity is one in which the limit of the function does not equal to the value of the function. This may be because the function does not exist at that point.<br /><b><br /></b> <b>Non-removable discontinuity</b><br /><b><br /></b> <b>A</b> non-removable discontinuity is a point at which a function is not continuous or is undefined. and cannot be made continuous by giving a new value at the point. A vertical asymptote and a jump are examples of non-removable discontinuity.<br /><br /><b>Intermediate value theorem</b><br /><b><br /></b>If a function f is continuous over an interval [a b] and V any number between f(a) and f(b), then there is a number c between a and b such as f(c) = V (that is f is taking any number between f(a) and f(b)). We can deduce from this theorem that if f(a) and f(b) have opposite signs, there is a number c such as f(c) - 0. This can be used to find the roots of a function,<br /><br />If you are interested in learning more about these concepts you can subscribe to this free <a href="https://center-for-integral-development.thinkific.com/courses/introduction-to-calculus">Introductory Calculus</a> course or this complete course <a href="https://center-for-integral-development.thinkific.com/courses/calculus-ab">Calculus AB</a><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/-gA1fCQsmBM" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/03/limits-and-continuity-vocabulary.htmltag:blogger.com,1999:blog-6995353831733168187.post-32135778042810665112017-01-28T22:21:00.000-08:002017-01-28T22:21:33.549-08:00Introduction to the notion of continuity of functions In general, something that is continuous continues without interruption. If during a jogging you run from point A to B without stopping your running is continuous. However, if you stop even once the running is discontinuous. A line of cars in traffic that never stop is continuous. If the cars stop the line is discontinuous. If you draw a straight line without lifting your pencil the line is continuous. If you draw a straight line with dots you lift your pencil several times. The line is discontinuous at every dot.<br /><br />The graph of the linear, parabolic, third-degree functions is an unbroken curve. It can be drawn without lifting the pencil from the paper. In general, the polynomial functions are continuous because their limit exists everywhere in the domain of the real numbers. A function of which the graph has holes, jumps or breaks is not continuous. Such functions are discontinuous.You have to lift your pencil to draw their graph.<br /><br />Watch these videos to get an idea of what it means for a function to be continuous.<br /><br /><br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Q7tEPyKS4Jg" width="500"></iframe> <iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/DCWeH62w-NA" width="500"></iframe><br /><br />Interested about an Introduction to Calculus course take this one for free <a href="http://center-for-integral-development.thinkific.com/courses/introduction-to-calculus">Introduction to Calculus</a> or if you prefer take the complete Calculus course<a href="http://center-for-integral-development.thinkific.com/courses/calculus-ab"> Calculus AB</a><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/aFPKGGn5s6E" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2017/01/introduction-to-notion-of-continuity-of_28.htmltag:blogger.com,1999:blog-6995353831733168187.post-46170481620343115572016-12-02T11:31:00.000-08:002016-12-05T10:08:37.135-08:00Notions of limits (written lesson)In the previous lesson on limits, I introduced the lesson by assigning some videos that you have to watch. Today we get to the written part of the lesson.<br /><br />I am going to describe this lesson and give you the assignments that you should do.<br /><br />Before I continue I have to tell you that your learning should not be limited to what your teacher gives you. There are plenty of resources that you can use to learn. You can learn from a teacher, from someone who knows a subject well and can teach it to others, from books, from electronic resources like CDs, from electronic communications like radio and televisions. More importantly today there is plenty of resources in the internet that you can use if you know how to access them. You should make yourself comfortable to all types of resources that you can use for learning. Videos are great to learn something but you can't limit yourself to this only. If you want to learn something deeply you have to get the written materials. The written materials allow you to get an overview of what you are going to learn and give you also the content. You can choose which parts to learn first or which parts to drop depending on your interests. The most important thing also is you can review the materials as much as you can. If your reading skills are good you can learn a lot from written materials but for math there isn't a lot to read. You have to do the reading and memorize certain things. You have to practice a lot.<br /><br />Here is the<a href="https://docs.google.com/document/d/1ZTICi3myfsHuHSWvBMhOp_6HkPSwZHreVE9lQ7bLT-c/edit"> link</a> for the lesson but before you start read the following;<br /><br /><b>Description of the lesson</b><br /><br />This lesson starts by a definition of limits and shows you the three methods of limits using examples. The lesson ends by giving you some problems to do. Below I give you the readings that you have to do under each sub-title and the tasks you have to do for each lesson.<br /><br /><b>Assignments</b><br /><b><br /></b><b>1. Objectives</b><br /><br />You should start by reading the objectives again then read the definition of limits. The first video that you watched on the previous lesson with videos gave you verbally an idea of what a limit is. Now you are going to have a written idea of limit and three methods that allow you to calculate a limit.<br /><b><br /></b><b>2. The Idea.</b><br /><br />You read the paragraph giving you an idea of what a limit is. You already have a video demonstration giving you the idea of a limit. Now you have a conceptual definition of a limit. You should try to state this definition either in your own words without compromising the concept or verbatim for more accuracy. Now that you have a definition of limit you are going to learn in written words how to find the limit of a function using three methods: <u>graph</u>, <u>table, algebra.</u><br /><u><br /></u><b>3. Methods for determining limits</b><br /><b><br /></b><b>a) The graph method</b><br /><b><br /></b>Under this title you should see a problem named "Example 1". This problem asks you to find three limits using the graph on the right. This problem is already solved for you. You are going to do two things with this problem.<br /><br /><b>Task I </b><br /><b><br /></b>Read the example and its solution. Read the explanations provided for the solution of the problem in case you don't understand it. Below is a guide to the explanations.<br /><br /><b>Explanation of the solution a) </b><br /><br />You should be able to understand the solution easily. I provide the explanations of the solution in case you don't understand it. I give you a method to understand the solution. It's graphic. You should read and do it.<br /><br /><b>Explanation of the solution of b) and c)</b><br /><br />The same method is used for the solution of a) and b)<br /><br /><b>Explanation of the solution of d)</b><br /><b><br /></b>You can use the same method for the solution of d) but this time notice that the function doesn't have a limit.<br /><br /><b>Task 2 </b><br /><b><br /></b>Do a pencil and paper to do the example yourself without referring to the solution. After you finish verify that your answers are correct.<b> </b><br /><br /><b>Task 3</b><br /><br />Do Practice I.<br /><br /><b>b) Table method</b><br /><br />In this method you are going to use two tables to find a limit. You start by giving x some values to the left of the given value of x and you group the values of x and f(x) in a table. You do the same thing to the right of the given value of x to have a second table. Even though I don't mention the tasks in the lesson you are going to do them in the same way you do for the previous problem.<br /><br /><b>Task I</b><br /><br />Start by reading the problem they ask you to find the solution. Then read the solution. I didn't provided any explanation of the solution because the solution is explanatory by itself. Below is a guided explanation<br /><br /><b>Explanation</b><br /><b><br /></b>You start by giving x a value less than 0 and closer to 0. This value has to be to the left of 0. Then you calculate the value of f(x). You give to x a second value and closer to zero than the previous one You calculate the second value of f(x). You continue to give some values to x closer and closer to x and calculate the corresponding values of x. You do a table grouping all the values of x and f(x) in a table. When you look at the table you notice that f(x) gets close to 1 to the left as x gets closer and closer to 0 to the left.<br /><br />Now you give x values to the right of 0 but closer to 0 and you calculate the corresponding values of f(x). You do a table grouping the values of x and f(x). When you look at the table you notice that as x gets closer and closer to to 0 to the right f(x) gets close to 1 to the right.<br /><br />Since f(x) gets close to 1 as x gets closer and closer to 0 both to the left and right to 0 we conclude the limit of f(x) is 1 when x gets close to 0.<br /><br /><b>Task 2</b><br /><br />Take a pencil and a piece of paper to do the problem by yourself.<br /><br /><b>Task 3</b><br /><br />Do the practice problem <br /><br /><b>Algebra method</b><br /><b><br /></b>This method is very simple but it involves some calculations to do. In this method you substitute x in the function<br /><br /><b>Task 1</b><br /><br />Read the problem first. Then write its solution. Below is a guided explanation.<br /><br />Explanation<br /><br />You substitute x in the function and you do the calculations to find f(x). The value of f(x) is the limit of the function<br /><br /><b>Task 2</b><br /><br />Do the problem by yourself<br /><br /><b>Task 3</b><br /><br />Do the practice problems<br /><br /><b>Review problems</b><br /><b><br /></b>Do the review problems involving the three methods.<br /><br />Interested in learning more about limits get this free course <a href="http://center-for-integral-development.thinkific.com/users/checkout/auth">Introduction to Calculus</a><br />You can also be enrolled in the complete<a href="http://center-for-integral-development.thinkific.com/users/checkout/auth"> Calculus</a> course<br /><br /><br /><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/5e92Vy0p7jU" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/12/notions-of-limits-written-lesson.htmltag:blogger.com,1999:blog-6995353831733168187.post-29810915836343055052016-11-04T11:29:00.000-07:002016-11-04T11:35:18.922-07:00How to learn a subject deeplyLearning a subject deeply means you know its theories and are able to apply it. Very often people learn a subject because they are required to without knowing its applications or if they would ever apply it. People learn practical subjects and are not able to apply them. These subjects require practice. But when you learn a subject deeply its practice becomes easy.<br /><br />To learn a subject deeply requires to know "how to learn". You start by learning the concepts or key words in the subject. Sometimes there are words that are not known or are not well understood. Having a clear definition of these words helps to learn the subject deeply. Besides knowing key vocabulary it is necessary to master the theories. It is also important to have a clear understanding of the concepts of the subject. This can be done by having a clear mental picture of these concepts in one's mind. If it's not possible to imagine the concepts one can try to represent them by a visual representation. In order to learn a subject deeply it has to be absorbed gradually. so that the previous concepts can be applied to the following ones. <br /><br />Deeper learning is the ability to apply knowledge to new situations. Deeper learning is associated with better life and work outcomes according to a 2012 report.<br /><br />Superficial learning is associated with poor performance. On the 2012 Program for International Student Assessment (PISA), a test that measures students' abilities to apply their knowledge to real-world problems U.S fifteen years old scored 26th of the 34 industrialized nations in mathematics.<br /><br />Schools that practice deeper learning have their students graduated and attended college at higher rates than schools that don't use deeper learning.<br /><br />Students who practice "deeper learning" take responsibility for their learning. In "deeper learning" students master their subjects deeply. They know the concepts, can apply them and reflect deeply on the subject.<br /><br />Deeper learning is defined by 6 competencies: mastering content, critical thinking, effective written and oral communication, collaboration, learning how to learn and developing academic mindsets.<br /><br />Deeper learning is associated with practice and reflection. In practical subjects learners can build things. Imagination, intuition and inspiration are some of the characteristics of deeper learning.<br /><br />Deeper learners cultivate academic mindsets. They make the most out of their learning experiences. They hold the following key beliefs:<br />"I can change my intelligence and abilities through effort"<br />"I can succeed"<br />"This work has value and purpose for me"<br /><br />Beliefs and learning skills bring success for learners.<br /><br />If you are interested in getting some help in learning Math, French, ESOL (English to Speakers of Other Languages), Spanish, visit New Direction Education Services at <a href="http://www.ndes.biz/">www.ndes.biz</a> to get the contact information . If you need help in AP Calculus take this <a href="http://center-for-integral-development.thinkific.com/courses/introduction-to-calculus">Introductory Course</a> for free. You can access the complete course <a href="http://center-for-integral-development.thinkific.com/courses/calculus-ab">here </a>(click the word "here") <img src="http://feeds.feedburner.com/~r/Alteredzine/~4/WJqnCfcEzwc" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/11/how-to-learn-subject-deeply.htmltag:blogger.com,1999:blog-6995353831733168187.post-9529543663149684572016-10-28T07:03:00.000-07:002016-10-28T07:29:25.617-07:00Notions of limits<b><br /></b><b>Lesson: Introduction to the notion of limits</b><br /><b><br /></b><b>Objectives</b>:<br /><br />At the end of this lesson the learner should be able to:<br /><br />1) Have an idea of what a limit is<br />2) Be able to calculate a limit using the graph, table and algebra method<br /><br />This lesson is part of a series of lessons on the AP Calculus course I will be teaching throughout this blog. This lesson is the first lesson on Chapter I of the course. It is made of two parts. The first part is made of video lectures. The second part consists of the written lesson and the activities.<br /><br /><b>Video lectures</b><br /><b><br /></b><b>1) Introduction to the notion of limit</b><br /><br />Here you'll have to watch this video that will introduce you the notion of limits. Here is the link: <a href="https://youtu.be/riXcZT2ICjA">Introduction to the notion of limit</a><br /><br /><b>2) Methods for determining limits</b><br /><br />The three methods for determining limits are: Graph, Table and Algebra method. You will have to watch three videos on the Graph method, two on the Table method and two on the Algebra method.<br /><br /><b>a) Graph method. </b><br /><br />Here are the links to watch the videos for this method:<br /><a href="https://www.youtube.com/watch?v=rccqylTloMs">Two-sided limits from graph</a><br /><a href="https://www.youtube.com/watch?v=GGQngIp0YGI">Limits examples Part I</a><br /><a href="https://www.youtube.com/watch?v=W0VWO4asgmk">Limits examples Part II</a><br /><br /><b>b) Table method</b><br /><b><br /></b>Here are the links to watch the links for the Table method<br /><a href="https://www.youtube.com/watch?v=cMNXs7JFC_g">Finding limits numerically with tables</a><br /><a href="https://www.youtube.com/watch?v=l7Tcay720vw">Determine a limit numerically</a><br /><br /><b>c) Algebra method</b><br /><b><br /></b>In the Algebra method you are going to watch two videos.<br /><br />1.<strong data-redactor-tag="strong" style="background-color: white; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px;"> In this video you are going to learn how to evaluate a limit using the substitution method and verifying the result using a graph. The notion of continuous functions is introduced to help to determine the limit. A continuous function is one that goes without interruption. The notion of continuity is introduced later in the Calculus course. Notice that the first function is a constant. As such the limit is a constant. This is one of the limit properties that will be introduced later. Since you don't know this property the limit is determined using a graph. The limits of the other 3 functions are calculated using the notion that if a function is continuous for a value x = c its limit is f(c). These 2 functions are continuous for any value of x. Therefore f(x) exists for any value of x and the direct substitution method is applied.</strong><br /><strong data-redactor-tag="strong" style="background-color: white; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px;"><br /></strong><strong data-redactor-tag="strong" style="background-color: white; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px;"> Here is the link of the video to watch;<a href="http://1.%20in%20this%20video%20you%20are%20going%20to%20learn%20how%20to%20evaluate%20a%20limit%20using%20the%20substitution%20method%20and%20verifying%20the%20result%20using%20a%20graph.%20the%20notion%20of%20continuous%20functions%20is%20introduced%20to%20help%20to%20determine%20the%20limit.%20a%20continuous%20function%20is%20one%20that%20goes%20without%20interruption.%20the%20notion%20of%20continuity%20is%20introduced%20later%20in%20the%20calculus%20course.%20notice%20that%20the%20first%20function%20is%20a%20constant.%20as%20such%20the%20limit%20is%20a%20constant.%20this%20is%20one%20of%20the%20limit%20properties%20that%20will%20be%20introduced%20later.%20since%20you%20don%27t%20know%20this%20property%20the%20limit%20is%20determined%20using%20a%20graph.%20the%20limits%20of%20the%20other%203%20functions%20are%20calculated%20using%20the%20notion%20that%20if%20a%20function%20is%20continuous%20for%20a%20value%20x%20%3D%20c%20its%20limit%20is%20f%28c%29.%20these%202%20functions%20are%20continuous%20for%20any%20value%20of%20x.%20therefore%20f%28x%29%20exists%20for%20any%20value%20of%20x%20and%20the%20direct%20substitution%20method%20is%20applied./"> Determining limit using direct substitution</a></strong><br /><br /><span style="font-family: "arial" , "helvetica" , "verdana" , "tahoma" , sans-serif;"><span style="background-color: white; font-size: 14px;">2.</span></span><span style="color: #555566; font-family: "arial" , "helvetica" , "verdana" , "tahoma" , sans-serif; font-size: 14px;">. In this video you are going to use the three methods to evaluate a limit</span><br /><div style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: none; box-shadow: none; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px; line-height: 1.6em; margin-bottom: 15px;">2.1 Direct substitution<br />2.2 Factoring<br />2.3 Conjugation</div><div style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: none; box-shadow: none; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px; line-height: 1.6em; margin-bottom: 15px;">Here is the link of the video to watch:<a href="http://www.calculus-help.com/how-do-you-evaluate-limits/"> How do you evaluate limits</a><br /><br />In the next post I will introduce you to the written lesson that includes the assignments that you will have to do. You can also subscribe to the<a href="http://center-for-integral-development.thinkific.com/courses/introduction-to-calculus"> free Introductory Calculus course</a>. If you want to get the complete Calculus course you can get it<a href="http://center-for-integral-development.thinkific.com/courses/calculus-ab"> here</a> (click on the highlighted word here)</div><div style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: none; box-shadow: none; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px; line-height: 1.6em; margin-bottom: 15px;"><br /></div><strong data-redactor-tag="strong" style="background-color: white; box-sizing: border-box; color: #555566; font-family: Arial, Helvetica, Verdana, Tahoma, sans-serif; font-size: 14px;"><br /></strong><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/2l4fJjRgj7Y" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/10/notions-of-limits.htmltag:blogger.com,1999:blog-6995353831733168187.post-46784745397381143722016-04-01T11:39:00.001-07:002016-04-01T11:44:22.291-07:00Learning Calculus by following a simple model of learningMany learners find it difficult to learn a subject or anything that they want to learn. The difficulties come from the fact that people have always thought that in order to learn something somebody has to teach it in the first place. Learning doesn't always come from someone else. One can learn by oneself. In fact learning happens throughout life mostly in the informal way. Life would be impossible without learning. Learning happens explicitly after birth. Babies learn to cry to get fed. This is a natural process of a simple stimulus-response conditioning. A natural stimulus is used in order to get a response. The baby cry is a natural stimulus to get a response which is food. Learning viewed this way is a change of behavior. Later comes complex stimulus-response conditioning. The complex stimulus-response conditioning is known as classical conditioning of Pavlov. In complex stimulus-response conditioning a second stimulus is introduced, which stimulus is neutral. Dog naturally salivate when they see meat but Pavlov was able to teach a dog to salivate at the sound of a bell by associating the sound of a bell to the presentation of the meat to the dog. By repeating several times the association meat with the sound of a bell the dog learns to salivate when the bell rings. This process of conditioned learning has been used by humans to live and to create different structures in society.<br /><br />Learning happens whether we want it or not. In order to learn more complex things ways of learning are necessary. One cannot depend exclusively one someone else to learn as if this person isn't present learning cannot take place. A teacher doesn't force learning to take place. He facilitates and creates conditions for learning. This starts by believing that you can learn. Then you learn the study skills and habits. You need to know the theories, rules and processes in order to learn math.<br /><br /> Mathematics play an important role in human activities. They are used from simple everyday activities such as personal budgeting, checkbook balancing, groceries shopping to more complicated disciplines such as Economy, Science, Computers, Engineering, etc. The buildings we live in the roads we use, the computers, cellphones, tablets, televisions, etc are designed by people who know math. Calculus is an important branch of mathematics used in various disciplines taught at the college level. The notions of limits are fundamental in understanding some very important notions in Calculus such as Continuity, Derivation and Integrals. I have designed two Calculus courses for learners taking AP Calculus or who will take it. If a student plans to take Calculus as their next math course it's good to start taking them now so that it doesn't look strange to them. They are also designed for students at the high school or college level who need a remediation course. The first one is a free Introductory Calculus course. The second is a complete Calculus course at an affordable price.<br /><br />The instruction process for this course is designed in the following manner:<br />1. Students will watch an introductory video. The videos introduce the lessons to the learner<br />2. There will be some readings to do. The readings expose the learners to the theories of different topics.<br />3. There will be some problems completely solved. Students should master the solution process of these problems.<br />3. They will have to solve practice problems demonstrating an understanding of the topics.<br /><div><br /></div><div>Courses in Basic Algebra, Algebra I & II, Geometry, Trigonometry, Pre-Calculus and math for adults are also available. Other face-to-face and online courses in French and English to Speakers of other Languages are available on demand. Online and face-to-face tutoring are also available in these subjects.For more information visit New Direction Education Services at <a href="http://www.ndes.biz/">www.ndes.biz</a>. If you are interested in the 2 Calculus courses, click on the link at the end of this post. If this is not for you please share the link to people who might be interested. Here is the link: <a href="https://sites.google.com/site/freetutoringbyemail/home/freeintroductorycalculuscoursepaidcalculuscourse">Free Introductory Calculus Course. Complete Calculus course.</a></div><div><br /></div><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/g3teh52SuBE" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/04/learning-calculus-by-following-simple.htmltag:blogger.com,1999:blog-6995353831733168187.post-19210317026198769212016-03-01T10:01:00.000-08:002016-03-01T10:13:11.537-08:00Mindsets impact mathematics achievementStudy done by the educator Carol Dweck and her colleagues shows that everyone has a learning mindset, a core belief about how they learn. People can have a growth mindset or a fixed mindset. In the Psychology of Learning a growth mindset is the attitude of people who believe that their intelligence can increase with hard work. The learning ability of people with a growth mindset tends therefore to increase. People with a fixed mindset believe that their intelligence is fixed and cannot go beyond their fixed levels. They think that their learning ability is limited. Because of this mindset they think that they can't learn a subject fully and realize great performances at it. These two types of mindsets lead to different kinds of learning behaviors and consequently to different learning outcomes. Learners with a fixed mindset give up easily while those with a growth mindset persist even though their work is hard.<br /><br />Mindsets impact math achievement. A survey was given to students in a 7th grade class to measure their mindset. The researchers monitored their math achievement over a two years period. The study yields to important results according to the type of students' mindsets. The math achievement of students with a growth mindset tends to progress increasingly while the math achievement of students with a fixed mindset stays constant.<br /><br />A study about the relationships between beliefs and brain activity shows that the brain of people with a growth mindset reacts differently than that of people with a fixed mindset when they make a mistake. Those with a growth mindset are more aware of their mistakes and willing to fix them. This attitude is different for those with a fixed mindset. Another study supports that students with a growth mindset experience heightened brain activity and are able to pay more attention to their mistakes.<br /><br />The brains of all participants to the latter study show some type of activity but the brain of those with a growth mindset is likely to show subsequent activities allowing them to be aware of their mistakes.<br /><br />What are the implications of these studies in learning math or any other subject? These studies show that it's not natural that some individuals are more intelligent than others. Beliefs and mindsets play a great role in people's level of intelligence and their ability to learn. People with a growth mindset or who believe that they can learn if they put some effort have have higher levels of intelligence and increased learning abilities. Those who have a fixed mindset think that their intelligence and leaning abilities are limited. Because of these beliefs they aren't making any effort to learn a subject.<br /><br />I presently teach and tutor face-to-face and online Math, French, ESL and Spanish. If you believe that you can't learn Math and any of the other subjects above I can work with you to help you to develop a growth mindset. I give away two freebies: a few tutoring sessions in any of the subjects mentioned above and a free Calculus course. For the free Calculus course click this link <a href="http://center-for-integral-development.thinkific.com/courses/notions-of-limits">Introduction to Calculus: notions of limits</a>. For free tutoring by email fill out this form <a href="http://eepurl.com/bs1wdr">Free Tutoring by Email</a><br />For paid tutoring and courses face-to-face and online visit New Direction Education Services at <a href="http://www.ndes.biz/">www.ndes.biz</a> and click on the contact information button. You can also reach me by email at pslvb34@gmail.com<img src="http://feeds.feedburner.com/~r/Alteredzine/~4/LejUTVu-ndU" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/03/mindsets-impact-mathematics-achievement.htmltag:blogger.com,1999:blog-6995353831733168187.post-79811360962858236242016-01-09T11:09:00.001-08:002016-01-16T12:16:12.342-08:00Visualization in mathematics helps students in math learningThere are different ways by which we acquire information. We mainly acquire information from our senses. The two senses mostly used in learning are the eyesight and the hearing. The multiple intelligences theory by Gardner. presently debated, show also other senses besides the traditional senses involved in learning. Teacher's lectures, videos, written materials, manipulatives are the primary ways by which we learn. Written information is widely used in learning and day-to-day activities. Reading and writing play an important part in learning and life. The command of these two techniques can help us tremendously in learning and life. Writiting comes as visual information in symbols. The comprehension of written information involves the mastery of different structures of a language. Visual information comes also in pictures and shapes that aid in the understanding of written information. The word "visualization" is a common word used in computer, psychotherapy, etc. Pictures that can come in different forms and shapes are easier to decode than symbols because they are more related to our personal experiences. Therefore they bring more clarity to coded information. In this article is highlighted the importance of visualization techniques to facilitate the learning of mathematics. We can approximately define "visual mathematics" as the represention of mathematics that are symbolic or not through shapes that correspond more to our actual experiences. Three main highlights are discussed in this article <br /><br /><b>Visual mathematics are used in basic and high levels of mathematics</b><br /><b><br /></b>Educators in beginning classes of mathematics use manipulatives, games, shapes and pictures to help learners to understand mathematics. Visualization techniques are also used in higher levels of mathematics. Mathematics don't deal only with numbers. Visual representation is a part of the structure of mathematics. Consider algebra that is mainly symbolic. Different shapes are used to represent abstract relations. Diagrams, tables, graphs are used to represent relations and functions . Visualization techniques can be used even in abstract theories and problems. One can invent pictures, graph or any sort of visualization technique to represent abstract situations. The visual representation can especially be useful when it facilitates understanding, higher order of thinking and develops ideas. <br /><br /><b>Brain research shows that visual mathematics improve student's math performance</b><br /><br /><b></b>Researchers found that when students used visual mathematics they activated another area of their brain besides the one used when using numbers and symbols. The communication and working of these two areas of the brain facilitate math learning. They even state that visualization techniques are more beneficial than numerical techniques of learning math even when students are essentially learning numerical mathematics. It's obvious that when the concrete is used to explain the abstract the understanding of the latter becomes clearer.<br /><br /><b>Visual mathematics help students to solve problems in different ways.</b> <br /><br />Visual mathematics are nothing but a visual representation of abstract mathematics. Visual mahematics facilitate individualized learning since students can have different views on visual representation. Not only visual representation facilitates understanding it develops imagination and allows communication to take place between students. They can compare their individual work between each other. They can also discuss problems together. Educators can favor this type of learning by asking students to come up with different ways of solving problems.<br /><br /><b>Conclusion</b><br /><b><br /></b>There is no doubt that visualization represents an important tool that can facilitate learning. However it can be used for some specific purposes but not as an obsession. Sometimes it might not be needed. When understanding is clear and there is no need for clarification and depth one can move further.<br /><br />It is also important to note that a math educator can use different learning techniques to facilitate student's learning comprehension. One is the use of sequential learning. Math is sequential meaning each concept is based on the previous one.It is important that students master previous concepts in order to understand the concept that is actually learned. The sequential nature of mathematics is obvious in the learning of the four basic arithmetic operations. The learning of subtraction is based on that of addition. Without knowing addition one can't do multiplication. Division implies the learning of multiplication and subtraction. As an educator I have found that students who have math learning difficulties don't master the basic calculations. They also don't love mathematics, which is linked to the learning deficiencies in the fundamental notions of mathematics. Study skills are also important in the study of mathematics. I have written about different study techniques in this blog. As a math educator and tutor. my primary task is to instill the love and usefulness of math in students. If you or your child is interested in math tutoring don't hesitate to contact me. You can also refer other learners to me.<br /><br /><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span><a href="http://eepurl.com/bs1wdr" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Free Tutoring by Email </a><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">. For paid tutoring and courses visit New Directions Education Services at</span><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;"> <a href="http://www.ndes.biz/">www.ndes.biz</a></span><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/SILCDxjxz2c" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2016/01/visualization-in-mathematics.htmltag:blogger.com,1999:blog-6995353831733168187.post-45258748344160264172015-12-19T08:28:00.000-08:002016-01-08T11:18:59.522-08:00How to remove obstacles to learning math<div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">I am adding a few comments about the article: Not a math person: "how to remove obstacles to learning math" written by Katrina Schwartz and published in the online magazine Mind/Shift concerning the pedagogical techniques to remove obstacles in learning math. The author of the article wrote: "Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance". Our brain is the command center of our physical, emotional, mental and spiritual life. The brain stores also a lot of information recorded consciously or unconsciously. It interprets our surrounding reality and draws its conclusions. One type of conclusions is "the beliefs and attitudes" mentioned in the quote. Our beliefs and attitudes strongly influence our actions and personality. If students believe they can't do math it's obvious that they are not going to make any efforts in order to learn the subject. Their actions will reflect negative views or attitudes about learning math. The beliefs and attitudes originate from the student himself and his social and educational environment. If the social environment including the family cannot do much about developing positive attitudes about learning math this is the role of the school system to favor the development of attitudes and beliefs necessary for learning math. Not only the math teacher develops techniques to help students to learn the subject but he encourages students to "like" the subject. I use the word "like" instead of any other complicated word because I found when a student likes a subject he tends to make efforts in order to perform strongly at it.<br /><span style="font-size: 1.375rem; letter-spacing: 0.01rem; line-height: 1.5;"><br /></span><span style="font-size: 1.375rem; letter-spacing: 0.01rem; line-height: 1.5;">"Neuroscientists now know that the brain has the abilities to grow and shrink". The fact that our brain grows simply means that we are able to use the brain to think, reflect and solve our problems. We know that every individual uses his brain to live. When we think for a certain period of time about something we solicit the brain's resources in order to help us in order to solve problems. We use a great portion of the "working memory" of our brain to the solution of these problems. However we need to use the brain resources and the "working memory" effectively. Background knowledge helps us to free some parts of the working memory in order to move quickly in the solution of problems. In fact a study done about the strong performance of chess players shows that the memorization of different chess positions helps players to think quicker and gain advantages on their competitor. The background knowledge is important in performing math. Background knowledge is the knowledge of facts and theories so that we don't have to demonstrate them each time we encouter them. The fact that we know the multiplication tables allow us to do the multiplication of different large numbers instead of figuring out each time the multiplication of single numbers. There are math problems that are so complicated that we have to know the method of solutions and the formulas instead of figuring out each time how to solve the same type of problems. Background knowledge of math theories help us to solve complicated math problems. A complicated and abstract subject like math cannot be mastered without knowing its theories. The other technique mentioned in the article is about visualization. Visualization allows to visualize a fact, theory, etc. It helps to figure out something more clearly. It is a tool. It doesn't substitute the knowledge of the subject. Here is an excerpt of the article: </span><br />Stanford math education professor <a href="https://ed.stanford.edu/faculty/joboaler" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">Jo Boaler </a>spends a lot of time worrying about how math education in the United States traumatizes kids. Recently, a colleague’s 7-year-old came home from school and announced he didn’t like math anymore. His mom asked why and he said, “math is too much answering and not enough learning.”</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">This story demonstrates how clearly kids understand that unlike their other courses, math is a performative subject, where their job is to come up with answers quickly. Boaler says that if this approach doesn’t change, the U.S. will always have weak math education.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">“There’s a widespread myth that some people are math people and some people are not,” Boaler told a group of parents and educators gathered at the <a href="https://www.innovativelearningconference.org/ehome/index.php?eventid=107259&" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">2015 Innovative Learning Conference</a>. “But it turns out there’s no such thing as a math brain.” Unfortunately, many parents, teachers and students believe this myth and it holds them up every day in their math learning.</div><br />There’s no such thing as a math brain.’<cite style="-webkit-font-smoothing: antialiased; box-sizing: border-box; display: block; font-size: 1.1rem; font-stretch: normal; font-style: normal; letter-spacing: 0.01rem; line-height: 1.4; margin-top: 0.625rem;">Jo Boaler, Stanford professor of math education</cite><br /><br /><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">“We live in a society with lots of kids who don’t believe they are good at math,” Boaler said at an Education Writers Association conference. “They’re put into low groups; they’re given low-level work and their pathway has been set.” But math education doesn’t have to look like this.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">Neuroscience research is now showing a strong connection between the attitudes and beliefs students hold about themselves and their academic performance. That’s a departure from the long-held traditional view that academic success is based only on the quality of the teacher and curriculum. But researchers like<a href="http://ww2.kqed.org/mindshift/2014/07/16/new-research-students-benefit-from-learning-that-intelligence-is-not-fixed/" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank"> Carol Dweck</a>, <a href="http://www.hewlett.org/uploads/documents/Academic_Mindsets_as_a_Critical_Component_of_Deeper_Learning_CAMILLE_FARRINGTON_April_20_2013.pdf" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">Camille Farrington</a> and <a href="http://www.utexas.edu/cola/prc/directory/faculty/profile.php?id=yeagerds" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;">David Yeager</a> have shown repeatedly that small interventions to change attitudes about learning can have an outsized effect on performance.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">Neuroscientists now know that the brain has the ability to grow and shrink. This was demonstrated in a<a href="http://www.scientificamerican.com/article/london-taxi-memory/" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">study of taxi drivers in London</a> who must memorize all the streets and landmarks in downtown London to earn a license. On average it takes people 12 tries to pass the test. Researchers found that the hippocampus of drivers studying for the test grew tremendously. But when those drivers retired, the brain shrank. Before this, no one knew the brain could grow and shrink like that.<br /><br /><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;"><span style="font-size: 1.375rem; letter-spacing: 0.01rem; line-height: 1.5;">“We now know that when you make a mistake in math, your brain grows,” Boaler said. Neuroscientists did MRI scans of students taking math tests and saw that when a student made a mistake a synapse fired, even if the student wasn’t aware of the mistake. “Your brain grows when you make a mistake, even if you’re not aware of it, because it’s a time when your brain is struggling,” Boaler said. “It’s the most important time for our brains.”</span></div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">A second synapse fires if the student recognizes his mistake. If that thought is revisited, the initial synapse firing can become a brain pathway, which is good for learning. If the thought isn’t revisited, that synapse will wash away.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">A recent <a href="http://www.nature.com/ncomms/2015/150930/ncomms9453/abs/ncomms9453.html" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">study of students with math learning disabilities</a> found in a scan that their brains did behave differently from kids without the disability. “What they saw was the brain lighting up in lots of different areas while working on math,” Boaler said. The children were recruiting parts of the brain not normally involved in math reasoning.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">The researchers tutored the group of students with math disabilities for eight weeks using the methods Boaler recommends like <a href="https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">visualizing math</a>, discussing problems and writing about math. At the end of the eight weeks, they scanned their brains again and found that the brains of the test group looked just like the kids who did not have math disabilities. This study shows that all kids can learn math when taught effectively. Boaler estimates that only 2 to 3 percent of people have such significant learning disabilities that they can’t learn math at the highest levels.</div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;">People who learned math the traditional way often push back against visual representations of math. That kind of thinking represents a deep misunderstanding of <a href="http://brannonlab.org.s84504.gridserver.com/wp-content/uploads/Park-Brannon-2013.pdf" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;" target="_blank">how the brain works</a>. “When you think visually about anything, different brain pathways light up than when we think numerically,” Boaler said. The more brain pathways a student engages on the same problem, the stronger the learning.</div><div class="wrap" style="background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 16px; line-height: 16px; margin: 0px; padding: 0px;"><br /><br /><br /><figure class="wp-caption aligncenter" id="attachment_42829" style="box-sizing: border-box; margin: 0.5rem auto 1.25rem; max-width: 923px;"><a href="https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/" style="background: transparent; box-sizing: border-box; color: #019cdc; line-height: inherit; text-decoration: none;"><img alt="An example of many ways to visually represent 18 x 5. " class="wp-image-42829 size-full" src="http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2015/11/Screen-shot-2015-11-18-at-2.46.29-PM.png" height="271" sizes="(max-width: 923px) 100vw, 923px" srcset="http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2015/11/Screen-shot-2015-11-18-at-2.46.29-PM-400x117.png 400w, http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2015/11/Screen-shot-2015-11-18-at-2.46.29-PM-800x235.png 800w, http://ww2.kqed.org/mindshift/wp-content/uploads/sites/23/2015/11/Screen-shot-2015-11-18-at-2.46.29-PM.png 923w" style="border: none; box-sizing: border-box; display: inline-block; height: auto; max-width: 100%; vertical-align: middle; width: 923px;" width="923" /></a></figure></div></div><div style="-webkit-font-smoothing: antialiased; background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 1.375rem; font-stretch: normal; letter-spacing: 0.01rem; line-height: 1.5; margin-bottom: 1.5em; padding: 0px; text-rendering: optimizeLegibility;"><div style="font-size: 1.375rem;">An example of many ways to visually represent 18*5 (Jo Boaler/YouCubed) (to be continued)</div><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span><a href="http://eepurl.com/bs1wdr" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Free Tutoring by Email </a><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">. For paid tutoring and courses visit New Directions Education Services at</span><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;"> <a href="http://www.ndes.biz/">www.ndes.biz</a></span><br /><div style="font-size: 1.375rem;"><br /></div></div><div class="wrap" style="background-color: white; box-sizing: border-box; color: #222222; font-family: freight-text-pro, Georgia, 'Times New Roman', Times, serif; font-size: 16px; line-height: 16px; margin: 0px; padding: 0px;"></div><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/an1oYuP2q1w" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2015/12/how-to-remove-obstacles-to-learning-math.htmltag:blogger.com,1999:blog-6995353831733168187.post-12019189079516178672015-12-02T09:00:00.000-08:002015-12-19T07:59:04.839-08:00Four best success skills in learning in school, outside and beyondSuccess skills in school and learning fall in different categories such as: study skills, organization, environment, etc. In this post I will develop four important skills for success in learning and one to avoid.<br /><br /><b>1. Study skills</b><br /><b><br /></b>Success in study require the ability to use some skills to master the subject. These skills are related to some reading and study techniques. Reading skills are important for success in study. Reading begins before study. In the reading phase some techniques are used such as previewing, skimming, scanning, etc, In the previewing phase one seeks to get a panoramic view of the material without reading the details. One looks at the title, headings, subheadings, pictures of a single piece of the study material. Then one reads the introduction, the first sentence of each paragraph and the conclusion. These techniques are not easy to use especially when one is accustomed in reading line by line. In acquiring the general view one can sketch an outline with the main points of the subject. Once one has finished with the general view one starts by reading the full text. This reading is active since it involves different activities such as outlining important details, thinking, reflecting, etc. The last phase consists in memorizing the important details of the subject.<br /><br /><b>2. Organization</b><br /><br />A calendar is important in order to find time to study. Everyone in school or not has different daily activities. It is important to schedule these activities in order to find time for study, If you are a student in grade school or at the university your time is divided in different blocks of activities, Classes take the majority of the time in school. Extra curricular activities and social events are also included in the school time. There are also personal activities outside the school. For adults who are in school they are very busy and share their time between work, personal activities and family responsibilities. One can use online calendars such as Google calendar to manage time for different activities. There are also some apps for reminders such as Google keep to remind about different activities. Students can use these reminders or hard and computer sticky notes to remind about textbook pages, assignments, etc. Google keep can also be used to take some notes.<br /><br /><b>3. Disconnection</b><br /><br />Disconnect from the internet is the most difficult thing to do since the internet is also used for classwork. In addition one is addicted to social medias, emails and text messages. One has to set the rule that when it's time to study one has to avoid logging in these things. One can set a time for study and another time for the internet to avoid doing both at the same time.<br /><br /><b>4. Environment</b><br /><br />Finding one's convenient environment is important. There are two elements in the environments: the place to study itself and the absence or not of complete quietness. A quite place such as the library or home place reserved especially for study is important. One should avoid to be bothered by other people, telephone, television, etc. If other people are doing other activities that prevent you from concentrating while you are studying this can distract you. If they ask you to do some things you lose time in your study. It doesn't mean that you can't interrupt your study to do some important things for someone else. But these things have to be really important and you are the sole person who can help in time and place. Some people like to learn while they listen to music, Others prefer not, The choice depends on your preferences. You can't study while watching the television, answering phones, sending text messages and logging in social medias. You decrease considerably the time dedicated to study and your concentration when you study and do these things at the same time. Observe the Ecclesiastes principle that says there is a time for each thing.<br /><br /><b>5. Cramming</b><br /><br />Cramming is the process of studying material that one hasn't studied before an examination. Study should be a daily activity. It extends over a certain period of time if one wants to master the material. It involves also constant reviewing. Mastering a certain amount of materials that one hasn't had the opportunity to digest during times where the study of materials is allotted can be hard to do. This requires the display of a considerable amount of energy and can lead to exhaustion.<br /><br /><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span><a href="http://eepurl.com/bs1wdr" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Free Tutoring by Email </a><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">. For paid tutoring and courses visit New Directions Education Services at</span><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;"> <a href="http://www.ndes.biz/">www.ndes.biz</a></span><br /><br /><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/G1T1pJsdmoQ" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2015/12/four-best-success-skills-in-learning-in.htmltag:blogger.com,1999:blog-6995353831733168187.post-81759639678399508982015-11-14T07:09:00.000-08:002015-11-14T07:19:33.681-08:00The role of the brain when kids learn mathAn article titled "Kids'brains reorganize when learning math", appeared in Associated Press in August 2014, describes the role of the long-term memory in the improvement of the learning of addition by kids. The article doesn't describe specifically what happens in the kid's brain when they automatically answer a question to a simple addition without counting on their fingers. However there are key elements in the article related to the role of the kids' brain in learning math and how this could help in learning math better. The discoveries are also related to the chid's cognitive development specifically how memories are recorded and retrieved in the brain.<br /><br />The scientists scanned the kids' brain to study how they were able to transition from counting to their fingers to simply answer a question related to a simple addition. They repeated the study a year later and did the same experiment in adolescents and adults. The results of the study are:<br /><br /><br /><ul><li>At some point kids makes the transition from counting from their fingers to automatically answer a question related to a simple addition. If they make this transition well their performance in the future learning of math will improve.</li><li> Older kids are able to do the simple addition quicker than the younger. In other words performance increases with age. </li><li>Being able to retrieve simple math addition in the memory helps kids to learn new math concepts. In other words this process allows children to use free space in the work memory in order to learn new math concepts.</li><li>This retrieval process improves the ability of the hippocampus which is the region of the brain where new memories come in before being transferred in long term memories.</li></ul>The implications of this study are important in learning math and in learning in general. It stresses the importance of the use of memory in learning math. Learning uses long-term memory as a storage where different concepts can be retrieved when one learns new things or things related to the previous concepts. Imagine that you are not able to remember your telephone number. For this reason you either write it somewhere or save it in your telephone memory. It would be annoying to loook for this number each time you have to dial it, However if you are able to remember the telephone number dial it each time becomes easier. In fact the use of our memory is a natural process in living. There are many things that are transferred in the long term memory without our conscious will. Once certain things are stored in the long term memory the retrieval becomes automatic after performing certain actions related to the information stored. When you learn to drive a car you store certain information in the brain. Once you know how to drive the car the retrieval information related to driving becomes automatic. You don't have to consciously remember the driving information.<br /><br />One tends to think that learning math is simply related to thinking and solving problems only. The role of the memory is important in these processes. We retrieve procedures, theorems, rules, etc in the memory in order to learn new concepts and solve problems. One would think that in using the memory one would record things without understanding them. Quite the contrary when you understand something you can better store it in the long term memory and retrieve it from there. The long term memory is where you store things that are important to you. Sometimes things are stored verbatim when it's necessary, For example you store formulas verbatim so that they can be retrieved for future use. Other times only representations of things are stored in the long memory. These representations are not exactly a reproduction of the reality but a way for you to figure them out. Learning involves processes in the brain. It is important to understand these processes to improve learning.<br />Source: <a href="http://www.deseretnews.com/article/765658477/Kids-brains-reorganize-when-learning-math-skills.html">http://www.deseretnews.com/article/765658477/Kids-brains-reorganize-when-learning-math-skills.html</a><br /> <span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span><a href="http://eepurl.com/bs1wdr" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Free Tutoring by Email </a><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;">. For paid tutoring and courses visit New Directions Education Services at</span><span style="font-family: "times new roman" , "times" , serif; line-height: 17.6px;"> </span><a href="http://www.ndes.wikidot.com/" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">www.ndes.wikidot.co</a>m<img src="http://feeds.feedburner.com/~r/Alteredzine/~4/gdIkg6stjl0" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2015/11/the-role-of-brain-when-kids-learn-math.htmltag:blogger.com,1999:blog-6995353831733168187.post-67841124882839085182015-11-06T15:28:00.000-08:002015-11-06T15:28:33.105-08:00"A.S.P.I.R.E." A Study SystemStudying effectively requires some good strategies. The acronym A.S.P.I.R.E allows to remember these strategies. "A" stands for "Approach/Attitude/Arrange". "S" stands for "Select/Survey/Scan". "P" stands for "Piece together the parts". "I" stands for "Investigate/Inquire/Inspect". "R" stands for "Re-examine/Reflect/Relay". "E" stands for "Evaluate/Examine/Explore".<br /><br /> <b style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">A: Approach/attitude/arrange</b><br /><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Approach your studies with a positive attitude</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Arrange your schedule to eliminate distractions</li></ul></ul><div style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin-bottom: 0.5em; outline: 0px; padding: 8px 0px 0px 10px; vertical-align: baseline;"><b style="background: transparent; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">S: Select/survey/scan</b></div><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Select a reasonable chunk of material to study</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Survey the headings, graphics, pre- and post questions to get an overview</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Scan the text for keywords and vocabulary: mark what you don’t understand</li></ul></ul><div style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin-bottom: 0.5em; outline: 0px; padding: 8px 0px 0px 10px; vertical-align: baseline;"><b style="background: transparent; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">P: Piece together the parts:</b></div><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Put aside your books and notes</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Piece together what you've studied, either alone, with a study pal or group:<br />summarize what you understand.</li></ul></ul><div style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin-bottom: 0.5em; outline: 0px; padding: 8px 0px 0px 10px; vertical-align: baseline;"><b style="background: transparent; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">I: Investigate/inquire/inspect:</b></div><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Investigate alternative sources of information you can refer to:<br />other text books, websites, experts, tutors, etc.</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Inquire from support professionals (academic support, librarians, tutors, teachers, experts,) and other resources for assistance</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Inspect what you did not understand.</li></ul></ul><div style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin-bottom: 0.5em; outline: 0px; padding: 8px 0px 0px 10px; vertical-align: baseline;"><b style="background: transparent; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">R: Reexamine/reflect/relay<br />Reexamine the content | Reflect on the material | Relay understanding</b></div><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Reexamine:<br />What questions are there yet to ask? Is there something I am missing?</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Reflect:<br />How can I apply this to my project? Is there a new application for it?</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Relay:<br />Can I explain this to my fellow students? Will they understand it better if I do?</li></ul></ul><div style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; margin-bottom: 0.5em; outline: 0px; padding: 8px 0px 0px 10px; vertical-align: baseline;"><b style="background: transparent; border: 0px; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">E: Evaluate/examine/explore:</b></div><ul style="background: rgb(255, 255, 255); border: 0px; font-family: 'Times New Roman', Times, serif; font-size: 16px; line-height: 1.1em; list-style: square; margin: 0px 0px 0px 10px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><ul style="background: transparent; border: 0px; line-height: 1.1em; list-style: disc; margin: 0px 0px 5px 15px; outline: 0px; padding: 0px 0px 5px 10px; vertical-align: baseline;"><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Evaluate your grades on tests and tasks: look for a pattern</li><li style="background: transparent; border: 0px; margin: 0px 0px 0px 25px; outline: 0px; padding: 1.1px 0px 0px; vertical-align: baseline;">Examine your progress: toward achieving your goals</li></ul></ul><span style="background-color: transparent; line-height: 1.1em;">Explore options: with a teacher, support professional, tutor, parent if you are not satisfied</span><span style="background-color: transparent; line-height: 1.1em;">. Source: <a href="http://www.studygs.net/aspire.htm">http://www.studygs.net/aspire.htm</a> </span><div><span style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span><a href="http://eepurl.com/bs1wdr" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">Free Tutoring by Email </a><span style="font-family: 'times new roman', times, serif; line-height: 17.6px;">. For paid tutoring and courses visit New Directions Education Services at</span><span style="font-family: 'times new roman', times, serif; line-height: 17.6px;"> </span><a href="http://www.ndes.wikidot.com/" style="font-family: 'times new roman', times, serif; line-height: 17.6px;">www.ndes.wikidot.com</a></div><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/mmtqCzP5wTc" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2015/11/aspire-study-system.htmltag:blogger.com,1999:blog-6995353831733168187.post-10648298351127999992015-10-30T10:27:00.000-07:002015-10-30T10:33:40.051-07:00Interest And Recognition Can Help A Student Become A ‘Math Person’In a new study published in the journal Child Development, Florida International University Professor Zahra Hazari found that <span class="IL_AD" id="IL_AD3">interest</span> and recognition can help a student become a "math person" and pursue a <span class="IL_AD" id="IL_AD6">STEM</span> career.<br /><br />Math isn't exactly every student's favorite subject, but those who have an affinity for it aren't necessarily born a "math person," as one might think.<br /><div dir="ltr">"Much of becoming a 'math person' and pursuing a related STEM (science, technology, engineering or math) career has to do with being recognized and becoming interested - not just being able to do it," said Hazari, who specializes in STEM Education at FIU's College of Education and STEM Transformation Institute, <a href="http://neatoday.org/2015/06/23/in-praise-of-the-math-person/" rel="nofollow">according to the NEA blog.</a> "This is important for promoting math education for everyone since it is not just about confidence and performance."</div><div dir="ltr">Hazari, who worked with colleagues Jennifer D. Cribbs from Western Kentucky University, and Philip M. Sadler and Gerhard Sonnert, both from <span class="IL_AD" id="IL_AD4">Harvard University</span> suggests that interest and recognition are key factors that can help students develop math skills.</div><div dir="ltr">The study, "<a href="http://onlinelibrary.wiley.com/doi/10.1111/cdev.12363/abstract" rel="nofollow">Establishing an Explanatory Model for Mathematics Identity</a>," suggests that students who feel confident in the subject won't necessarily become engaged in it, as previous studies have suggested.</div><div dir="ltr">The team surveyed more than 9,000 college calculus students from across the country. They found that students in the high-level course wanted to pursue math mainly because they'd received recognition for their abilities and also found it interesting.</div><div dir="ltr">In <span class="IL_AD" id="IL_AD5">the survey</span>, students were asked if they thought parents, friends, relatives, and math teachers saw them as a "math person."</div><div dir="ltr">Those who responded "yes" were <span class="IL_AD" id="IL_AD1">classified</span> as feeling recognized.</div><div dir="ltr">In other words, what motivates a student to pursue a career in STEM and encourages them to continue along this path is interest, recognition, and engagement.</div>"It is surprising that a student who becomes confident in his math abilities will not necessarily develop a math identity," Hazari said. "We really have to engage students in more meaningful ways through their own interests and help them overcome challenges and recognize them for doing so. If we want to empower students and provide access to STEM careers, it can't just be about confidence and performance. Attitudes and personal motivation matters immensely."<br />Source:<a href="http://www.ischoolguide.com/articles/15754/20150624/study-help-student-math-stem-career.htm"> http://www.ischoolguide.com/articles/15754/20150624/study-help-student-math-stem-career.htm</a><br /> <span style="font-family: Times New Roman, Times, serif;"><span style="line-height: 17.6px;">Interested in learning to use effective study skills? For free tutoring by email fill out this form:</span></span><br /><span style="font-family: Times New Roman, Times, serif;"><span style="line-height: 17.6px;"><a href="http://eepurl.com/bs1wdr"> Free Tutoring by Email </a>. For paid tutoring and courses visit New Directions Education Services at <a href="http://www.ndes.wikidot.com/">www.ndes.wikidot.com</a></span></span><br /><br /><br /><img src="http://feeds.feedburner.com/~r/Alteredzine/~4/rhs044Oxj1Y" height="1" width="1" alt=""/>Yves Simonhttps://plus.google.com/114465475940553888784noreply@blogger.com0http://alteredzine.blogspot.com/2015/10/interest-and-recognition-can-help.html