Calculus Page: Calculus problems designed for high school students currently enrolled in a first year Calculus class 2007-11-13T09:55:53Z Douglas Twitchell MrT http://www.theproblemsite.com/problems/calculus/2006/Oct_31.asp 2006-10-31T00:00:00Z 2006-10-31T00:00:00Z

Estimate the surface area of the lake

A certain lake has an irregular shape. In order to estimate the area, the North-South distance across the lake is measured every 100 feet down an East-West line. The results are shown in the table below. Use an appropriate technique to estimate the number of acres that the lake covers. Give a complete explanation of how you computed the area. <table> <tr> <td> East-West position (ft)</td> <td> Distance across lake (ft)</td> </tr> <tr> <td>0</td> <td>900</td> </tr> <td>100</td> <td>1500</td> </tr> <td>200</td> <td>1800</td> </tr> <td>300</td> <td>2000</td> </tr> <td>400</td> <td>2500</td> </tr> <td>500</td> <td>2700</td> </tr> <td>600</td> <td>2700</td> </tr> <td>700</td> <td>2700</td> </tr> <td>800</td> <td>2700</td> </tr> <td>900</td> <td>2700</td> </tr> <td>1000</td> <td>2700</td> </tr> <td>1100</td> <td>2700</td> </tr> <td>1200</td> <td>2700</td> </tr> <td>1300</td> <td>2700</td> </tr> <td>1400</td> <td>2600</td> </tr> <td>1500</td> <td>2600</td> </tr> <td>1600</td> <td>2600</td> </tr> <td>1700</td> <td>2600</td> </tr> <td>1800</td> <td>2600</td> </tr> <td>1900</td> <td>2600</td> </tr> <td>2000</td> <td>2500</td> </tr> <td>2100</td> <td>2500</td> </tr> <td>2200</td> <td>2500</td> </tr> <td>2300</td> <td>2500</td> </tr> <td>2400</td> <td>2500</td> </tr> <td>2500</td> <td>2500</td> </tr> <td>2600</td> <td>2400</td> </tr> <td>2700</td> <td>2300</td> </tr> <td>2800</td> <td>2300</td> </tr> <td>2900</td> <td>1800</td> </tr> <td>3000</td> <td>1600</td> </tr> <td>3100</td> <td>1500</td> </tr> <td>3200</td> <td>1400</td> </tr> <td>3300</td> <td>1400</td> </tr> <td>3400</td> <td>1300</td> </tr> <td>3500</td> <td>1300</td> </tr> <td>3600</td> <td>1100</td> </tr> <td>3700</td> <td>800</td> </tr> <td>3800</td> <td>0</td> </tr> </table> <br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2006/Oct_31_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2006/Sep_20.asp 2006-09-20T00:00:00Z 2006-09-20T00:00:00Z

Find a limit algebraically

Find the following limit algebraically. In your solution, describe the process used. Do NOT simply plug the expression into a grapher to find the limit.<br><br><img src="http://www.theproblemsite.com/problems/calculus/images/img_154_1.gif"><br><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2006/Sep_20_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2003/May_5.asp 2003-05-05T00:00:00Z 2003-05-05T00:00:00Z

Finding the anti-derivative.

Find the antiderivative shown WITHOUT the use of a calculator or computer system that does algebra. Explain your method clearly.<br><br><img src="http://www.theproblemsite.com/problems/calculus/images/img_118_1.gif"><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2003/May_5_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2003/Mar_27.asp 2003-03-27T00:00:00Z 2003-03-27T00:00:00Z

Finding the volume of fuel oil in a tanker

A fuel-oil tank is 10 feet long and has flat ends that are perpendicular to the ground surface. Cross-sections parallel to the flat ends have the shape of the ellipse x²/9 + y²/36 = 1. If the fuel oil in the tank is 9 feet deep, what is the volume of the fuel oil in the tank? Show the integrals you use to solve this problem. You MAY do a numerical integration, rather than applying the Fundamental Theorem.<br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2003/Mar_27_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2003/Jan_18.asp 2003-01-18T00:00:00Z 2003-01-18T00:00:00Z

Growth of Mold proportional to the amount present

A mold grows at a rate proportional to the amount present. Initially, its weight is 3 grams. After 2 days, its weight is 5 grams. How much does it weigh after 12 days?<br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2003/Jan_18_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2002/Dec_4.asp 2002-12-04T00:00:00Z 2002-12-04T00:00:00Z

Distance traveled; Upper Riemann Sum, a Lower Riemann Sum

There is more than one way to solve this particular problem. Please feel free to suggest an alternate solution, IN ADDITION to the method I have requested. If you provide an appropriate alternate solution, I may be inclined to give additional bonus points! <br><br>Please provide an explanation other than "I put the data in my calculator and ran a program."<br><br><hr><br><br>The velocity from a speedometer on a car was recorded over a one-hour time span. The results appear in the table below.<br><br><img src="http://www.theproblemsite.com/problems/calculus/images/img_116_1.gif"><br><br>Compute an approximation to the distance traveled by computing an Upper Riemann Sum, a Lower Riemann Sum, and then finding the mean of those two values. Please give the values of the two computed sums, in addition to your final answer, as part of your solution.<br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2002/Dec_4_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2002/Nov_1.asp 2002-11-01T00:00:00Z 2002-11-01T00:00:00Z

Find the rate at which the area defined by the poles is changing

A particular dance is performed with four moving poles. At any one instant in time, these poles form a rectangle. The poles forming each pair of opposite sides move closer together and farther apart throughout the course of the dance. The dancer is to step into this rectangle and out again in such a way that he or she never gets pinched by the poles. In the diagram shown, assume that the Length of the rectangle varies according to the equation L = 4 + 3cos t, and the Width of the rectangle varies according to the equation W = 3 + 3sin t, where t is in seconds, and the dimensions are in feet. The area, A, then varies as a function of time. <img src="http://www.theproblemsite.com/problems/calculus/images/img_115_1.gif"> <br><br>Find the rate at which the area is changing at time t = 5 seconds. Be sure to indicate whether the area is increasing or decreasing.<br><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2002/Nov_1_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2002/Oct_1.asp 2002-10-01T00:00:00Z 2002-10-01T00:00:00Z

Line segment connected to a parabola

The graph shown below consists of a line segment connected to a piece of a parabola, with the intersection at (0, 2). The derivative at the point of intersection is continuous, and each tick mark on the axes represents one unit. Find the equation of the parabola in the form p(x) = ax² + bx + c.<br><br><img src="http://www.theproblemsite.com/problems/calculus/images/img_114_1.gif"><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2002/Oct_1_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2002/Sep_3.asp 2002-09-03T00:00:00Z 2002-09-03T00:00:00Z

Functions defined in pieces.

Given the function defined by:<br><img src="http://www.theproblemsite.com/problems/calculus/images/img_113_1.gif"><br><br>Determine the ordered triple (a, b, c) so that all of the following conditions are true:<br><br>1. f (x) is continuous<br>2. f (3) = 20<br>3. <img src="http://www.theproblemsite.com/problems/calculus/images/img_113_2.gif"><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2002/Sep_3_solution.asp" target="_blank">Read The Solution</a></i> MrT http://www.theproblemsite.com/problems/calculus/2002/May_28.asp 2002-05-28T00:00:00Z 2002-05-28T00:00:00Z

Find the value of the integral shown

Find the value of the following integral, or show that it diverges.<br><br><img src="http://www.theproblemsite.com/problems/calculus/images/img_112_1.gif"><br><br><i><a href="http://www.theproblemsite.com/problems/calculus/2002/May_28_solution.asp" target="_blank">Read The Solution</a></i>