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Learning Activity-Foundations of functions

published on August 31st, 2008 . by Vanaja

I have made some learning activities for my students who learn the basics of functions. Here I share those activities with you. 8-)

Curriculum Standards: The student describes independent and dependent quantities in functional relationships.

Performance Objectives: The student identifies independent and dependent quantities in functional relationships.

Activity

Group Size: 2-3

Material: Index cards

Instructions:

Copy the following on the index cards. Shuffle the cards. Distribute 9 cards for each student. Place the remaining cards face down on the pile. Each student should make three sets of cards in which each set contain three cards, one a functional relationship and other two its dependent and independent variable. The players can take one card at a time from the pile. If the card matches with any other card with you, you can take it and leave a card you least want. The player who makes the three sets first, will be the winner.

Area of the circle A=pir2

Perimeter of sphere p= 4a

Volume of a sphere V=4/3pir3

The rent of taxi increases with distance traveled

Time taken for doing a job decreases with number of workers

The speed of a chemical reaction s= 1.5 t2 where t is the time

The weight of students increases with height of the students

 

Area A dependent variable

Area A independent variable

Radius of the circle is the dependent variable

Radius of the circle is the independent variable

Perimeter P is the dependent variable

Perimeter P is the independent variable

Side a is the dependent variable

Side a is the independent variable

Volume of the sphere V is the dependent variable

Volume of the sphere V is the independent variable

Radius of the sphere is the dependent variable

Radius of the sphere is the independent variable

The rent of the taxi is the dependent variable

The rent of the taxi is the independent variable

Distance traveled is the dependent variable

Distance traveled is the independent variable

Time taken for doing a job is the dependent variable

Time taken for doing a job is the independent variable

Number of workers is the dependent variable

Number of workers is the independent variable

The speed of a chemical reaction s is the dependent variable

The speed of a chemical reaction s is the independent variable

The time t is the dependent variable

The time t is the independent variable

The weight of students is the dependent variable

The weight of students is the independent variable

Height of the students is the dependent variable

Height of the students is the independent variable

Identity Function

published on March 22nd, 2008 . by Vanaja

The function that associates each real number to itself is called the identity function and is usually denoted by I.

So, the function f:R->R defined by f(x)=x for all x in R is called the identity function.

From the knowledge of coordinate geometry, y=x represents a straight line passing through the origin and inclined at angle 45° with the x axis.

identiti-function.JPG

Clearly the domain and range of the identity function are both equal to R.

We can observe that it is a bijection.

Graph of Functions (Constant Function)

published on March 15th, 2008 . by Vanaja

A function of the type y=f(x)=k where k is a fixed real number.

Graph of constant function.
The graph of the constant function is a straight line parallel to x axis, which is above or below according to k is positive or negative. That is if k>0 the graph will be above x axis and at a distance k units above it. If k<0, then it will be k units below it. If k=0, then the graph will coincide with the x axis.

The domain of the constant function f(x)=k is the set R of all real numbers and range of the function is the singleton set {k}

So, we can see a constant function is a many-one into function.

Two Similar Cylinders

published on March 5th, 2008 . by Vanaja


The two cylindrical pans are similar. The diameter of the smaller pan is equal to the radius of the larger pan. How many of these smaller cans could fill the larger can?

similar-cylinders.JPG

Hint: Since the two cylinders are similar, their dimensions are in the same ratio. It is given that the diameter of the smaller pan is same as the radius of the larger pan. That is the radius of the two pans are in the ratio 2:1. In other words we can say if the radius of the larger pan is ‘r’ the radius of the smaller pan is r/2 and since they are similar their heights are also in the same ratio 2:1. So, if h is the height of the larger triangle, h/2 is the height of the smaller triangle.

Now,

Volume of the larger cylinder Vl= pi r2h

Volume of the smaller cylinder Vs=pi(r/2)2(h/2) =(pir2h )/8 = Vl/8

i.e Vs =Vl /8

Therefore 8 smaller pans can fill one larger pan.

Black & White

published on February 27th, 2008 . by Vanaja

The following figures represent a relationship between two variables.

linear.JPG

Which rule relates x the number of dark squares to y, the number of white squares?

 

Ans:

y=2x+3

A Sales Chart

published on February 17th, 2008 . by Vanaja

Sarah went to a grocery shop. There was a sales chart.

Items Price
Cabbage $1.8 for 2
Carrot $0.6 for 1.k.g
Onion $1.75 for 1 carton
Potato $2.05 for 1 carton
Tomato $1 for 1 k.


Sarah bought 4 carton potatoes and 1 cabbage. If she gave $10, how much money she got back?



Hint: 10-{(4×2.05)+.9}

Limit of a function at a point.

published on February 16th, 2008 . by Vanaja

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

It is very important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus.

Definition

Let f(x) be a function of x. Let a and l be constants such that as x\rightarrow a, we have f\left(x \right)\rightarrow l. In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

\lim_{x\rightarrow a}f\left(x \right)=l

In case , no such number l exist, then we say that \lim_{x\rightarrow a}f\left(x \right)does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have \lim_{n\rightarrow \infty } (Area of the polygon of n sides)=Area of the circle.

The Fundamental Principle of Counting

published on January 13th, 2008 . by Vanaja

If an event can happen in exactly m ways, and if following it, a second event can happen in exactly n ways, then the two events in succession can happen in exactly mn ways.

Illustration.

Suppose there are five routs from A to B and three routs from B to C. In how many ways a person can go from A to C?

Since there are five different routs from A to b, the person can go the first part of his journey in 5 different ways. Having completed in any one of the 5 different ways , he has 3 different ways to complete the second part of the journey fro B to C. Thus each way of going from A to B give rise to 3 different ways of going from B to C.

There fore the total number of ways of completing the whole journey = number of ways for the first part x number of ways for the second part.
= 5 x 3=15.

Generalisation

If an event can occur in m different ways, a second event in n different ways, a third event in exactly p different ways and so on, then the total number of ways in which all events can occur in succession is mnp….

Some defnitions of average

published on January 10th, 2008 . by Vanaja

One of the most important objectives of statistical analysis is to get one single value that describes the characteristic of the entire mass of unwieldy data. Such a value is called central value or an average or the expected value of the variable.

The term ” average ” has been defined by various authors. Some important definitions are given below.

  • “Average is an attempt to find one single figure to describe whole of figures” -Clark
  • ” An average is a single value selected from a group of values to represent them in some way-a value which is supposed to stand for whole group, of which it is a part, as typical of all the values in he group” -A.E Waugh
  • ” An average is a typical value in the sense that it is sometimes employed to represent all the individual values in a series or of a variable” -Ya-Lun-Chou
  • “An average is a single value within the range of the data that is used to represent all the values in the series.Since an average is somewhere within the range of the data. It is also called a measure of central value” - Croxton &Cowden

Belated New Year Wishes

published on January 6th, 2008 . by Vanaja

I had been busy with some personal matters for the last five six weeks. In between gone for a vacation as well. But the new year day was a bit dull as my kids and I was not well after coming back from the vacation.

I wish you all a very prosperous and happy new year.

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