## Math 130: Calculus I

Lab 2, February 1, 2001

## Part 1: Exercises for your lab report

This part of the lab contains several exercises that you should write up and turn in as your lab report next Wednesday. These exercises are numbered and are shown in blue.

The first exercise is about linear models. That is, we have some reason to believe that two quantities are related by a linear function of the form

y = m*x + b,for some numbersmandb.We want to use the relationship to make predictions about the world. Of course, when we deal with the real world, a model is generally only approximate and might apply only in a limited domain.The following data were observed in a biology experiment [Croxton, Cowden, and Klein,

Applied General Statistics,Prentice-Hall, 1967]. The data indicate that there is a relationship between temperature and the number of times a cricket chirps per minute. (When it's warmer, the cricket chirps more frequently.) We will assume that a linear model applies. That is, the number of chirps per minute is a linear function of the temperature in degrees Fahrenheit.

Temperature,^{o}FChirps per minute46 40 51 55 56 70 61 96 66 113 71 137

Exercise 1:

(a)Carefully graph this data and find an equation of a line that approximates the linear relation between temperature,x,and chirps per minute,y.(b)Using this relation, what number of chirps would you expect to hear per minute if the temperature is 40 degrees?(c)If you heard a cricket chirp 86 times in one minute, what is the temperature?(d)What would you say if you heard a cricket chirp 86 times in fifteen seconds?

A function

y=f(x)is said to be anevenfunction iff(-x)=f(x)for eachxin the domain off.The graph of an even function is symmetrical about the vertical axis. (The name comes from the fact that the even powers ofx--y=x^{2},y=xand so on -- are even functions.) A function^{4},y=f(x)is said to be anevenfunction iff(-x)=-f(x)for eachxin the domain off.The graph of an even function is symmetrical about the origin. (The odd powers ofx--y=x^{3},y=xand so on -- are odd functions.)^{5},Here is the same graphing program that you used last week in lab. Use it to check that the functions

y=xand^{2}y=xare even. You should see that whenever a point^{4}(a,b)is on the graph, so is(-a,b).You can also check that combinations of even powers, such asy=xgive even functions. Similarly, look at the odd functions^{4}-2*x^{2},y=x^{3},y=xand^{5},y=xYou should see that whenever a point^{5}-2*x^{3}.(a,b)is on the graph, so is(-a,-b).

You can check that

y=sin(x)is an odd function, whiley=cos(x)is even. However, most functions are neither even nor odd. For example, look aty=xHowever, it is interesting that every function can be written as the sum of an even function and an odd function (along as its domain is symmetrical about zero).^{4}-2*x^{3}+3*x-1.

Exercise 2:Lety=f(x)be any function that is defined for all values ofx

(a)Show that the functiony = (f(x)+f(-x))/2is an even function.(b)Show that the functiony = (f(x)-f(-x))/2is an odd function.(c)Show thatfcan be written as the sum of an odd function and an even function.(d)For the polynomial functionxwhat are the odd and even functions that you get?^{4}-2*x^{3}+3*x-1,The SimpleGraph program that you used above has a particular function

y=g(x)built-in. You can graph it by enteringg(x)in the function input box. Try it. Try entering(g(x)+g(-x))/2and check that it is an even function. Try entering(g(x)-g(-x))/2and check that it is an odd function. (By the way, this version of the program will let you move the graph around by right-clicking and dragging it. Sometimes, this can be a useful way to look at parts of the graph that are not visible.)

In class, we talked about transformations of functions such as

y=k*f(x),y=f(k*x),y=f(x)+k,y=f(x+k),and combinations of such transformations. However, instead of just looking at one transformed function such asy=f(3*x), it's useful to think of all the possible functionsy=f(k*x)for different values ofkand see what happens to the graph askvaries. Here is a new program that lets you do this. Launch it and then move the slider at the bottom back and forth to see what happens to the graph ofy=sin(k*x)as the value ofkis varied between 0 and 5. Also look atk*sin(x),sin(x+k),andsin(x)+kand see what happens to them askis varied. You might want to try other functions besidessin(x),such astan(x)orx^{2}.

Exercise 3:Write a short essay explaining what happens, dynamically, askis varied for each of the functionssin(k*x),k*sin(x),sin(x+k),andsin(x)+k. Explainwhyyou see the behavior that you see!

The next little program that you will look at is an applet that should help you understand composition of functions. Given two functions

y=f(x)andy=g(x),this applet shows the graph ofy=g(f(x))(along with the graphs offandg.) There is a small red square on the x-axis in the graph off.You can drag this square back and forth. The applet marks the corresponding point(x,f(x))on the graph off.It also marks(f(x),g(f(x))on the graph ofgand(x,g(f(x)))on the graph ofgof.Make sure that you understand the point of this (which is to show howg(f(x))is computed and graphed). Try out the applet with a few different functions. Note that you can right-click-and-drag on any of the graphs to move it around. (On a Macintosh, hold down the Command key and click.) Also, you can resize the window to make it bigger.`

For the program that you are using in this course, the absolute value function should be written as

abs(x)rather than |x|. Ify=h(x)is any function, we can form the composition functionsy=abs(h(x))andy=h(abs(x)). Use the Composition Applet to look at these compositions for the functionsin(x).That is, you want to consider the compositionsy=abs(sin(x))andy=sin(abs(x)).To work withy=abs(sin(x)),set up the applet withf(x)=sin(x)andg(x)=abs(x).Drag the red square back and forth to help you understand how the graph ofabs(sin(x))is created. Then reverse the definitions off(x)andg(x)to work withsin(abs(x)).

Exercise 4:

(a)Sketch the graphs ofabs(sin(x))andsin(abs(x)),andexplainwhy the graphs have the shapes they do.(b)Suppose thathis any function. Explain in words how the graph ofy=abs(h(x))can be obtained from the graph ofy=h(x).(Of course, you could obtain the graph by plotting points. But I want a simple geometric construction that will make the graph ofy=abs(h(x))from the graph ofy=h(x).)(c)Suppose thathis any function. Explain in words how the graph ofy=h(abs(x))can be obtained from the graph ofy=h(x).(Again, give a geometric description.)

That's the end of the exercises that you have to turn in next Wednesday. However, here, just for fun, is another version of the Composition Applet. In this version, there are not formulas for the functions. You can define the functions

y=f(x)andy=g(x)interactively, by dragging the little circular points on the graph up and down. Play with this feature to see how the composition functiony=g(f(x))is affected. You can still drag the red square back-and-forth to see how the value ofg(f(x))is computed for all the possible values ofx. (This is a useful exercise because it's important to realize that functions can be defined by graphs as well as by formulas, and operations can be performed on graphs as well as on formulas.)

## Part 2: Practice Problems

Here are a few more problems that you can work on during the lab. These problems are not part of the lab report.

(1)Suppose that the following picture representshalfof the graph of anevenfunction. Sketch the complete graph. What about if it's anoddfunction?

(2)Draw a graph that might reasonably represent your heightHin feet as a function of your agetin years. What is the domain ofH? What is the range? How does the rate of increase ofHchange with time? What sort of behavior would you expect forH(t)in the future (assuming that you live forever)?

(3)A rational function is one that can be written as a quotient of two polynomials. For example, the functionsf(x) = (x+1)/(xand^{2}-3x)g(x) = (2x)/(x+1)are rational functions. Computef(x)+g(x)for this pair of functions and show that it is rational by writing it in the form of one polynomial divided by another polynomial. Also compute the compositionf(g(x))for this pair of functions and show that it is rational by writing it in the form of one polynomial divided by another polynomial. (If fact, iff(x)andg(x)areanyrational functions, then the compositionf(g(x))is also a rational function.)

(4)You will very soon need to know all about lines, their slopes, and their equations. Here are three problems that you should be able to do quickly, without having to think too much about them:

- Find an equation for the line through the point (3,4) that has slope -1/2.
- Find an equation for the line that passes through the points (-1,3) and (2,-7).
- Find the slope of the line that has equation
3x-2y=7.

(5)You will also need to understand what slope means geometrically. Estimate the slope of each of the following lines. Assume that the scales on the horizontal and vertical axes are equal. (And make sure that you know why this is important.)

(6)Iffis a function, it's possible to take the composition offwith itself. The formula for this function isy=f(f(x)). For example, we could considersin(sin(x))orabs(abs(x)). Leth(x)be the functionh(x)=1/x.What ish(h(x))? Try looking at this composition in the Function Composition Applet, with bothf(x)andg(x)set equal to1/x.Explain the graph ofg(f(x))in this case. What happens in this case atx=0?