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 numbers m and b. 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, oF Chirps per minute 46 40 51 55 56 70 61 96 66 113 71 137
- (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 an even function if f(-x)=f(x) for each x in the domain of f. The graph of an even function is symmetrical about the vertical axis. (The name comes from the fact that the even powers of x -- y=x2, y=x4, and so on -- are even functions.) A function y=f(x) is said to be an even function if f(-x)=-f(x) for each x in the domain of f. The graph of an even function is symmetrical about the origin. (The odd powers of x -- y=x3, y=x5, and so on -- are odd functions.)
Here is the same graphing program that you used last week in lab. Use it to check that the functions y=x2 and y=x4 are even. You should see that whenever a point (a,b) is on the graph, so is (-a,b). You can also check that combinations of even powers, such as y=x4-2*x2, give even functions. Similarly, look at the odd functions y=x3, y=x5, and y=x5-2*x3. You should see that whenever a point (a,b) is on the graph, so is (-a,-b).
You can check that y=sin(x) is an odd function, while y=cos(x) is even. However, most functions are neither even nor odd. For example, look at y=x4-2*x3+3*x-1. However, 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).
Exercise 2: Let y=f(x) be any function that is defined for all values of x
- (a) Show that the function y = (f(x)+f(-x))/2 is an even function.
- (b) Show that the function y = (f(x)-f(-x))/2 is an odd function.
- (c) Show that f can be written as the sum of an odd function and an even function.
- (d) For the polynomial function x4-2*x3+3*x-1, what are the odd and even functions that you get?
The SimpleGraph program that you used above has a particular function y=g(x) built-in. You can graph it by entering g(x) in the function input box. Try it. Try entering (g(x)+g(-x))/2 and check that it is an even function. Try entering (g(x)-g(-x))/2 and 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 as y=f(3*x), it's useful to think of all the possible functions y=f(k*x) for different values of k and see what happens to the graph as k varies. 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 of y=sin(k*x) as the value of k is varied between 0 and 5. Also look at k*sin(x), sin(x+k), and sin(x)+k and see what happens to them as k is varied. You might want to try other functions besides sin(x), such as tan(x) or x2.
Exercise 3: Write a short essay explaining what happens, dynamically, as k is varied for each of the functions sin(k*x), k*sin(x), sin(x+k), and sin(x)+k. Explain why you 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) and y=g(x), this applet shows the graph of y=g(f(x)) (along with the graphs of f and g.) There is a small red square on the x-axis in the graph of f. You can drag this square back and forth. The applet marks the corresponding point (x,f(x)) on the graph of f. It also marks (f(x),g(f(x)) on the graph of g and (x,g(f(x))) on the graph of gof. Make sure that you understand the point of this (which is to show how g(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|. If y=h(x) is any function, we can form the composition functions y=abs(h(x)) and y=h(abs(x)). Use the Composition Applet to look at these compositions for the function sin(x). That is, you want to consider the compositions y=abs(sin(x)) and y=sin(abs(x)). To work with y=abs(sin(x)), set up the applet with f(x)=sin(x) and g(x)=abs(x). Drag the red square back and forth to help you understand how the graph of abs(sin(x)) is created. Then reverse the definitions of f(x) and g(x) to work with sin(abs(x)).
- (a) Sketch the graphs of abs(sin(x)) and sin(abs(x)), and explain why the graphs have the shapes they do.
- (b) Suppose that h is any function. Explain in words how the graph of y=abs(h(x)) can be obtained from the graph of y=h(x). (Of course, you could obtain the graph by plotting points. But I want a simple geometric construction that will make the graph of y=abs(h(x)) from the graph of y=h(x).)
- (c) Suppose that h is any function. Explain in words how the graph of y=h(abs(x)) can be obtained from the graph of y=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) and y=g(x) interactively, by dragging the little circular points on the graph up and down. Play with this feature to see how the composition function y=g(f(x)) is affected. You can still drag the red square back-and-forth to see how the value of g(f(x)) is computed for all the possible values of x. (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 represents half of the graph of an even function. Sketch the complete graph. What about if it's an odd function?
(2) Draw a graph that might reasonably represent your height H in feet as a function of your age t in years. What is the domain of H? What is the range? How does the rate of increase of H change with time? What sort of behavior would you expect for H(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 functions f(x) = (x+1)/(x2-3x) and g(x) = (2x)/(x+1) are rational functions. Compute f(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 composition f(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, if f(x) and g(x) are any rational functions, then the composition f(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) If f is a function, it's possible to take the composition of f with itself. The formula for this function is y=f(f(x)). For example, we could consider sin(sin(x)) or abs(abs(x)). Let h(x) be the function h(x)=1/x. What is h(h(x))? Try looking at this composition in the Function Composition Applet, with both f(x) and g(x) set equal to 1/x. Explain the graph of g(f(x)) in this case. What happens in this case at x=0?