Math 130, Spring 2001, Lab 13

Exercise 1

The following applet has several example functions. You can load the examples by selecting from the pop-up menu at the top of the applet. The examples are meant to illustrate a variety of properties, such as cusps and asymptotes of various sorts. You might be able to combine and manipulate these functions to answer some of the problems that follow the applet. At least, you can get some hints. So, take some time to look through the functions. (In this applet, you can use the mouse to do zooming, and you can right-click and drag with the mouse to slide the graph around. You can get a better view of cusps, in particular, if you zoom in on them.)

In each of the following problems, you should find a formula for a function, f, that has the specified properties. You can test your answers by plugging your formulas into the applet. (Note that in the applet, you should enter "cubert(x)" to represent the cube root of $ x$. Use "cubert(x^2)" to represent $ x^{2/3}$. The problem with the formulas $ x^{1/3}$ and $ x^{2/3}$ is that in the applet, these formulas are only defined for $ x\ge0$.)

(a) $ f(x)$ is defined for all $ x$; $ \displaystyle \lim_{x\to-\infty} = -1$; and $ \displaystyle \lim_{x\to\infty} = \infty$.

(b) $ f(x)$ and $ f'(x)$ are defined for all $ x$; $ f$ has critical points at $ x=-1$, $ x=0$, and $ x=2$ and at no other points. (When you do this exercise, you should encounter the problem of finding a function that has a given derivative. In this case, where the function is just a polynomial, you should be able to figure out an answer.)

(c) $ f$ has a vertical asymptote at $ x=1$ and $ f(x)$ is defined for all other values of $ x$; $ \displaystyle \lim_{x\to-\infty} = 0$; and $ \displaystyle \lim_{x\to\infty} = 2$.

(d) $ f$ is defined for all $ x$; $ f$ has cusps at $ x=-1$ and at $ x=1$; $ \displaystyle \lim_{x\to-\infty} = \lim_{x\to\infty} = 0$.

(e) $ f$ has a vertical asymptotes at $ x=0$ and $ x=2$ and is defined for all other values of $ x$; and $ f$ has $ y=-x$ as an oblique asymptote.

(f) $ f(x)$ and $ f'(x)$ are defined for all $ x$; $ f$ has $ y=2x-1$ as an oblique asymptote; and the graph of $ f$ does not intersect the line $ y=2x-1$.

Exercise 2

Many max/min problems deal with finding a maximum or a minimum distance. The distance between two points $ (x_1,y_1)$ and $ (x_2,y_2)$ is given by the distance formula as $ \sqrt{(x_1-x_2)^2+(y_1-y_2)^2\,}$. So, you end up working with a formula that involves a square root, which can be unpleasant when you have to take the derivative. In fact, though, you can get away without the square root.

Suppose that $ f(x)>0$ for all $ x$ (or at least on the interval that we are interested in). Show that the functions $ f(x)$ and $ \sqrt{f(x)\,}$ have exactly the same critical points. Explain why $ f(x)$ has a maximum value at $ x=a$ if and only if $ \sqrt{f(x)\,}$ has a maximum value at $ x=a$. (The same is also true for minimum values.) You don't have to give a complete, formal proof, but try to give a convincing explanation.

Exercise 3

Don't confuse related rate problems with max/min problems! Solve the following problem. Include a diagram and English explanations of your work. Here's a hint: express the distance between the ships as a function of $ t$.

At 12:00 noon, Ship A is 20 miles west of a certain point in the ocean, and it is sailing east at 12 miles per hour. At the same time, Ship B is 35 miles south of that point and is sailing north at 15 miles per hour. Assume that both ships continue to sail in the same direction at a constant speed. What is the closest that the ships come to each other? At what time do they make this closest approach?

David Eck, April 2001