Chapter 5

Which of the following are equal to \(g^{\prime}(3) ?\) A sketch with labeled points will be useful. (a) \(\lim _{h \rightarrow 0} \frac{g(3+h)-g(3)}{h}\) (b) \(\lim _{x \rightarrow 0} \frac{g(x)-g(3)}{x-3}\) (c) \(\lim _{x \rightarrow 3} \frac{g(x)-g(3)}{x-3}\) (d) \(\lim _{s \rightarrow 3} \frac{g(3)-g(s)}{3-s}\) (e) \(\lim _{\Delta x \rightarrow 3} \frac{g(3+\Delta x)-g(3)}{\Delta x}\) (f) \(\lim _{\Delta x \rightarrow 0} \frac{g(3+\Delta x)-g(3)}{\Delta x}\) (g) \(\lim _{\Delta x \rightarrow 0} \frac{g(3)-g(3+\Delta x)}{-\Delta x}\)

Suppose a ball is thrown straight up from a height of 48 feet and given an initial upward velocity of \(8 \mathrm{ft} / \mathrm{sec} .\) Then its height at time \(t, t\) in seconds, is given by \(h(t)=-16 t^{2}+8 t+48\), for \(t \in[0,2] .\) In this problem we will look at the ball s velocity at \(t=1\). (a) At \(t=1\), is the ball heading up, or down? Explain your reasoning. (b) By calculating the average rate of change of height with respect to time, \(\frac{\Delta h}{\Delta t}\), on the intervals \([0.9,1]\) and \([1,1.1]\), give bounds for the ball s velocity at \(t=1\). (c) Improve your bounds by using the intervals \([0.99,1]\) and \([1,1.01]\). (d) Use the limit de nition of \(f^{\prime}(1)\) to nd the ball s instantaneous velocity at \(t=1\).

Use the limit de nition of derivative to show that the derivative of the linear function \(f(x)=a x+b\) is \(a\). Why is this exactly what you would expect? You have shown that the derivative of a constant is zero. Explain, and explain why this is exactly what you would expect.

An orange is growing on a tree. Assume that the orange is always spherical, and that it has not yet reached its mature size. Its current radius is \(r \mathrm{~cm}\). (a) If the radius increases by \(0.5 \mathrm{~cm}\), what is the corresponding increase in volume? What is \(\frac{\Delta V}{\Delta r}\) ? (b) If the radius of the orange increases by \(\Delta r\), what is the corresponding increase in volume? What is \(\frac{\Delta V}{\Delta r} ?\) (Please simplify your answer.) (c) Show that \(\lim _{\Delta r \rightarrow 0} \frac{\Delta V}{\Delta r}=4 \pi r^{2}\). Conclude that for \(\Delta r\) very small \(\Delta V \approx\left(4 \pi r^{2}\right) \Delta r\). (d) The surface area of a sphere is \(4 \pi r^{2}\). Explain, in terms of an orange, why the approximation \(\Delta V \approx\left(4 \pi r^{2}\right) \Delta r\) make sense.

Let \(f(x)=\frac{1}{x} .\) In this problem we will look at the slope of the tangent line to \(f(x)\) at point \(P=\left(\frac{1}{2}, 2\right)\) (a) Is the slope of the tangent line to \(f\) at \(P\) positive, or negative? (b) By calculating the slope of the secant line through \(P\) and a nearby point on the graph of \(f\), approximate \(f^{\prime}\left(\frac{1}{2}\right)\). First choose the point with an \(x\) -coordinate of \(0.49 .\) Next choose the point with an \(x\) -coordinate of \(0.501 .\) Now produce an approximation that is better than either of the previous two. (c) By calculating the limit of the difference quotient, nd \(f^{\prime}\left(\frac{1}{2}\right)\). (d) Find the equation of the tangent line to \(f\) at \(P\).

Use the limit de nition of derivative to nd the derivative of \(f(x)=k x^{2}\).

Let \(f(x)=x^{3}\) and \(P\) be the point \((1,1)\) on the graph of \(f\). (a) Approximate the slope of the line tangent to \(f\) at \(P\) by looking at the slope of the secant line through \(P\) and \(Q\), where \(Q=(1+h, f(1+h)\) ). Calculate the difference quotient for various values of \(h\), both positive and negative. See if your calculator or computer will produce a table of values. (b) Calculate \(f^{\prime}(1)\) by computing the limit of the difference quotient. (c) For what values of \(h\) is the difference quotient greater than \(f^{\prime}(1) ?\) For what values of \(h\) is the difference quotient less than \(f^{\prime}(1) ?\) Make sense out of this by looking at the graph of \(x^{3}\).

Let \(f(x)=x^{2}\). Find the point at which the line tangent to \(f(x)\) at \(x=2\) intersects the line tangent to \(f(x)\) at \(x=-1\).

Let \(f(x)=x^{3}\) and \(P\) be the point \((0,0)\) on the graph of \(f\). (a) Approximate \(f^{\prime}(0)\) by looking at the slope of the secant line through \(P\) and a nearby point \(Q\) on the graph of \(f\). Use a calculator or computer to get a sequence of successive approximations corresponding to allowing the point \(Q=(h, f(h))\) to slide along the graph of \(f\) toward \(P\). Choose both positive and negative values of \(h\). (b) Use the results of part (a) to guess the slope of the line tangent to \(x^{3}\) at \(x=0\). (c) Calculate \(f^{\prime}(0)\) by computing the limit of the difference quotient \(\frac{f(0+h)-f(0)}{h}\). (d) Challenge question: In the previous three problems, by varying the sign of \(h\) in the difference quotient \(\frac{f(c+h)-f(c)}{h}\) we obtain upper and lower bounds for \(f^{\prime}(c) .\) In this problem, the difference quotient computed in part (a) is always larger than \(f^{\prime}(0)\). Explain what is going on by looking at the graphs of the various functions.

The graph of \(g(x)=x^{2}-4\) looks just like the graph of \(f(x)=x^{2}\) shifted vertically downward 4 units. Will \(f^{\prime}\) and \(g^{\prime}\) be the same or different? Explain your reasoning.

Use the limit de nition of derivative to nd the derivative of \(f(x)=x^{3}\).

This problem deals with the effect of altitude on how far a batted ball will travel. The drag resistance on the ball is proportional to the density of the air, i.e., the barometric pressure if the temperature is held constant. Let us take as an example a 400 -foot home run in Yankee Stadium, which is approximately at sea level. On average, an increase in altitude of 275 feet would increase the length of this drive by 2 feet. (Adair, Robert K. The Physics of Baseball. New York: Harper \& Row, 1990.) Let \(B(a)\) be the distance this ball would travel as a function of the altitude of the ballpark in which it is hit. Assume the relationship between altitude and distance is linear. (a) What is the meaning of \(\frac{d B}{d a} ?\) What are its units? (b) What is the numerical value of \(\frac{d B}{d a}\) ? (c) Write an equation for \(B(a)\). (d) Prior to major league baseball's 1993 expansion into Denver, Atlanta, which has an altitude of 1050 feet, was the highest city in the majors. How far would this 400-foot Yankee Stadium drive travel in Atlanta? (e) How far would it travel in Denver (altitude 5280 feet)?

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\). $$ f(x)=\sqrt{x} \text { . Approximate } f^{\prime}(9) \text { . } $$

Using the limit de nition of the derivative, nd \(f^{\prime}(x)\) if \(f(x)=(x-1)^{2}\).

Suppose that \(A(p)\) gives the number of pounds of apples sold as a function of the price (in dollars) per pound. (a) What are the units of \(\frac{d A}{d p}\) ? (b) Do you expect \(\frac{d A}{d p}\) to be positive? Why or why not? (c) Interpret the statement \(A^{\prime}(0.88)=-5\).

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\). $$ f(x)=\frac{1}{\sqrt{x}} \text { . Approximate } f^{\prime}(4) $$

Let \(g(x)=\frac{x}{2 x+5}\). Using the limit de nition of derivative, nd \(g^{\prime}(x)\).

Between 1940 and 1995 the size of the average farm in America increased from 174 acres to 469 acres. (Facts from the World Almanac and Book of Facts 1997.) Suppose that \(A(t)\) gives the average number of acres of an American farm \(t\) years after 1940 . \(A(t)\) is an increasing function. (a) What are the units of \(\frac{d A}{d t}\) ? (b) Average farm size increased much more dramatically in the 50 s than in the 80 s. Which is larger, \(A(12)\) or \(A(43)\) ? Which do you think is larger, \(A^{\prime}(12)\) or \(A^{\prime}(43)\) ?

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\).\ $$ f(x)=x^{4} \text { . Approximate } f^{\prime}(1) \text { . } $$

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=3 x+5 $$

A baked apple is taken out of the oven and put into the refrigerator. The refrigerator is kept at a constant temperature. Newton's Law of Cooling says that the difference between the temperature of the apple and the temperature of the refrigerator decreases at a rate proportional to itself. That is, the apple cools down most rapidly at the outset of its stay in the refrigerator, and cools increasingly slowly as time goes by. You have the following pieces of information: At the moment the apple is put in the refrigerator its temperature is 110 degrees and is dropping at a rate of 4 degrees per minute. Twenty minutes later the temperature of the apple is 70 degrees. (a) Let \(T\) be the temperature of the apple at time \(t\), where \(t\) is measured in minutes and \(t=0\) is when the apple is put in the refrigerator. Express the three bits of information provided above in functional notation. Sketch a graph of \(T\) versus \(t\). (b) Using the same set of axes as you did in part (a), draw the cooling curve the baked apple would have if it were cooling linearly, with the initial temperature of 110 degrees and initial rate of cooling of 4 degrees per minute. What would the temperature of the apple be after 15 minutes using this linear model? In reality, is the temperature more or less? (c) Since the apple's temperature dropped from 110 degrees to 70 degrees in twenty minutes, the average rate of change of temperature over the first twenty minutes is \(\frac{-40 \text { degrees }}{20 \text { minutes }}\) or \(-2 \frac{\text { degrees }}{\text { minute }} .\) Using the same set of axes as you did in parts (a) and (b), draw the cooling curve the baked apple would have if it were cooling linearly, with the initial temperature of 110 degrees and rate of cooling of 2 degrees per minute. What would the temperature of the apple be after 15 minutes using this linear model? In reality, is the temperature more or less?

In Problems 5 through 8 , estimate \(f^{\prime}(c)\) by calculating the difference quotient \(\frac{f(c+h)-f(c)}{h}\) for successively smaller values of \(|h| .\) Use both positive and negative values of \(h .\). $$ f(x)=\sqrt[3]{x} \text { . Approximate } f^{\prime}(8) $$

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\pi x-\sqrt{3} $$

A hot-air balloonist is taking a balloon trip up a river valley. The trip begins at the mouth of the river. The balloon's altitude varies throughout the trip. Suppose that \(A(t)\), is the function that gives the balloon's height (in feet) above the ground at time \(t\), where \(t\) is the time from the start of the trip measured in hours. (a) Suppose that at time \(t=4\) hours \(A^{\prime}(4)\) is 70 . Interpret what \(A^{\prime}(4)=70\) tells us in words. (b) Let \(f\) be the function that takes as input \(x\), where \(x\) is the balloon's horizontal distance from the mouth of the river \((x\) measured in feet \()\) and gives as output the time it has taken the balloon to make it from the mouth of the river to this point. In other words, if \(f(1000)=4\) then the balloon has taken 4 hours to travel 1000 feet up the river bank. i. Let \(h(x)=A(f(x))\), where \(f\) and \(A\) are the functions given above. Describe in words the input and output of the function \(h\). ii. Interpret the statement \(h(700)=100\) in words. iii. Interpret the statement \(h^{\prime}(700)=60\) in words.

Let \(f(x)=\frac{6}{x+1}\). Let \(P\) and \(Q\) be points on the graph of \(f\) with \(x\) -coordinates 3 and \(3+h\), respectively. (a) Sketch the graph of \(f\) and the secant lines through \(P\) and \(Q\) for \(h=1\) and \(h=0.1\). (b) Find the slope of the secant line through \(P\) and \(Q\) for \(h=1, h=0.1\), and \(h=0.01\). (c) Find the slope of the tangent line to \(f\) at point \(P\) by calculating the appropriate limit. (d) Find the equation of the line tangent to \(f\) at point \(P\).

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\frac{2 x-5}{3} $$

How are the graphs of \(f^{\prime}\) and \(g^{\prime}\) related in each of the following situations? Explain your reasoning. (a) \(g(x)=f(x)+3\) (b) \(g(x)=f(x+3)\) (c) \(g(x)=3 f(x)\)

Suppose that \(C(s)\) gives the number of calories that an average adult burns by walking at a steady speed of \(s\) miles per hour for one hour. (a) What are the units of \(\frac{d C}{d s}\) ? (b) Do you expect \(\frac{d C}{d s}\) to be positive? Why or why not? (c) Interpret the statement \(C^{\prime}(3)=25\). Hint: If you are having difficulties with this problem, consider sketching a graph. What are the labels on the axes? (That is, what are the independent and dependent variables?) Thinking about these variables, what should the graph look like? How do your assumptions about the graph relate to the questions posed above?

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\pi(x+7)-2 $$

You have formed a study group with a few of your friends. One of the people in your study group has been ill for the past \(2 \frac{1}{2}\) weeks and is concerned about the upcoming examination. She needs to understand the main ideas of the past few weeks. Your essay should be designed to help her. (a) Explain the relationship between average rate of change and instantaneous rate of change, and between secant lines and tangent lines. Your classmate is unclear why the definition of derivative involves some limit with \(h\) going to zero. She wants to know why you can't just set \(h=0\) to begin with and be done with it. Why does calculus involve this limit business? (b) She also has one specific question. She is not clear about how you get the graph of \(f^{\prime}\) from the graph of \(f\). Just before she got sick we were doing that. She believes that if the graph of \(f^{\prime}\) is increasing, then the graph of \(f\) is also increasing. Her Course Assistant says this is wrong, but she claims that sometimes she gets the right answer using this reasoning. What's the story? How to write this essay: First, think about your friend's position. Particularly in part (a) see if you can understand what is confusing her and how to clarify it for her. Outline your answer. Then write your essay. Use words precisely. Try to avoid pronouns. For example, do not say "it" is increasing; be specific \(-\) what is increasing? Use words to say precisely what you mean. The real purpose of this essay: These are issues that are important for you to understand. We want you to put together what you have learned in your own words. We also want you to learn to write about mathematics by using words precisely to say exactly what you mean.

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=x^{2} $$

In Problems 11 and 12, estimate the slope of the tangent line at point \(P\) on the graph of \(f(x)\) by choosing a nearby point \(Q\) on the graph of \(f, Q=(a, f(a))\) and finding the slope of the secant line through \(P\) and \(Q .\) By choosing \(Q\) successively closer to \(P\), guess the slope of the tangent line at \(P\). $$ f(x)=x^{5} . P=(2,32) . \text { Use your calculator to construct a table. } $$

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\frac{1}{x} $$

For Problems 7 through 13, find \(f^{\prime}(x), f^{\prime}(0), f^{\prime}(2)\), and \(f^{\prime}(-1) .\) $$ f(x)=\frac{x+\pi}{2} $$

If we have a formula for \(f\) we can get quite good numerical estimates of the slope of the tangent line to \(f\) at a particular point. In this exercise we will do that. Below is the graph of \(y=f(x)=x^{2}-4\) (a) Goal: We want to estimate the value of \(f^{\prime}(2)\), i.e., to approximate the slope of the tangent line to the graph at the point \((2,0)\). To this end, do the following: (b) Goal: We want to estimate the value of \(f^{\prime}(1) ;\) that is, we want to approximate the slope of the tangent line to the graph at the point \((1,-3)\). (c) Goal: We want to estimate the value of \(f^{\prime}(0) ;\) i.e., we want to approximate the slope of the tangent line to the graph at the point \((0,-4)\). i. Sketch the tangent line to the graph at \((0,-4)\). ii. Find the slope of the line through \((0,-4)\) and \((0+h, f(0+h))\). iii. Use your answer to part ii to estimate the slope of the tangent line. (d) Goal: To estimate the value of \(f^{\prime}(\mathrm{c})\) for \(\mathrm{c}\) a constant. i. Find the slope of the line through (c, \(f(\mathrm{c}))\) and \((\mathrm{c}+h, f(\mathrm{c}+h))\). ii. Use your answer to the previous question to nd the slope of the tangent line to \(f\) at \(x=\mathrm{c}\). iii. Sketch the graph of \(f^{\prime}(x)\).

Let \(f(x)=2 x^{2}-5 x-1 .\) Let \(P\) and \(Q\) be points on the graph of \(f\) with \(x\) -coordinates 3 and \(3+h\), respectively. (a) Sketch the graph of \(f\) and the secant lines through \(P\) and \(Q\) for \(h=1\) and \(h=0.1\). (b) Find the slope of the secant line through \(P\) and \(Q\) for \(h=1, h=0.1\), and \(h=0.01\). (c) Find the slope of the tangent line to \(f\) at point \(P\) by calculating the appropriate limit. (d) Find the equation of the line tangent to \(f\) at point \(P\).

Find the equation of the line tangent to \(y=\frac{2}{x+1}\) at the point \((1,1)\).

Let \(f(x)=\frac{1}{x^{2}}\). Let \(P\) and \(Q\) be points on the graph of \(f\) with coordinates \((x, f(x))\) and \((x+\Delta x, f(x+\Delta x))\), respectively. (a) Find the slope of the secant line through \(P\) and \(Q\). Simplify your answer as much as possible. (b) By calculating the appropriate limit, nd the slope of the tangent line to \(f(x)\) at point \(P\)

We showed that the derivative of \(\sqrt{x}\) (or \(x^{\frac{1}{2}}\) ) is \(\frac{1}{2} \frac{1}{\sqrt{x}}\) (or \(\left.\frac{1}{2} x^{-\frac{1}{2}}\right) .\) Here we focus on \(f(x)=\sqrt{x-1}\) (a) How is the graph of \(\sqrt{x-1}\) related to that of \(\sqrt{x}\) ? (b) How is the graph of the derivative of \(\sqrt{x-1}\) related to that of the derivative of \(\sqrt{x} ?\) Illustrate with a rough sketch. (c) Given your answer to part (b) explain why \(\left.\frac{d}{d x} \sqrt{x}\right|_{x=4}=\left.\frac{d}{d x} \sqrt{x-1}\right|_{x=5}\). In other words, explain why the derivative of \(\sqrt{x}\) at \(x=4\) is equal to the derivative of \(\sqrt{x-1}\) evaluated at \(x=5\) (d) Show that \(f^{\prime}(5)=\frac{1}{4}\) using the limit de nition of derivative: $$ f^{\prime}(5)=\lim _{x \rightarrow 5} \frac{f(x)-f(5)}{x-5} $$

Let \(g\) be a function that is locally linear. We know that \(g^{\prime}(a)\) is the slope of the tangent line to the graph of \(g\) at point \(A=(a, g(a))\). Let \(Q=(t, g(t))\) be an arbitrary point on the graph of \(g, Q\) distinct from \(A\). (a) Write a difference quotient (i.e., an expression of the form \(\frac{?-?}{2-?}\), the quotient of two differences) that gives the slope of the secant line through points \(A\) and \(Q\). (b) Take the appropriate limit of the difference quotient in part (a) to arrive at an expression for \(g^{\prime}(a)\).

Show that \(\frac{d}{d x} \sqrt{x+8}=\frac{1}{2 \sqrt{x+8}}\) using the limit de nition of derivative. You ll use different versions of the de nition in parts (a) and (b). In both cases it will be necessary to rationalize the numerator in order to evaluate the limit. (a) \(f^{\prime}(x)=\lim _{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}\) (b) \(f^{\prime}(x)=\lim _{b \rightarrow x} \frac{f(b)-f(x)}{b-x}\)

Let \(f(x)=\frac{3}{x-5}\). (a) Using the limit de nition of derivative, nd \(f^{\prime}(2)\). (b) Find two ways of checking whether or not your answer is reasonable. These methods should not involve simply checking your algebra. They can be numerical or graphical use your ingenuity.

Let \(f(x)=x^{-\frac{1}{2}}\). Use the limit de nition of derivative to show that \(f^{\prime}(x)=-\frac{1}{2} x^{-\frac{3}{2}}\).

For Problems 21 through 24 use the limit definition of \(f^{\prime}(a)\) to find the derivative of \(f\) at the point indicated. $$ f(x)=\frac{3}{2-x} \text { at } x=1 $$

For Problems 21 through 24 use the limit definition of \(f^{\prime}(a)\) to find the derivative of \(f\) at the point indicated. $$ f(x)=x(x+3) \text { at } x=2 $$

For Problems 21 through 24 use the limit definition of \(f^{\prime}(a)\) to find the derivative of \(f\) at the point indicated. $$ f(x)=\frac{x}{2}+\frac{2}{x} \text { at } x=1 $$

For Problems 21 through 24 use the limit definition of \(f^{\prime}(a)\) to find the derivative of \(f\) at the point indicated. $$ f(x)=(x-3)^{2} \text { at } x=3 $$