Chapter 4

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=\frac{x^{4}+1}{x^{3}} $$

Use the Mean Value Theorem to show that $$ |\sin x-\sin y| \leq|x-y| $$

Consider \(f(x)=A x^{2}+B x+C\) with \(A>0 .\) Show that \(f(x) \geq 0\) for all \(x\) if and only if \(B^{2}-4 A C \leq 0\).

Use a graphing calculator or a computer to do Problems \(45-48\) Let \(f(x)=\sin x+\cos (x / 2)\) on the interval \(I=(-2,7)\). (a) Draw the graph of \(f\) on \(I .\) (b) Use this graph to estimate where \(f^{\prime}(x)<0\) on \(I\). (c) Use this graph to estimate where \(f^{\prime \prime}(x)<0\) on \(I\). (d) Plot the graph of \(f^{\prime}\) to confirm your answer to part (b). (e) Plot the graph of \(f^{\prime \prime}\) to confirm your answer to part (c).

Let \(E\) be a differentiable function satisfying \(E(u+v)=E(u) E(v)\) for all \(u\) and \(v .\) Find a formula for \(E(x) .\) Hint: First find \(E^{\prime}(x) .\)

\(f^{\prime \prime}(x)\) is given. Find \(f(x)\) by antidifferentiating twice. Note that in this case your answer should involve two arbitrary constants, one from each antidifferentiation. For example, if \(f^{\prime \prime}(x)=x\), then \(f^{\prime}(x)=x^{2} / 2+C_{1}\) and \(f(x)=\) \(x^{3} / 6+C_{1} x+C_{2} .\) The constants \(C_{1}\) and \(C_{2}\) cannot be combined because \(C_{1} x\) is not a constant. $$ f^{\prime \prime}(x)=2 \sqrt[3]{x+1} $$

Suppose that in a race, horse \(A\) and horse \(B\) begin at the same point and finish in a dead heat. Prove that their speeds were identical at some instant of the race.

Consider \(f(x)=A x^{3}+B x^{2}+C x+D\) with \(A>0\). Show that \(f\) has one local maximum and one local minimum if and only if \(B^{2}-3 A C>0\).

Using the same axes, draw the graphs for \(0 \leq t \leq 100\) of the following two models for the growth of world population (both described in this section). (a) Exponential growth: \(y=6.4 e^{0.0132 t}\) (b) Logistic growth: \(y=102.4 /\left(6+10 e^{-0.030 t}\right)\) Compare what the two models predict for world population in 2010,2040, and 2090 . Note: Both models assume that world population was \(6.4\) billion in \(2004(t=0)\).

Prove the formula $$ \int\left[f(x) g^{\prime}(x)+g(x) f^{\prime}(x)\right] d x=f(x) g(x)+C $$ Hint: See the box in the margin next to Theorem A.

A clock has hour and minute hands of lengths \(h\) and \(m\), respectively, with \(h \leq m .\) We wish to study this clock at times between \(12: 00\) and \(12: 30 .\) Let \(\theta, \phi\), and \(L\) be as in Figure 33 and note that \(\theta\) increases at a constant rate. By the Law of Cosines, \(L=L(\theta)=\left(h^{2}+m^{2}-2 h m \cos \theta\right)^{1 / 2}\), and so $$ L^{\prime}(\theta)=h m\left(h^{2}+m^{2}-2 h m \cos \theta\right)^{-1 / 2} \sin \theta $$ (a) For \(h=3\) and \(m=5\), determine \(L^{\prime}, L\), and \(\phi\) at the instant when \(L^{\prime}\) is largest. (b) Rework part (a) when \(h=5\) and \(m=13\). (c) Based on parts (a) and (b), make conjectures about the values of \(L^{\prime}, L\), and \(\phi\) at the instant when the tips of the hands are separating most rapidly. (d) Try to prove your conjectures.

What conclusions can you draw about \(f\) from the information that \(f^{\prime}(c)=f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)>0 ?\)

Use a graphing calculator or a computer to do. Let \(f^{\prime}(x)=x^{3}-5 x^{2}+2\) on \(I=[-2,4]\). Where on \(I\) is \(f\) increasing?

The graph of \(y=f(x)\) depends on a parameter c. Using a CAS, investigate how the extremum and inflection points depend on the value of \(c .\) Identify the values of \(c\) at which the basic shape of the curve changes. $$ f(x)=c+\sin c x $$

Prove the formula $$ \int \frac{g(x) f^{\prime}(x)-f(x) g^{\prime}(x)}{g^{2}(x)} d x=\frac{f(x)}{g(x)}+C $$

Use the Mean Value Theorem to show that the graph of a concave up function \(f\) is always above its tangent line; that is, show that $$ f(x)>f(c)+f^{\prime}(c)(x-c), \quad x \neq c $$

Use a graphing calculator or a computer to do. Let \(f^{\prime \prime}(x)=x^{4}-5 x^{3}+4 x^{2}+4\) on \(I=[-2,3]\). Where on \(I\) is \(f\) concave down?

What conclusions can you draw about \(f\) from the information that \(f^{\prime}(c)=f^{\prime \prime}(c)=0\) and \(f^{\prime \prime \prime}(c)>0 ?\)

Prove that if \(|f(y)-f(x)| \leq M(y-x)^{2}\) for all \(x\) and \(y\) then \(f\) is a constant function.

The earth's position in the solar system at time \(t\) can be described approximately by \(P(93 \cos (2 \pi t), 93 \sin (2 \pi t))\), where the sun is at the origin and distances are measured in millions of miles. Suppose that an asteroid has position \(Q(60 \cos [2 \pi(1.51 t-1)], 120 \sin [2 \pi(1.51 t-1)]) .\) When, over the time period \([0,20]\) (i.e., over the next 20 years), does the asteroid come closest to the earth? How close does it come?

Translate each of the following into the language of derivatives of distance with respect to time. For each part, sketch a plot of the car's position \(s\) against time \(t\), and indicate the concavity. (a) The speed of the car is proportional to the distance it has traveled. (b) The car is speeding up. (c) I didn't say the car was slowing down; I said its rate of increase in speed was slowing down. (d) The car's speed is increasing 10 miles per hour every minute. (e) The car is slowing very gently to a stop. (f) The car always travels the same distance in equal time intervals.

Let \(g(x)\) be a function that has two derivatives and satisfies the following properties: (a) \(g(1)=1\); (b) \(g^{\prime}(x)>0\) for all \(x \neq 1\); (c) \(g\) is concave down for all \(x<1\) and concave up for all \(x>1\) (d) \(f(x)=g\left(x^{4}\right)\); Sketch a possible graph of \(f(x)\) and justify your answer.

Give an example of a function \(f\) that is continuous on \([0,1]\), differentiable on \((0,1)\), and not differentiable on \([0,1]\), and has a tangent line at every point of \([0,1]\).

An advertising flyer is to contain 50 square inches of printed matter, with 2 -inch margins at the top and bottom and 1-inch margins on each side. What dimensions for the flyer would use the least paper?

Let \(H(x)\) have three continuous derivatives, and be such that \(H(1)=H^{\prime}(1)=H^{\prime \prime}(1)=0\), but \(H^{\prime \prime \prime}(1) \neq 0 .\) Does \(H(x)\) have a local maximum, local minimum, or a point of inflection at \(x=1\) ? Justify your answer

Find \(\int f^{\prime \prime}(x) d x\) if \(f(x)=x \sqrt{x^{3}+1}\)

Translate each of the following statements into mathematical language, sketch a plot of the appropriate function, and indicate the concavity. (a) The cost of a car continues to increase and at a faster and faster rate. (b) During the last 2 years, the United States has continued to cut its consumption of oil, but at a slower and slower rate. (c) World population continues to grow, but at a slower and slower rate. (d) The angle that the Leaning Tower of Pisa makes with the vertical is increasing more and more rapidly. (e) Upper Midwest firm's profit growth slows. (f) The XYZ Company has been losing money, but will soon turn this situation around.

In each case, is it possible for a function \(F\) with two continuous derivatives to satisfy the following properties? If so sketch such a function. If not, justify your answer. (a) \(F^{\prime}(x)>0, F^{\prime \prime}(x)>0\), while \(F(x)<0\) for all \(x\). (b) \(F^{\prime \prime}(x)<0\), while \(F(x)>0\). (c) \(F^{\prime \prime}(x)<0\), while \(F^{\prime}(x)>0\).

Brass is produced in long rolls of a thin sheet. To monitor the quality, inspectors select at random a piece of the sheet, measure its area, and count the number of surface imperfections on that piece. The area varies from piece to piece. The following table gives data on the area (in square feet) of the selected piece and the number of surface imperfections found on that piece. $$ \begin{array}{ccc} \hline \text { Piece } & \begin{array}{c} \text { Area in } \\ \text { Square Feet } \end{array} & \begin{array}{c} \text { Number of } \\ \text { Surface Imperfections } \end{array} \\ \hline 1 & 1.0 & 3 \\ 2 & 4.0 & 12 \\ 3 & 3.6 & 9 \\ 4 & 1.5 & 5 \\ 5 & 3.0 & 8 \\ \hline \end{array} $$ (a) Make a scatter plot with area on the horizontal axis and number of surface imperfections on the vertical axis. (b) Does it look like a line through the origin would be a good model for these data? Explain. (c) Find the equation of the least-squares line through the origin. (d) Use the result of part (c) to predict how many surface imperfections there would be on a sheet with area \(2.0\) square feet

A car is stationary at a toll booth. Eighteen minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Sketch a possible graph of \(v\) versus \(t\). Sketch a possible graph of the distance traveled \(s\) against \(t .\) Use the Mean Value Theorem to show that the car must have exceeded the 60 mile per hour speed limit at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Translate each statement from the following newspaper column into a statement about derivatives. (a) In the United States, the ratio \(R\) of government debt to national income remained unchanged at around \(28 \%\) up to 1981 , but (b) then it began to increase more and more sharply, reaching \(36 \%\) during \(1983 .\)

Use a graphing calculator or a CAS to plot the graphs of each of the following functions on the indicated interval. Determine the coordinates of any of the global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. Restrict the \(y\) -axis window to \(-5 \leq y \leq 5.\) (a) \(f(x)=x^{2} \tan x ;\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (b) \(f(x)=x^{3} \tan x ;\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) (c) \(f(x)=2 x+\sin x ;[-\pi, \pi]\) (d) \(f(x)=x-\frac{\sin x}{2} ;[-\pi, \pi]\)

Suppose that every customer order taken by the XYZ Company requires exacty 5 hours of labor for handling the paperwork; this length of time is fixed and does not vary from lot to lot. The total number of hours \(y\) required to manufacture and sell a lot of size \(x\) would then be \(y=(\) number of hours to produce a lot of size \(x)+5\) Some data on XYZ's bookcases are given in the following table. $$ \begin{array}{ccc} \hline & & \text { Total Labor } \\ \text { Order } & \text { Lot Size } x & \text { Hours } y \\ \hline 1 & 11 & 38 \\ 2 & 16 & 52 \\ 3 & 8 & 29 \\ 4 & 7 & 25 \\ 5 & 10 & 38 \\ \hline \end{array} $$ (a) From the description of the problem, the least-squares line should have 5 as its \(y\) -intercept. Find a formula for the value of the slope \(b\) that minimizes the sum of squares $$ S=\sum_{i=1}^{n}\left[y_{i}-\left(5+b x_{i}\right)\right]^{2} $$ (b) Use this formula to estimate the slope \(b\). (c) Use your least-squares line to predict the total number of labor hours to produce a lot consisting of 15 bookcases.

A car is stationary at a toll booth. Twenty minutes later at a point 20 miles down the road the car is clocked at 60 miles per hour. Explain why the car must have exceeded 60 miles per hour at some time after leaving the toll booth, but before the car was clocked at 60 miles per hour.

Each of the following functions is periodic. Use a graphFicing calculator or a CAS to plot the graph of each of the following functions over one full period with the center of the interval located at the origin. Determine the coordinates of any of the global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=2 \sin x+\cos ^{2} x\) (b) \(f(x)=2 \sin x+\sin ^{2} x\) (c) \(f(x)=\cos 2 x-2 \cos x\) (d) \(f(x)=\sin 3 x-\sin x\) (e) \(f(x)=\sin 2 x-\cos 3 x\)

Evaluate the indefinite integral $$ \int \sin ^{3}\left[\left(x^{2}+1\right)^{4}\right] \cos \left[\left(x^{2}+1\right)^{4}\right]\left(x^{2}+1\right)^{3} x d x $$ Hint: Let \(u=\sin \left(x^{2}+1\right)^{4}\).

The fixed monthly cost of operating a plant that makes Zbars is \(\$ 7000\), while the cost of manufacturing each unit is \(\$ 100\). Write an expression for \(C(x)\), the total cost of making \(x\) Zbars in a month.

Show that if an object's position function is given by \(s(t)=a t^{2}+b t+c\), then the average velocity over the interval \([A, B]\) is equal to the instantaneous velocity at the midpoint of \([A, B] .\)

Evaluate \(\int|x| d x\).

The manufacturer of Zbars estimates that 100 units per month can be sold if the unit price is \(\$ 250\) and that sales will increase by 10 units for each \(\$ 5\) decrease in price. Write an expression for the price \(p(n)\) and the revenue \(R(n)\) if \(n\) units are sold in one month, \(n \geq 100\)

Evaluate \(\int \sin ^{2} x d x\).

Some software packages can evaluate indefinite integrals. Use your software on each of the following. (a) \(\int 6 \sin (3(x-2)) d x\) (b) \(\int \sin ^{3}(x / 6) d x\) (c) \(\int\left(x^{2} \cos 2 x+x \sin 2 x\right) d x\)

Let \(F_{0}(x)=x \sin x\) and \(F_{n+1}(x)=\int F_{n}(x) d x\). (a) Determine \(F_{1}(x), F_{2}(x), F_{3}(x)\), and \(F_{4}(x)\). (b) On the basis of part (a), conjecture the form of \(F_{16}(x)\).

The total cost of producing and selling \(x\) units of Xbars per month is \(C(x)=100+3.002 x-0.0001 x^{2} .\) If the production level is 1600 units per month, find the average cost, \(C(x) / x\), of each unit and the marginal cost.

The total cost of producing and selling \(n\) units of a certain commodity per week is \(C(n)=1000+n^{2} / 1200 .\) Find the average cost, \(C(n) / n\), of each unit and the marginal cost at a production level of 800 units per week.

Use a graphing calculator or a CAS to plot the graph of each of the following functions on \([-1,7]\). Determine the coordinates of any global extrema and any inflection points. You should be able to give answers that are accurate to at least one decimal place. (a) \(f(x)=x \sqrt{x^{2}-6 x+40}\) (b) \(f(x)=\sqrt{|x|}\left(x^{2}-6 x+40\right)\) (c) \(f(x)=\sqrt{x^{2}-6 x+40} /(x-2)\) (d) \(f(x)=\sin \left[\left(x^{2}-6 x+40\right) / 6\right]\)

The total cost of producing and selling \(100 x\) units of a particular commodity per week is $$ C(x)=1000+33 x-9 x^{2}+x^{3} $$ Find (a) the level of production at which the marginal cost is a minimum, and (b) the minimum marginal cost.

A price function, \(p\), is defined by $$ p(x)=20+4 x-\frac{x^{2}}{3} $$ where \(x \geq 0\) is the number of units. (a) Find the total revenue function and the marginal revenue function. (b) On what interval is the total revenue increasing? (c) For what number \(x\) is the marginal revenue a maximum?

For the price function defined by $$ p(x)=(182-x / 36)^{1 / 2} $$ find the number of units \(x_{1}\) that makes the total revenue a maximum and state the maximum possible revenue. What is the marginal revenue when the optimum number of units, \(x_{1}\), is sold?

A riverboat company offers a fraternal organization a Fourth of July excursion with the understanding that there will be at least 400 passengers. The price of each ticket will be \(\$ 12.00\), and the company agrees to discount the price by \(\$ 0.20\) for each 10 passengers in excess of \(400 .\) Write an expression for the price function \(p(x)\) and find the number \(x_{1}\) of passengers that makes the total revenue a maximum.