Chapter 4

Verify the formulas by differentiation. $$\int(3 x+5)^{-2} d x=-\frac{(3 x+5)^{-1}}{3}+C$$

The 800 -room Mega Motel chain is filled to capacity when the room charge is \(\$ 50\) per night. For each \(\$ 10\) increase in room charge, 40 fewer rooms are filled each night. What charge per room will result in the maximum revenue per night?

a. Construct a polynomial \(f(x)\) that has zeros at \(x=-2,-1,0\) 1, and 2. b. Graph \(f\) and its derivative \(f^{\prime}\) together. How is what you see related to Rolle's Theorem? c. Do \(g(x)=\sin x\) and its derivative \(g^{\prime}\) illustrate the same phenomenon as \(f\) and \(f^{\prime} ?\)

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\). $$f(x)=-2 \cos x-\cos ^{2} x, \quad-\pi \leq x \leq \pi$$

Find the extreme values (absolute and local) of the function over its natural domain, and where they occur. $$y=\frac{x+1}{x^{2}+2 x+2}$$

Verify the formulas by differentiation. $$\int \sec ^{2}(5 x-1) d x=\frac{1}{5} \tan (5 x-1)+C$$

Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Also assume that \(f(a)\) and \(f(b)\) have opposite signs and that \(f^{\prime} \neq 0\) between \(a\) and \(b\). Show that \(f(x)=0\) exactly once between \(a\) and \(b\)

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2 / 3}(x+2)$$

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\). $$f(x)=\csc ^{2} x-2 \cot x, \quad 0

Verify the formulas by differentiation. $$\int \csc ^{2}\left(\frac{x-1}{3}\right) d x=-3 \cot \left(\frac{x-1}{3}\right)+C$$

a. When we cough, the trachea (windpipe) contracts to increase the velocity of the air going out. This raises the questions of how much it should contract to maximize the velocity and whether it really contracts that much when we cough. Under reasonable assumptions about the elasticity of the tracheal wall and about how the air near the wall is slowed by friction, the average flow velocity \(v\) can be modeled by the equation $$v=c\left(r_{0}-r\right) r^{2} \mathrm{cm} / \mathrm{s}, \quad \frac{r_{0}}{2} \leq r \leq r_{0}$$ where \(r_{0}\) is the rest radius of the trachea in centimeters and \(c\) is a positive constant whose value depends in part on the length of the trachea. Show that \(v\) is greatest when \(r=(2 / 3) r_{0} ;\) that is, when the trachea is about \(33 \%\) contracted. The remarkable fact is that X-ray photographs confirm that the trachea contracts about this much during a cough. b. Take \(r_{0}\) to be 0.5 and \(c\) to be 1 and graph \(v\) over the interval \(0 \leq r \leq 0.5 .\) Compare what you see with the claim that \(v\) is at a maximum when \(r=(2 / 3) r_{0}\)

Assume that \(f\) and \(g\) are differentiable on \([a, b]\) and that \(f(a)=g(a)\) and \(f(b)=g(b) .\) Show that there is at least one point between \(a\) and \(b\) where the tangents to the graphs of \(f\) and \(g\) are parallel or the same line. Illustrate with a sketch.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2 / 3}\left(x^{2}-4\right)$$

a. Find the local extrema of each function on the given interval, and say where they occur. b. Graph the function and its derivative together. Comment on the behavior of \(f\) in relation to the signs and values of \(f^{\prime}\). $$f(x)=\sec ^{2} x-2 \tan x, \quad \frac{-\pi}{2}

Verify the formulas by differentiation. $$\int \frac{1}{(x+1)^{2}} d x=-\frac{1}{x+1}+C$$

Show that if \(a, b, c,\) and \(d\) are positive integers, then $$\frac{\left(a^{2}+1\right)\left(b^{2}+1\right)\left(c^{2}+1\right)\left(d^{2}+1\right)}{a b c d} \geq 16$$

Suppose that \(f^{\prime}(x) \leq 1\) for \(1 \leq x \leq 4 .\) Show that \(f(4)-\) \(f(1) \leq 3\)

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x \sqrt{4-x^{2}}$$

Verify the formulas by differentiation. $$\int \frac{1}{(x+1)^{2}} d x=\frac{x}{x+1}+C$$

a. Show that $$f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}$$ is an increasing function of \(x\) b. Show that $$g(x)=\frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}}$$ is a decreasing function of \(x\) c. Show that $$\frac{d t}{d x}=\frac{x}{c_{1} \sqrt{a^{2}+x^{2}}}-\frac{d-x}{c_{2} \sqrt{b^{2}+(d-x)^{2}}}$$ is an increasing function of \(x\)

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=x^{2} \sqrt{3-x}$$

Show that the functions in Exercises 61 and 62 have local extreme values at the given values of \(\theta\), and say which kind of local extreme the function has. $$h(\theta)=5 \sin \frac{\theta}{2}, \quad 0 \leq \theta \leq \pi, \quad \text { at } \theta=0 \text { and } \theta=\pi$$

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int x \sin x d x=\frac{x^{2}}{2} \sin x+C\) b. \(\int x \sin x d x=-x \cos x+C\) c. \(\int x \sin x d x=-x \cos x+\sin x+C\)

Show that \(|\cos x-1| \leq|x|\) for all \(x\) -values.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} 4-2 x, & x \leq 1 \\ x+1, & x > 1 \end{array}\right.$$

Sketch the graph of a differentiable function \(y=f(x)\) through the point (1,1) if \(f^{\prime}(1)=0\) and a. \(f^{\prime}(x)>0\) for \(x<1\) and \(f^{\prime}(x)<0\) for \(x>1\) b. \(f^{\prime}(x)<0\) for \(x<1\) and \(f^{\prime}(x)>0\) for \(x>1\) c. \(f^{\prime}(x)>0\) for \(x \neq 1\) d. \(f^{\prime}(x)<0\) for \(x \neq 1\)

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{\sec ^{3} \theta}{3}+C\) b. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \tan ^{2} \theta+C\) c. \(\int \tan \theta \sec ^{2} \theta d \theta=\frac{1}{2} \sec ^{2} \theta+C\)

You have been asked to determine whether the function \(f(x)=\) \(3+4 \cos x+\cos 2 x\) is ever negative. a. Explain why you need to consider values of \(x\) only in the interval \([0,2 \pi]\) b. Is \(f\) ever negative? Explain.

Show that for any numbers \(a\) and \(b\), the sine inequality \(| \sin b-\) \(\sin a|\leq| b-a |\) is true.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} 3-x, & x < 0 \\ 3+2 x-x^{2}, & x \geq 0 \end{array}\right.$$

Sketch the graph of a differentiable function \(y=f(x)\) that has a. a local minimum at (1, 1) and a local maximum at (3, 3); b. a local maximum at (1, 1) and a local minimum at (3, 3); c. local maxima at ( 1,1 ) and (3,3) d. local minima at ( 1,1 ) and (3,3)

Right, or wrong? Say which for each formula and give a brie reason for each answer. a. \(\int(2 x+1)^{2} d x=\frac{(2 x+1)^{3}}{3}+C\) b. \(\int 3(2 x+1)^{2} d x=(2 x+1)^{3}+C\) c. \(\int 6(2 x+1)^{2} d x=(2 x+1)^{3}+C\)

a. The function \(y=\cot x-\sqrt{2} \csc x\) has an absolute maximum value on the interval \(0

If the graphs of two differentiable functions \(f(x)\) and \(g(x)\) start at the same point in the plane and the functions have the same rate of change at every point, do the graphs have to be identical? Give reasons for your answer.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} -x^{2}-2 x+4, & x \leq 1 \\ -x^{2}+6 x-4, & x>1 \end{array}\right.$$

Sketch the graph of a continuous function \(y=g(x)\) such that a. \(g(2)=2,02,\) and \(g^{\prime}(x) \rightarrow-1^{+}\) as \(x \rightarrow 2^{+}\) b. \(g(2)=2, g^{\prime}<0\) for \(x<2, g^{\prime}(x) \rightarrow-\infty\) as \(x \rightarrow 2^{-}\) \(g^{\prime}>0\) for \(x>2,\) and \(g^{\prime}(x) \rightarrow \infty\) as \(x \rightarrow 2^{+}\)

Right, or wrong? Say which for each formula and give a brief reason for each answer. a. \(\int \sqrt{2 x+1} d x=\sqrt{x^{2}+x+C}\) b. \(\int \sqrt{2 x+1} d x=\sqrt{x^{2}+x}+C\) c. \(\int \sqrt{2 x+1} d x=\frac{1}{3}(\sqrt{2 x+1})^{3}+C\)

If \(|f(w)-f(x)| \leq|w-x|\) for all values \(w\) and \(x\) and \(f\) is a differentiable function, show that \(-1 \leq f^{\prime}(x) \leq 1\) for all \(x\) -values.

Find the critical points, domain endpoints, and extreme values (absolute and local) for each function. $$y=\left\\{\begin{array}{ll} -\frac{1}{4} x^{2}-\frac{1}{2} x+\frac{15}{4}, & x \leq 1 \\ x^{3}-6 x^{2}+8 x, & x > 1 \end{array}\right.$$

Right, or wrong? Give a brief reason why. $$\int \frac{-15(x+3)^{2}}{(x-2)^{4}} d x=\left(\frac{x+3}{x-2}\right)^{3}+C$$

a. How close does the curve \(y=\sqrt{x}\) come to the point \((3 / 2,0) ?\) (Hint: If you minimize the square of the distance, you can avoid square roots.) b. Graph the distance function \(D(x)\) and \(y=\sqrt{x}\) together and reconcile what you see with your answer in part (a).

Assume that \(f\) is differentiable on \(a \leq x \leq b\) and that \(f(b)

Give reasons for your answers. Let \(f(x)=(x-2)^{2 / 3}\) a. Does \(f^{\prime}(2)\) exist? b. Show that the only local extreme value of \(f\) occurs at \(x=2\) c. Does the result in part (b) contradict the Extreme Value Theorem? d. Repeat parts (a) and (b) for \(f(x)=(x-a)^{2 / 3},\) replacing 2 by \(a\)

Right, or wrong? Give a brief reason why. $$\int \frac{x \cos \left(x^{2}\right)-\sin \left(x^{2}\right)}{x^{2}} d x=\frac{\sin \left(x^{2}\right)}{x}+C$$

a. How close does the semicircle \(y=\sqrt{16-x^{2}}\) come to the point \((1, \sqrt{3}) ?\) b. Graph the distance function and \(y=\sqrt{16-x^{2}}\) together and reconcile what you see with your answer in part (a).

Let \(f\) be a function defined on an interval \([a, b] .\) What conditions could you place on \(f\) to guarantee that $$\min f^{\prime} \leq \frac{f(b)-f(a)}{b-a} \leq \max f^{\prime}$$ where \(\min f^{\prime}\) and \(\max f^{\prime}\) refer to the minimum and maximum values of \(f^{\prime}\) on \([a, b] ?\) Give reasons for your answers.

Give reasons for your answers. Let \(f(x)=\left|x^{3}-9 x\right|\) a. Does \(f^{\prime}(0)\) exist? b. Does \(f^{\prime}(3)\) exist? c. Does \(f^{\prime}(-3)\) exist?

Determine the values of constants \(a\) and \(b\) so that \(f(x)=\) \(a x^{2}+b x\) has an absolute maximum at the point (1,2).

Which of the following graphs shows the solution of the initial value problem $$\frac{d y}{d x}=-x, \quad y=1 \text { when } x=-1 ?$$ Give reasons for your answer.

If an even function \(f(x)\) has a local maximum value at \(x=c,\) can anything be said about the value of \(f\) at \(x=-c ?\) Give reasons for your answer.