Chapter 16

Just outside Newburgh, the New York State Thruway (I-87), running north-south, intersects Interstate 84, which runs east-west. At noon a car is at this intersection and traveling north at a constant speed of 55 miles per hour. At this moment a Greyhound bus is 150 miles west of the intersection and traveling east at a steady pace of 65 miles per hour. (a) When will the bus and the car be closest to one another? (b) What is the minimum distance between the two vehicles? (c) How far away from the intersection is the bus at this time?

Find \(y^{\prime}\). \(y=x^{2 \pi}+2 \pi^{x}\)

(a) Which of the following are equal to \((\ln x)^{2} ?\) i. \((\ln x)(\ln x)\) ii. \(\ln x^{2} \quad\) iii. \(\ln [(x)(x)] \quad\) iv. \(2 \ln x\) (b) Which of the following are equal to \(2 \ln x\) ? i. \((\ln x)(\ln x) \quad\) ii. \(\ln x^{2} \quad\) iii. \(\ln [(x)(x)]\) (c) Differentiate \(y=(\ln x)^{2}\). (Do this twice, first using the product rule and then using the Chain Rule.) (d) Differentiate \(y=\ln x^{2}\). (Do this twice, first using the log rules and the derivative of \(\ln x\) and then using the Chain Rule.)

Draw a semicircle of radius \(2 .\) Inscribe a rectangle as shown. What are the dimensions of the rectangle of the largest area? What is the largest area?

Find \(y^{\prime}\). \(y=\left(2 x^{2}+1\right)^{\sqrt{3}}\)

In Problems 2 through 20, find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. (a) \(f(x)=3(x+2)^{-5}\) (b) \(f(x)=2(3 x+7)^{-8}\)

Find \(y^{\prime}\). \(y=(3 x)^{\sqrt{2}+1}+\frac{1}{\sqrt{\pi x}}\)

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. \(f(x)=\ln \sqrt{\pi x+1}+\sqrt{\pi x}+(\pi x+\pi)^{5}+\frac{1}{\left(\pi x^{2}+1\right)^{3}}\) Hint: Use log operations to simplify the first term.)

Differentiate. (a) \(y=\frac{1}{x \ln 2+1}\) (b) \(y=\ln \left(5 x^{3}+8 x\right)\) (c) \(y=\left(2^{x}\right)\left(x^{2}+x\right)^{7}\) (d) \(y=\sqrt{\ln (5 x)+e^{6 x}}\) (e) \(y=\frac{7}{\sqrt{\ln x}}\) (f) \(y=\left(4^{x / 3}\right) \ln 3 x\)

Differentiate the following, simplifying the expression first if useful. (a) \(y=\pi e^{3 t^{2}+\pi}\) (b) \(y=\ln \left(e^{t}+1\right)\) (c) \(y=\frac{\pi^{2}}{\sqrt{x^{2}+4}}\) (d) \(y=\frac{1}{(\ln x)^{2.6}}\) (e) \(y=\frac{1}{\left(\ln x^{2}\right)^{1.5}}\) (f) \(y=\sqrt[3]{\ln \left(e^{t}+1\right)}\)

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{x}{\left(x^{3}+7 x\right)^{4}} $$

Differentiate. (a) \(y=\left(2^{2}\right)^{x}\) (b) \(y=2^{2^{x}}\) (c) \(y=\frac{e^{\pi x}}{x}\) (d) \(y=\frac{x^{3}+1}{x^{2}+1}\) (e) \(y=5 \ln \left(\frac{5 x+3}{\sqrt{x}}\right)\) (f) \(y=\frac{3}{2 \ln \left(8 x^{2}+1\right)}\)

Find \(y^{\prime}\), simplifying the expression first where useful. (a) \(y=e^{x} x^{e}\) (b) \(y=e^{1 / x}\) (c) \(y=\sqrt{e^{-x} x}\) (d) \(y=[\ln \sqrt{1-x}]^{-3.5}\) (e) \(y=\ln \left(\frac{x+1}{x-1}\right)\) (f) \(y=(1-\ln x)^{5 / 4}\) (g) \(y=\ln \sqrt{x(x+1)}\) (h) \(y=\frac{5}{\sqrt{\frac{1}{e^{6 x}}+x}}\)

What is the global maximum value of the function \(f(x)=\frac{3}{\sqrt{x^{2}+1}}\) and where is it attained? Instructions: First just look at this function. Without any calculus, try to figure out the answer. (It may be useful to check symmetry considerations.) Now use the first derivative to support your answer.

Identify and classify all critical points of the function \(f(x)=\left(x^{2}-4\right) x^{\pi+1}\) for \(x>0\).

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=e^{5 x}(1+2 x)^{6} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\left(1-\frac{1}{x}\right) e^{-x} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\ln \left(\sqrt{x^{3}}\right) e^{6 x} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=5 \ln \left(2 x^{2}+3 x\right) $$

\(g(x)\) is a continuous function with exactly two zeros, one at \(x=1\) and the other at \(x=4 . g(x)\) has a local minimum at \(x=3\) and a local maximum at \(x=7\). These are the only local extrema of \(g\). Let \(f(x)=[g(x)]^{4}\). (a) Find \(f^{\prime}(x)\) in terms of \(g\) and its derivatives. (b) Can we determine (definitively) whether \(g\) has an absolute minimum value on \((-\infty, \infty) ?\) If we can, where is that absolute minimum value attained? Can we determine (definitively) whether \(g\) has an absolute maximum value? If we can, where is that absolute maximum value attained? (c) What are the critical points of \(f ?\) (d) On what intervals is the graph of \(f\) increasing? On what intervals is it decreasing? (e) Identify the local maximum and minimum points of \(f\). (f) Can we determine (definitively) whether \(f\) has an absolute minimum value? If so, can we determine what that value is? If you haven't already done so, step back, take a good look at the problem (a bird's-eye view) and make sure your answers make sense.

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\left(3 x^{3}+2 x\right)^{13} $$

Assume that \(f, g\), and \(h\) are differentiable. Differentiate \(p(x)\) where (a) \(p(x)=f(x) g(x) h(x) .\) (Hint: Use the Product Rule twice.) (b) \(p(x)=\sqrt{g(x)+\ln f(x)}\)

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{\pi^{2}}{3\left(x^{3}+2\right)^{6}} $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=f(\ln x)-\ln (f(x)) $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{1}{x^{3}+7 x+5} $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=\sqrt{f(x) g(x)} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{3^{x}}{2^{x+1}} $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=\frac{1}{\sqrt{f(g(x))}} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=x 5^{\frac{1-1}{2}} $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=f\left(x^{2}\right) e^{g(x)} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\frac{4}{\sqrt{e^{x}+1}} $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=\frac{1}{[f(x)]^{2}}+f\left(\frac{1}{x^{2}}\right) $$

In Problems 13 through 18, find \(h^{\prime}(x) .\) Assume that \(f\) and \(g\) are differentiable on \((-\infty, \infty)\). Your answers may be in terms of \(f, g, f^{\prime}\), and \(g^{\prime}\). $$ h(x)=[f(x)]^{3} g(2 x) $$

In Problems 19 through 22, find \(\frac{d y}{d x}\). Take the time to prepare the expression so that it is as simple as possible to differentiate. $$ y=3 \ln \left(\frac{x^{2}-1}{x+2}\right) $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=\ln \left(e^{x}+x^{2}\right) $$

In Problems 19 through 22, find \(\frac{d y}{d x}\). Take the time to prepare the expression so that it is as simple as possible to differentiate. $$ y=\sqrt{\left(x^{2}+3\right)^{5}} $$

Find \(f^{\prime}(x) .\) Do these problems without using the Quotient Rule. $$ f(x)=x \ln \left(\frac{x}{x^{2}+1}\right) $$

In Problems 19 through 22, find \(\frac{d y}{d x}\). Take the time to prepare the expression so that it is as simple as possible to differentiate. $$ y=\frac{5 x^{\pi}-x^{3}-1}{x^{2}} $$

Find a formula for \(\frac{d y}{d x}\) if \(y=f(g(h(x)))\), where \(f, g\), and \(h\) are differentiable everywhere.

In Problems 19 through 22, find \(\frac{d y}{d x}\). Take the time to prepare the expression so that it is as simple as possible to differentiate. $$ y=5 \ln \sqrt{\frac{3 x}{x^{2}+1}} $$

In Problems 22 through 25, graph \(f(x)\), labeling the \(x\) -coordinates of all local extrema. Strategize. Is it more convenient to keep expressions factored? $$ f(x)=x(x+3)^{2} $$

In Problems 23 through 29, differentiate. In Problems 23 through 25, assume \(f\) is differentiable. Your answers may be in terms of \(f\) and \(f^{\prime} .\) $$ y=\ln \left(\frac{x}{f\left(x^{2}\right)}\right) $$

Graph \(f(x)\), labeling the \(x\) -coordinates of all local extrema. Strategize. Is it more convenient to keep expressions factored? $$ f(x)=(x-2)(x+1)^{2} $$

In Problems 23 through 29, differentiate. In Problems 23 through 25, assume \(f\) is differentiable. Your answers may be in terms of \(f\) and \(f^{\prime} .\) $$ y=[f(x)]^{2}+2^{f(x)} $$

Graph \(f(x)\), labeling the \(x\) -coordinates of all local extrema. Strategize. Is it more convenient to keep expressions factored? $$ f(x)=(3-x)^{2}(x-1) $$

Graph \(f(x)\), labeling the \(x\) -coordinates of all local extrema. Strategize. Is it more convenient to keep expressions factored? $$ f(x)=e^{x}(x-3)^{3} $$

In Problems 23 through 29, differentiate. In Problems 23 through 25, assume \(f\) is differentiable. Your answers may be in terms of \(f\) and \(f^{\prime} .\) $$ f(x)=\ln \left(e^{(x+5)^{2}}\right) $$

Prove that if \(f(x)=(x-a)^{3}(x-b)\) where \(a, b>0\) and \(a \neq b\), then \(f\) has a point of inflection at \(x=a\).

(a) Which of the following are equal to \(e^{-x^{2}} ?\) Identify all correct answers. i. \(e^{-(x)(x)}\) ii. \(\left(e^{-x}\right)^{x}\) iii. \(\left(\frac{1}{e^{x}}\right)^{x}\) iv. \(\left(\frac{1}{e^{x}}\right)^{2}\) v. \(\left(e^{-x}\right)^{2}\) vi. \(\left(e^{-2}\right)^{x}\) vii. \(e^{-2 x}\) viii. \(e^{(-x)^{2}}\) (b) Differentiate \(e^{-x^{2}}\).

In Problems 23 through 29, differentiate. In Problems 23 through 25, assume \(f\) is differentiable. Your answers may be in terms of \(f\) and \(f^{\prime} .\) $$ f(x)=\left(x^{3}+e\right)^{\pi} $$