Chapter 4

Use (4.14) to find the derivative of the inverse at the indicated point. Denote the inverse of \(y=\sin x,-\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\), by \(y=\arcsin x\), \(-1 \leq x \leq 1\). Show that $$ \frac{d}{d x} \arcsin x=\frac{1}{\sqrt{1-x^{2}}}, \quad-1

Differentiate the functions with respect to the independent variable. \(f(x)=\cos \left(e^{x}\right)\)

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=(1-x)\left(2-x^{2}\right)\), at \(x=2\)

Differentiate the functions given with respect to the independent variable. $$ f(x)=\pi^{3} x-x^{2} \pi $$

Molecules of \(\mathrm{A}\) and \(\mathrm{B}\) react to produce products \(\mathrm{C}\) and \(\mathrm{D}\) $$ \mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}+\mathrm{D} $$ The rate constant for this reaction is \(k .\) We denote by \([\mathrm{A}]\) the amount of A present, and so on. (a) Explain why the amount of \(\mathrm{C}\) present obeys the differential equation $$ \frac{d[\mathrm{C}]}{d t}=k[\mathrm{~A}][\mathrm{B}] $$ (b) Find similar differential equations for \([\mathrm{A}],[\mathrm{B}]\), and \([\mathrm{D}]\).

Assume that the radius \(r\) and the volume \(V=\frac{4}{3} \pi r^{3}\) of a sphere are differentiable functions of \(t .\) Express \(d V / d t\) in terms of \(d r / d t\).

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(x)=\sqrt[3]{1+2 x} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{-x}\) at \(a=0\)

(a) Use the formal definition to find the derivative of \(y=\) \(1-x^{3}\) at \(x=2\) (b) Show that the point \((2,-7)\) is on the graph of \(y=1-x^{3}\), and find the equation of the normal line at the point \((2,-7)\). (c) Graph \(y=1-x^{3}\) and the tangent line at the point \((2,-7)\) in the same coordinate system.

Find the derivative with respect to the independent variable. $$ f(x)=4 \cos ^{2} x+2 \cos x^{4} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (x+1) $$

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=5(1-2 x)(x+1)-3\), at \(x=0\)

Differentiate $$ f(x)=a x^{3} $$ with respect to \(x\). Assume that \(a\) is a constant.

Which of the following statements is true? (A) If \(f(x)\) is continuous, then \(f(x)\) is differentiable. (B) If \(f(x)\) is differentiable, then \(f(x)\) is continuous.

Suppose that water is stored in a cylindrical tank of radius \(5 \mathrm{~m}\). If the height of the water in the tank is \(h\), then the volume of the water is \(V=\pi r^{2} h=\left(25 \mathrm{~m}^{2}\right) \pi h=25 \pi h \mathrm{~m}^{2} .\) If we drain the water at a rate of 250 liters per minute, what is the rate at which the water level inside the tank drops? (Note that 1 cubic meter contains 1000 liters.)

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ f(x)=\sqrt[7]{x^{2}-2 x+1} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{-3 x}\) at \(a=0\)

(a) Use the formal definition to find the derivative of \(y=\frac{1}{x}\) at \(x=2\). (b) Show that the point \(\left(2, \frac{1}{2}\right)\) is on the graph of \(y=\frac{1}{x}\), and find the equation of the normal line at the point \(\left(2, \frac{1}{2}\right)\). (c) Graph \(y=\frac{1}{x}\) and the tangent line at the point \(\left(2, \frac{1}{2}\right)\) in the same coordinate system.

Find the derivative with respect to the independent variable. $$ f(x)=-3 \cos ^{2}\left(3 x^{2}-4\right) $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (3 x+4) $$

Apply the product rule to find the normal line, in slope-intercept form, of \(y=f(x)\) at the specified point. \(f(x)=\frac{(2-x)(3-x)}{4}\), at \(x=-1\)

Differentiate $$ f(x)=x^{3}+a $$ with respect to \(x\). Assume that \(a\) is a constant.

Suppose that we pump water into an inverted right circular conical tank at the rate of 5 cubic feet per minute (i.e., the tank stands with its point facing downward). The tank has a height of 6 ft and the radius on top is \(3 \mathrm{ft}\). What is the rate at which the water level is rising when the water is 2 ft deep? (Note that the volume of a right circular cone of radius \(r\) and height \(h\) is \(V=\frac{1}{3} \pi r^{2} h .\) )

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=e^{x-1}\) at \(a=1\)

Use the formal definition to find the derivative of \(y=\sqrt{x}\) at \(x=2\)

Find the derivative with respect to the independent variable. $$ f(x)=2 \tan \left(1-x^{2}\right) $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (1-2 x) $$

Apply the product rule for the product of three functions to find the derivative of \(y=f(x)\). \(f(x)=(x-3)(2-3 x)(5-x)\)

Differentiate $$ f(x)=a x^{2}-2 a $$ with respect to \(x\). Assume that \(a\) is a constant.

If \(f(x)\) is differentiable for all \(x \in \mathbf{R}\) except at \(x=c\), is it true that \(f(x)\) must be continuous at \(x=c\) ? Justify your answer.

Two people start biking from the same point. One heads east at \(15 \mathrm{mph}\), the other south at \(18 \mathrm{mph}\). What is the rate at which the distance between the two people is changing after 20 minutes and after 40 minutes?

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ g(s)=\left(3 s^{7}-7 s\right)^{3 / 2} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ . \(f(x)=e^{2 x+1}\) at \(a=-1 / 2\)

Use the formal definition to find the derivative of \(f(x)=\frac{1}{x+1}\) at \(x=0\).

Find the derivative with respect to the independent variable. $$ f(x)=-\cos \left(3 x^{3}-4 x\right) $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln (4-3 x) $$

Apply the product rule for the product of three functions to find the derivative of \(y=f(x)\). \(f(x)=(2 x-1)(3 x+4)(1-x)\)

Differentiate $$ f(x)=a^{2} x^{4}-2 a x^{2} $$ with respect to \(x\). Assume that \(a\) is a constant.

In Problems 26-39, graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=|x-2| $$

Skull Size Allometric equations describe the scaling relationship between two measurements, such as skull length versus body length. In vertebrates, we typically find that [skull length] \(\propto[\text { body length }]^{a}\) for \(0

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(t)=\left(t^{4}-5 t\right)^{5 / 2} $$

Calculate the linear approximation for \(f(x)\) : $$f(x) \approx f(a)+f^{\prime}(a)(x-a)$$ \(f(x)=(1+x)^{-n}\) at \(a=0 .\) (Assume that \(n\) is a positive integer.)

Find the equation of the tangent line to the curve \(y=3 x^{2}+1\) at the point \((0,1)\).

Find the derivative with respect to the independent variable. $$ f(x)=\sin \sqrt{x} $$

Differentiate the functions with respect to the independent variable. (Note that log denotes the logarithm to base 10.) $$ f(x)=\ln x^{2} $$

Apply the product rule for the product of three functions to find the derivative of \(y=f(x)\). \(f(x)=(x-3)\left(2 x^{2}+1\right)\left(1-x^{2}\right)\)

Differentiate $$ h(s)=r s^{2}-r $$ with respect to \(s\). Assume that \(r\) is a constant.

In Problems , graph each function and, on the basis of the graph, guess where the function is not differentiable. (Assume the largest possible domain.) $$ y=-|x+5| $$

Metabolism West, Brown, and Enquist (1997) argued that because of the distribution of blood vessels through mammalian bodies, the energy needs \(E\) of mammals increase with the \(3 / 4\) power of their mass, \(M ;\) i.e., $$ E=c M^{3 / 4} $$ for some constant \(c\). As a mammal grows, \(M\) increases. Show how \(d E / d t\) is related to \(d M / d t\) according to the theory of West, Brown, and Enquist.

In Problems 1-28, differentiate the functions with respect to the independent variable. $$ h(t)=\left(3 t+\frac{3}{t}\right)^{2 / 5} $$