Chapter 3

Evaluate the given limit without using L'Hôpital's rule, and then check that your answer is correct using L'Hôpital's rule. (a) \(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x^{2}+2 x-8}\) (b) \(\lim _{x \rightarrow+\infty} \frac{2 x-5}{3 x+7}\)

Both \(x\) and \(y\) denote functions of \(t\) that are related by the given equation. Use this equation and the given derivative information to find the specified derivative. $$ \begin{array}{l}{\text { Equation: } y=3 x+5 .} \\ {\text { (a) Given that } d x / d t=2, \text { find } d y / d t \text { when } x=1 .} \\ {\text { (b) Given that } d y / d t=-1, \text { find } d x / d t \text { when } x=0 \text { . }}\end{array} $$

(a) Find \(d y / d x\) by differentiating implicitly. (b) Solve the equation for \(y\) as a function of \(x,\) and find \(d y / d x\) from that equation. (c) Confirm that the two results are consistent by expressing the derivative in part (a) as a function of \(x\) alone. \(x+x y-2 x^{3}=2\)

Find \(d y / d x\) $$ y=\ln \frac{x}{3} $$

Evaluate the given limit without using L'Hôpital's rule, and then check that your answer is correct using L'Hôpital's rule. (a) \(\lim _{x \rightarrow 0} \frac{\sin x}{\tan x}\) (b) \(\lim _{x \rightarrow 1} \frac{x^{2}-1}{x^{3}-1}\)

Both \(x\) and \(y\) denote functions of \(t\) that are related by the given equation. Use this equation and the given derivative information to find the specified derivative. $$ \begin{array}{l}{\text { Equation: } x+4 y=3 .} \\ {\text { (a) Given that } d x / d t=1, \text { find } d y / d t \text { when } x=2 .} \\ {\text { (b) Given that } d y / d t=4, \text { find } d x / d t \text { when } x=3 \text { . }}\end{array} $$

Let \(f(x)=x^{3}+2 e^{x}\) (a) Show that \(f\) is one-to-one and confirm that \(f(0)=2\). (b) Find \(\left(f^{-1}\right)^{\prime}(2)\).

(a) Find \(d y / d x\) by differentiating implicitly. (b) Solve the equation for \(y\) as a function of \(x,\) and find \(d y / d x\) from that equation. (c) Confirm that the two results are consistent by expressing the derivative in part (a) as a function of \(x\) alone. \(\sqrt{y}-\sin x=2\)

Find \(d y / d x\) $$ y=\ln |1+x| $$

(a) Find the local linear approximation of the function \(f(x)=\sqrt{1+x}\) at \(x_{0}=0,\) and use it to approximate \(\sqrt{0.9}\) and \(\sqrt{1.1}\) (b) Graph \(f\) and its tangent line at \(x_{0}\) together, and use the graphs to illustrate the relationship between the exact values and the approximations of \(\sqrt{0.9}\) and \(\sqrt{1.1}\)

Both \(x\) and \(y\) denote functions of \(t\) that are related by the given equation. Use this equation and the given derivative information to find the specified derivative. $$ \begin{array}{l}{\text { Equation: } 4 x^{2}+9 y^{2}=1} \\ {\text { (a) Given that } d x / d t=3, \text { find } d y / d t \text { when }} \\ {\qquad(x, y)=\left(\frac{1}{2 \sqrt{2}}, \frac{1}{3 \sqrt{2}}\right)}\end{array} $$ $$ \begin{array}{l}{\text { (b) Given that } d y / d t=8, \text { find } d x / d t \text { when }} \\ {\qquad(x, y)=\left(\frac{1}{3},-\frac{\sqrt{5}}{9}\right)}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{2}+y^{2}=100\)

Find \(d y / d x\) $$ y=\ln (2+\sqrt{x}) $$

Determine whether the statement is true or false. Explain your answer. For any polynomial \(p(x), \lim _{x \rightarrow+\infty} \frac{p(x)}{e^{x}}=0\).

Both \(x\) and \(y\) denote functions of \(t\) that are related by the given equation. Use this equation and the given derivative information to find the specified derivative. $$ \begin{array}{l}{\text { Equation: } x^{2}+y^{2}=2 x+4 y} \\ {\text { (a) Given that } d x / d t=-5 \text { , find } d y / d t \text { when }} \\ {(x, y)=(3,1) .} \\ {\text { (b) Given that } d y / d t=6, \text { find } d x / d t \text { when }} \\ {\quad(x, y)=(1+\sqrt{2}, 2+\sqrt{3})}\end{array} $$

Find \(\left(f^{-1}\right)^{\prime}(x)\) using Formula (2), and check your answer by differentiating \(f^{-1}\) directly. $$ f(x)=\ln (2 x+1) $$

Find \(d y / d x\) by implicit differentiation. \(x^{3}+y^{3}=3 x y^{2}\)

Find \(d y / d x\) $$ y=\ln \left|x^{2}-1\right| $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ (1+x)^{15} \approx 1+15 x $$

Let \(A\) be the area of a square whose sides have length \(x,\) and assume that \(x\) varies with the time \(t\) (a) Draw a picture of the square with the labels \(A\) and \(x\) placed appropriately. (b) Write an equation that relates \(A\) and \(x\). (c) Use the equation in part (b) to find an equation that relates \(d A / d t\) and \(d x / d t\). (d) At a certain instant the sides are \(3 \mathrm{ft}\) long and increasing at a rate of \(2 \mathrm{ft} / \mathrm{min}\). How fast is the area increasing at that instant?

Determine whether the function \(f\) is one-to-one by examining the sign of \(f^{\prime}(x)\). $$ \begin{array}{l}{\text { (a) } f(x)=x^{2}+8 x+1} \\ {\text { (b) } f(x)=2 x^{5}+x^{3}+3 x+2} \\ {\text { (c) } f(x)=2 x+\sin x} \\ {\text { (d) } f(x)=\left(\frac{1}{2}\right)^{x}}\end{array} $$

Find \(d y / d x\) by implicit differentiation. \(x^{2} y+3 x y^{3}-x=3\)

Find \(d y / d x\) $$ y=\ln \left|x^{3}-7 x^{2}-3\right| $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ \frac{1}{\sqrt{1-x}} \approx 1+\frac{1}{2} x $$

Determine whether the statement is true or false. Explain your answer. $$\lim _{x \rightarrow 0^{+}}(\sin x)^{1 / x}=0$$

In parts (a)-(d), let \(A\) be the area of a circle of radius \(r,\) and assume that \(r\) increases with the time \(t\) (a) Draw a picture of the circle with the labels \(A\) and \(r\) placed appropriately. (b) Write an equation that relates \(A\) and \(r .\) (c) Use the equation in part (b) to find an equation that relates \(d A / d t\) and \(d r / d t\) (d) At a certain instant the radius is \(5 \mathrm{cm}\) and increasing at the rate of \(2 \mathrm{cm} / \mathrm{s}\). How fast is the area increasing at that instant?

Determine whether the function \(f\) is one-to-one by examining the sign of \(f^{\prime}(x)\). $$ \begin{array}{l}{\text { (a) } f(x)=x^{3}+3 x^{2}-8} \\ {\text { (b) } f(x)=x^{5}+8 x^{3}+2 x-1} \\ {\text { (c) } f(x)=\frac{x}{x+1}} \\ {\text { (d) } f(x)=\log _{b} x, \quad 0

Find \(d y / d x\) by implicit differentiation. \(x^{3} y^{2}-5 x^{2} y+x=1\)

Find \(d y / d x\) $$ y=\ln \left(\frac{x}{1+x^{2}}\right) $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ \tan x \approx x $$

Find the limits. $$ \lim _{x \rightarrow 0} \frac{e^{x}-1}{\sin x} $$

Let \(V\) be the volume of a cylinder having height \(h\) and radius \(r,\) and assume that \(h\) and \(r\) vary with time. (a) How are \(d V / d t, d h / d t,\) and \(d r / d t\) related? (b) At a certain instant, the height is 6 in and increasing at 1 in \(/ \mathrm{s}\), while the radius is 10 in and decreasing at \(1 \mathrm{in} / \mathrm{s}\). While the radius is \(10 \mathrm{in}\) and decreasing at \(1 \mathrm{in} / \mathrm{s}\). How fast is the volume changing at that instant? Is the volume increasing or decreasing at that instant?

Find \(d y / d x\) by implicit differentiation. \(\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{y}}=1\)

Find \(d y / d x\) $$ y=\ln \left|\frac{1+x}{1-x}\right| $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ \frac{1}{1+x} \approx 1-x $$

Find the limits. $$ \lim _{x \rightarrow 0} \frac{\sin 2 x}{\sin 5 x} $$

Let \(l\) be the length of a diagonal of a rectangle whose sides have lengths \(x\) and \(y,\) and assume that \(x\) and \(y\) vary with time. (a) How are \(d l / d t, d x / d t,\) and \(d y / d t\) related? (b) If \(x\) increases at a constant rate of \(\frac{1}{2} \mathrm{ft} / \mathrm{s}\) and \(y\) decreases at a constant rate of \(\frac{1}{4} \mathrm{ft} / \mathrm{s}\), how fast is the size of the diagonal changing when \(x=3 \mathrm{ft}\) and \(y=4 \mathrm{ft} ?\) Is the diagonal increasing or decreasing at that instant?

Find \(d y / d x\) by implicit differentiation. \(x^{2}=\frac{x+y}{x-y}\)

Find \(d y / d x\) $$ y=\ln x^{2} $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ e^{x} \approx 1+x $$

Find the limits. $$ \lim _{\theta \rightarrow 0} \frac{\tan \theta}{\theta} $$

Let \(\theta\) (in radians) be an acute angle in a right triangle, and let \(x\) and \(y\), respectively, be the lengths of the sides adjacent to and opposite \(\theta .\) Suppose also that \(x\) and \(y\) vare with time. (a) How are \(d \theta / d t, d x / d t,\) and \(d y / d t\) related? (b) At a certain instant, \(x=2\) units and is increasing at 1 unit/s, while \(y=2\) units and is decreasing at \(\frac{1}{4}\) unit/s. How fast is \(\theta\) changing at that instant? Is \(\theta\) increasing or decreasing at that instant?

Find \(d y / d x\) by implicit differentiation. \(\sin \left(x^{2} y^{2}\right)=x\)

Find \(d y / d x\) $$ y=(\ln x)^{3} $$

Confirm that the stated formula is the local linear approximation at \(x_{0}=0\) $$ \ln (1+x) \approx x $$

Find the limits. $$ \lim _{t \rightarrow 0} \frac{t e^{t}}{1-e^{t}} $$

Suppose that \(z=x^{3} y^{2},\) where both \(x\) and \(y\) are changing with time. At a certain instant when \(x=1\) and \(y=2, x\) is decreasing at the rate of 2 units \(/ \mathrm{s}\), and \(y\) is increasing at the rate of 3 units \(/ \mathrm{s}\). How fast is \(z\) changing at this instant? Is \(z\) increasing or decreasing?

Find \(d y / d x\) by implicit differentiation. \(\cos \left(x y^{2}\right)=y\)

Find \(d y / d x\) $$ y=\sqrt{\ln x} $$

Confirm that the stated formula is the local linear approximation of \(f\) at \(x_{0}=1,\) where \(\Delta x=x-1\) $$ f(x)=x^{4} ;(1+\Delta x)^{4} \approx 1+4 \Delta x $$