Chapter 4

For the following exercises, draw the situations and solve the related-rate problems. You are walking to a bus stop at a right-angle corner. You move north at a rate of 2 m/sec and are 20 m south of the intersection. The bus travels west at a rate of 10 m/sec away from the intersection – you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are 20 m south of the intersection and the bus is 10 m west of the intersection?

Consider a pool shaped like the bottom half of a sphere, that is being filled at a rate of 25 \(\mathrm{ft}^{3} / \mathrm{min}\). The radius of the pool is \(10 \mathrm{ft}\). You are walking to a bus stop at a right-angle corner. You move north at a rate of \(2 \mathrm{~m} / \mathrm{sec}\) and are \(20 \mathrm{~m}\) south of the intersection. The bus travels west at a rate of \(10 \mathrm{~m} /\) sec away from the intersection - you have missed the bus! What is the rate at which the angle between you and the bus is changing when you are \(20 \mathrm{~m}\) south of the intersection and the bus is \(10 \mathrm{~m}\) west of the intersection?

What is the linear approximation for any generic linear function \(y=m x+b ?\)

Explain why the linear approximation becomes less accurate as you increase the distance between \(x\) and \(a\). Use a graph to prove your argument.

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=x+x^{4}, a=0 $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=x+x^{4}, a=0\)

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=\frac{1}{x}, a=2 $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=\frac{1}{x}, a=2\)

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=\tan x, a=\frac{\pi}{4} $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=\tan x, a=\frac{\pi}{4}\)

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=\sin x, a=\frac{\pi}{2} $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=\sin x, a=\frac{\pi}{2}\)

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=x \sin x, a=2 \pi $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=x \sin x, a=2 \pi\)

For the following exercises, find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. $$ f(x)=\sin ^{2} x, a=0 $$

Find the linear approximation \(L(x)\) to \(y=f(x)\) near \(x=a\) for the function. \(f(x)=\sin ^{2} x, a=0\)

For the following exercises, compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\) , and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\) . Check your answer using a calculator. $$ (2.001)^{6} $$

Compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\), and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\). Check your answer using a calculator. [T] \((2.001)^{6}\)

Compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\), and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\). Check your answer using a calculator. [T] \(\sin (0.02)\)

For the following exercises, compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\) , and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\) . Check your answer using a calculator. $$ \cos (0.03) $$

Compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\), and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\). Check your answer using a calculator. \([\mathrm{T}] \cos (0.03)\)

Compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\), and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\). Check your answer using a calculator. \([\mathrm{T}](15.99)^{1 / 4}\)

For the following exercises, compute the values given within 0.01 by deciding on the appropriate \(f(x)\) and \(a\) , and evaluating \(L(x)=f(a)+f^{\prime}(a)(x-a)\) . Check your answer using a calculator. $$ \sin (3.14) $$

For the following exercises, determine the appropriate \(f(x)\) and \(a, \quad\) and evaluate \(L(x)=f(a)+f^{\prime}(a)(x-a)\) Calculate the numerical error in the linear approximations that follow. $$ (\sin (0.01))^{2} $$

For the following exercises, determine the appropriate \(f(x)\) and \(a, \quad\) and evaluate \(L(x)=f(a)+f^{\prime}(a)(x-a)\) Calculate the numerical error in the linear approximations that follow. $$ \left(1+\frac{1}{10}\right)^{10} $$

For the following exercises, determine the appropriate \(f(x)\) and \(a, \quad\) and evaluate \(L(x)=f(a)+f^{\prime}(a)(x-a)\) Calculate the numerical error in the linear approximations that follow. $$ \sqrt{8.99} $$

For the following exercises, find the differential of the function. $$ y=3 x^{4}+x^{2}-2 x+1 $$

For the following exercises, find the differential of the function. $$ y=x \cos x $$

For the following exercises, find the differential of the function. $$ y=\sqrt{1+x} $$

Determine the appropriate \(f(x)\) and \(a,\) and evaluate \(L(x)=f(a)+f^{\prime}(a)(x-a) .\) Calculate the numerical error in the linear approximations that follow. \(y=\sqrt{1+x}\)

For the following exercises, find the differential of the function. $$ y=\frac{x^{2}+2}{x-1} $$

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=3 x^{2}-x+6, \quad x=2, \quad d x=0.1 $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=3 x^{2}-x+6, \quad x=2, \quad d x=0.1\)

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=\frac{1}{x+1}, \quad x=1, \quad d x=0.25 $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=\frac{1}{x+1}, \quad x=1, \quad d x=0.25\)

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=\tan x, \quad x=0, \quad d x=\frac{\pi}{10} $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=\tan x, \quad x=0, \quad d x=\frac{\pi}{10}\)

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=\frac{3 x^{2}+2}{\sqrt{x+1}}, \quad x=0, \quad d x=0.1 $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=\frac{3 x^{2}+2}{\sqrt{x+1}}, \quad x=0, \quad d x=0.1\)

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=\frac{\sin (2 x)}{x}, \quad x=\pi, \quad d x=0.25 $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=\frac{\sin (2 x)}{x}, \quad x=\pi, \quad d x=0.25\)

For the following exercises, find the differential and evaluate for the given \(x\) and \(d x .\) $$ y=x^{3}+2 x+\frac{1}{x}, \quad x=1, \quad d x=0.05 $$

Find the differential and evaluate for the given \(x\) and \(d x\). \(y=x^{3}+2 x+\frac{1}{x}, \quad x=1, \quad d x=0.05\)

For the following exercises, find the change in volume \(d V\) or in surface area \(d A\) $$ d V \text { if the sides of a cube change from } 10 \text { to } 10.1 $$

Find the change in volume \(d V\) or in surface area \(d A\). \(d V\) if the sides of a cube change from 10 to 10.1 .

For the following exercises, find the change in volume \(d V\) or in surface area \(d A\) $$ d A \text { if the sides of a cube change from } x \text { to } x+d x $$

Find the change in volume \(d V\) or in surface area \(d A\). \(d A\) if the sides of a cube change from \(x\) to \(x+d x\).

For the following exercises, find the change in volume \(d V\) or in surface area \(d A\) $$ d A \text { if the radius of a sphere changes from } r \text { by } d r $$

\(d V\) if a circular cylinder with \(r=2\) changes height from 3 \(\mathrm{cm}\) to 3.05 \(\mathrm{cm} .\)

Find the change in volume \(d V\) or in surface area \(d A\). \(d V\) if a circular cylinder with \(r=2\) changes height from \(3 \mathrm{~cm}\) to \(3.05 \mathrm{~cm}\).