3.2E: Exercises

Exercise $\PageIndex{1}$

For the following exercises, assume that $f(x)$ and $g(x)$ are both differentiable functions for all $x$. Find the derivative of each of the functions $h(x)$.

1) $h(x)=4f(x)+\frac{g(x)}{7}$

2) $h(x)=x^3f(x)$

3) $h(x)=\frac{f(x)g(x)}{2}$

4) $h(x)=\frac{3f(x)}{g(x)+2}$

Answers to even numbered questions

2. $h′(x)=3x^2f(x)+x^3f′(x)$

4. $h′(x)=\frac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}$

Exercise $\PageIndex{2}$

For the following exercises, assume that f(x) and g(x) are both differentiable functions with values as given in the following table. Use the following table to calculate the following derivatives.

$x$	1	2	3	4
$f(x)$	3	5	−2	0
$g(x)$	2	3	−4	6
$f′(x)$	−1	7	8	−3
$g′(x)$	4	1	2	9

1) Find $h′(1)$ if $h(x)=x f(x)+4g(x)$.

2) Find $h′(2)$ if $h(x)=\frac{f(x)}{g(x)}$.

3) Find $h′(3)$ if $h(x)=2x+f(x)g(x)$.

4) Find $h′(4)$ if $h(x)=\frac{1}{x}+\frac{g(x)}{f(x)}$.

Answers to even numbered questions

2. $\frac{16}{9}$

4. Undefined

Exercise $\PageIndex{3}$

For the following exercises, use the following figure to find the indicated derivatives, if they exist.

1) Let $h(x)=f(x)+g(x)$. Find

a. $h′(1)$,

b. $h′(3)$, and

c. $h′(4)$.

2) Let $h(x)=f(x)g(x).$ Find

a. $h′(1),$

b. $h′(3)$, and

c. $h′(4).$

3) Let $h(x)=\frac{f(x)}{g(x)}.$ Find

a. $h′(1),$

b. $h′(3)$, and

c. $h′(4).$

Solution to even numbered question:

Let $h(x)=f(x)g(x).$  Notice that from the graph,

a) $f(1)=3,f(3)=1,$ and $f(4)=2$

b)$g(1)=3,g(3)=2.5,$ and $g(4)=2.5$

c) The rate of change of $f(x)$ (slope),$f′(x)$, is $-1$ when $x=1$,  is $DNE$ when $x=3$ and is $1$ when $x=4$.

d) Also, the rate of change of $g(x)$ (slope) is $1$ when $x=1$,  is $0$ when $x=3$ and is $0$ when $x=4$.

Now we can solve the problem:

a. $h′(1)=f′(1)g(1)+f(1)g′(1)=(-1)(1)+(3)(1)=2,$

b. $h′(3)=f′(3)g(3)+f(3)g′(3)=DNE$, and

c. $h′(4)=f′(4)g(4)+f(4)g′(4)=(1)(2.5)+(2)(0)=2.5.$

Answers to even numbered questions

2a. $2$

b. does not exist

c. $2.5$

Exercise $\PageIndex{4}$

For the following exercises,

a. evaluate $f′(a)$, and

b. graph the function $f(x)$ and the tangent line at $x=a.$

1) $f(x)=2x^3+3x−x^2,a=2$

2) $f(x)=\frac{1}{x}−x^2,a=1$

3) $f(x)=x^2−x^{12}+3x+2,a=0$

4) $f(x)=\frac{1}{x}−x^{2/3},a=−1$

Answers to odd numbered questions

1a. 23

b. $y=23x−28$

3a. 3

b. $y=3x+2$

Exercise $\PageIndex{5}$

Find the derivative of the following functions and simplify your answer:

1) $f(x)=x^7+10$

2) $f(x)=5x^3−x+1$

3) $f(x)=4x^2−7x$

4) $f(x)=8x^4+9x^2−1$

5) $f(x)=x^4+2x$

6) $f(x)=3x(18x^4+\frac{13}{x+1})$

7) $f(x)=(x+2)(2x^2−3)$

8) $f(x)=x^2(\frac{2}{x^2}+\frac{5}{x^3})$

9) $f(x)=\frac{x^3+2x^2−4}{3}$

10) $f(x)=\frac{4x^3−2x+1}{x^2}$

11) $f(x)=\frac{x^2+4}{x^2−4}$

12) $f(x)=\frac{x+9}{x^2−7x+1}$

13) $\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}$.

14) $\displaystyle{y(x)=\frac{3x^2+5}{2x^2+x-3}}$

Answers to even numbered questions

2. $f′(x)=15x^2−1$

4. $f′(x)=32x^3+18x$

6.$f′(x)=270x^4+\frac{39}{(x+1)^2}$

8. $f′(x)=\frac{−5}{x^2}$

10. $f′(x)=\frac{4x^4+2x^2−2x}{x^4}$

12. $f′(x)=\frac{−x^2−18x+64}{(x^2−7x+1)^2}$

Exercise $\PageIndex{6}$

For the following exercises, find the equation of the tangent line $T(x)$ to the graph of the given function at the indicated point. Use a graphing calculator to graph the function and the tangent line.

1) $y=3x^2+4x+1$ at $(0,1)$

2) $y=2\sqrt{x}+1$ at $(4,5)$

3) $y=\frac{2x}{x−1}$ at $(−1,1)$

4) $y=\frac{2}{x}−\frac{3}{x^2}$ at $(1,−1)$

Answers to even numbered questions

2. $T(x)=\frac{1}{2}x+3$,

4. $T(x)=4x−5$

Exercise $\PageIndex{7}$

1) Find the equation of the tangent line to the graph of $f(x)=2x^3+4x^2−5x−3$ at $x=−1.$

2) Find the equation of the tangent line to the graph of $f(x)=x^2+\frac{4}{x}−10$ at $x=8$.

3) Find the equation of the tangent line to the graph of $f(x)=(3x−x^2)(3−x−x^2)$ at $x=1$.

4) Find the point on the graph of $f(x)=x^3$ such that the tangent line at that point has an x intercept of 6.

5) Find the equation of the line passing through the point $P(3,3)$ and tangent to the graph of $f(x)=\frac{6}{x−1}$.

Answers to odd numbered questions

1. $y=−7x−3$

3. $y=−5x+7$

5. $y=−\frac{3}{2}x+\frac{15}{2}$

Exercise $\PageIndex{8}$

Determine all points on the graph of $f(x)=x^3+x^2−x−1$ for which the slope of the tangent line is

a. horizontal

b. −1.

Answer

Under Construction

 

Exercise $\PageIndex{9}$

Find a quadratic polynomial such that $f(1)=5,f′(1)=3$ and $f''(1)=−6.$

Answer

$y=−3x^2+9x−1$

Exercise $\PageIndex{10}$

A car driving along a freeway with traffic has traveled $s(t)=t^3−6t^2+9t$ meters in $t$ seconds.

a. Determine the time in seconds when the velocity of the car is 0.

b. Determine the acceleration of the car when the velocity is 0.

Answer

Under Construction

Exercise $\PageIndex{11}$

A herring swimming along a straight line has traveled $s(t)=\frac{t^2}{t^2+2}$ feet in $t$ seconds.

Determine the velocity of the herring when it has traveled 3 seconds.

Answer

$\frac{12}{121}$ or 0.0992 ft/s

Exercise $\PageIndex{12}$

The population in millions of arctic flounder in the Atlantic Ocean is modeled by the function $P(t)=\frac{8t+3}{0.2t^2+1}$, where $t$ is measured in years.

a. Determine the initial flounder population.

b. Determine $P′(10)$ and briefly interpret the result.

Answer

 

Under Construction

 

Exercise $\PageIndex{12}$

The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C(t)=\frac{2t^2+t}{t^3+50}$, where $C$ is measured in milligrams per liter of blood.

a. Find the rate of change of $C(t).$

b. Determine the rate of change for $t=8,12,24$,and $36$.

c. Briefly describe what seems to be occurring as the number of hours increases.

Answer

$a. \frac{−2t^4−2t^3+200t+50}{(t^3+50)^2}$ $b. −0.02395$ mg/L-hr, −0.01344 mg/L-hr, −0.003566 mg/L-hr, −0.001579 mg/L-hr c. The rate at which the concentration of drug in the bloodstream decreases is slowing to 0 as time increases.mplate active on the page.

Exercise $\PageIndex{13}$

A book publisher has a cost function given by $C(x)=\frac{x^3+2x+3}{x^2}$, where x is the number of copies of a book in thousands and C is the cost per book, measured in dollars. Evaluate $C′(2)$ and explain its meaning.

Answer

Under Construction

Exercise $\PageIndex{14}$

According to Newton’s law of universal gravitation, the force $F$ between two bodies of constant mass $m_1$ and $m_2$ is given by the formula $F=\frac{Gm_1m_2}{d^2}$, where $G$ is the gravitational constant and $d$ is the distance between the bodies.

a. Suppose that $G,m_1,$ and $m_2$ are constants. Find the rate of change of force $F$ with respect to distance $d$.

b. Find the rate of change of force $F$ with gravitational constant $G=6.67×10^{−11} Nm^2/kg^2$, on two bodies 10 meters apart, each with a mass of 1000 kilograms.

Answer

a. $F'(d)=\frac{−2Gm_1m_2}{d_3}$ '

b. $−1.33×10^{−7} N/m$

Contributors and Attributions

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.