3.4E: Exercises for Section 3.3

In exercises 1 - 12, find $f'(x)$ for each function.

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

2) $f(x)=5x^3−x+1$

Answer

$f'(x)=15x^2−1$

3) $f(x)=4x^2−7x$

4) $f(x)=8x^4+9x^2−1$

Answer

$f'(x) = 32x^3+18x$

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

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

Answer

$f'(x) = 270x^4+\dfrac{39}{(x+1)^2}$

7) $f(x)=(x+2)(2x^2−3)$

8) $f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)$

Answer

$f'(x) = \dfrac{−5}{x^2}$

9) $f(x)=\dfrac{x^3+2x^2−4}{3}$

10) $f(x)=\dfrac{4x^3−2x+1}{x^2}$

Answer

$f'(x) = \dfrac{4x^4+2x^2−2x}{x^4}$

11) $f(x)=\dfrac{x^2+4}{x^2−4}$

12) $f(x)=\dfrac{x+9}{x^2−7x+1}$

Answer

$f'(x) = \dfrac{−x^2−18x+64}{(x^2−7x+1)^2}$

In exercises 13 - 16, 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.

13) [T] $y=3x^2+4x+1$ at $(0,1)$

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

Answer

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

15) [T] $y=\dfrac{2x}{x−1}$ at $(−1,1)$

16) [T] $y=\dfrac{2}{x}−\dfrac{3}{x^2}$ at $(1,−1)$

Answer

$T(x)=4x−5$

In exercises 17 - 20, assume that $f(x)$ and $g(x)$ are both differentiable functions for all $x$. Find the derivative of each of the functions $h(x)$.

17) $h(x)=4f(x)+\dfrac{g(x)}{7}$

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

Answer

$h'(x)=3x^2f(x)+x^3f′(x)$

19) $h(x)=\dfrac{f(x)g(x)}{2}$

20) $h(x)=\dfrac{3f(x)}{g(x)+2}$

Answer

$h'(x)=\dfrac{3f′(x)(g(x)+2)−3f(x)g′(x)}{(g(x)+2)^2}$

For exercises 21 - 24, 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

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

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

Answer

$h'(2) =\frac{16}{9}$

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

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

Answer

$h'(4)$ is undefined.

In exercises 25 - 27, use the following figure to find the indicated derivatives, if they exist.

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

a) $h′(1)$,

b) $h′(3)$, and

c) $h′(4)$.

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

a) $h′(1),$

b) $h′(3)$, and

c) $h′(4).$

Answer

a. $h'(1) = 2$,
b. $h'(3)$ does not exist,
c. $h'(4) = 2.5$

27) Let $h(x)=\dfrac{f(x)}{g(x)}.$ Find

a) $h′(1),$

b) $h′(3)$, and

c) $h′(4).$

In exercises 28 - 31,

a) evaluate $f′(a)$, and

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

28) [T] $f(x)=2x^3+3x−x^2, \quad a=2$

Answer

a. 23
b. $y=23x−28$

29) [T] $f(x)=\dfrac{1}{x}−x^2, \quad a=1$

30) [T] $f(x)=x^2−x^{12}+3x+2, \quad a=0$

Answer

a. $3$
b. $y=3x+2$

31) [T] $f(x)=\dfrac{1}{x}−x^{2/3}, \quad a=−1$

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

Answer

$y=−7x−3$

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

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

Answer

$y=−5x+7$

35) Find the point on the graph of $f(x)=x^3$ such that the tangent line at that point has an $x$-intercept of $(6,0)$.

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

Answer

$y=−\frac{3}{2}x+\frac{15}{2}$

37) 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.

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

Answer

$y=−3x^2+9x−1$

39) 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.

40) [T] A herring swimming along a straight line has traveled $s(t)=\dfrac{t^2}{t^2+2}$ 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

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

a. Determine the initial flounder population.

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

42) [T] The concentration of antibiotic in the bloodstream $t$ hours after being injected is given by the function $C(t)=\dfrac{2t^2+t}{t^3+50}$, 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. $\dfrac{−2t^4−2t^3+200t+50}{(t^3+50)^2}$
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.

43) A book publisher has a cost function given by $C(x)=\dfrac{x^3+2x+3}{x^2}$, 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.

44) [T] 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 $F=\dfrac{Gm_1m_2}{d^2}$, 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 $G=6.67×10^{−11} \text{Nm}^2/\text{kg}^2$, on two bodies 10 meters apart, each with a mass of 1000 kilograms.

Answer

a. $F'(d)=\dfrac{−2Gm_1m_2}{d_3}$
b. $−1.33×10^{−7}$ N/m