3.2E: Exercises for Section 3.2

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In exercises 1 - 8, 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) $f(x)=x^2\left(\dfrac{2}{x^2}+\dfrac{5}{x^3}\right)$

Answer: $f'(x) = \dfrac{−5}{x^2}$

8) $f(x)=\dfrac{x^3+2x^2−4}{3}$

In exercises 9 - 12, 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.

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

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

Answer

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

$This graph has a straight line with y intercept near 0 and slope slightly less than 3.$

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

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

Answer

$T(x)=4x−5$

$The graph y is a two crescents with the crescent in the third quadrant sloping gently from (−3, −1) to (−1, −5) and the other crescent sloping more sharply from (0.8, −5) to (3, 0.2). The straight line T(x) is drawn through (0, −5) with slope 4.$

In exercise 13, assume that $f(x)$ and $g(x)$ are both differentiable functions for all $x$. Find the derivative of each of the functions $h(x)$.

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

In exercises 14-17,

a) evaluate $f′(a)$, and

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

14) [T] $f(x)=2x^3+3x−x^2, \quad a=2$

Answer

a. 23
b. $y=23x−28$

$The graph is a slightly deformed cubic function passing through the origin. The tangent line is drawn through (0, −28) with slope 23.$

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

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

Answer

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

$The graph starts in the third quadrant, increases quickly and passes through the x axis near −0.9, then increases at a lower rate, passes through (0, 2), increases to (1, 5), and then decreases quickly and passes through the x axis near 1.2.$

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

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

Answer: $y=−7x−3$

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

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

21) 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: $y=−\frac{3}{2}x+\frac{15}{2}$

22) 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.

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

Answer: $y=−3x^2+9x−1$

24) 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.

25) 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.

26) 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.

27) [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

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.

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