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3.10: Chapter 3 Review Exercises

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Mar 31, 2025
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- 3.9E: Exercises for Section 3.10
- 4: Applications of Derivatives

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Exercises 1 - 4

True or False? Justify the answer with a proof or a counterexample.

Every function has a derivative.

Answer: False

A continuous function has a continuous derivative.
A continuous function has a derivative.

Answer: False

If a function is differentiable, it is continuous.

Exercises 5 - 6

Use the limit definition of the derivative to exactly evaluate the derivative.

$f(x)=\sqrt{x+4}$

Answer: $f'(x) = \dfrac{1}{2\sqrt{x+4}}$

$f(x)=\dfrac{3}{x}$

Exercises 7 - 14

Find the derivatives of the given functions.

$f(x)=3x^3−\dfrac{4}{x^2}$

Answer: $f'(x) = 9x^2+\frac{8}{x^3}$

$f(x)=(4−x^2)^3$
$f(x)=e^{\sin x}$

Answer: $f'(x) = e^{\sin x}\cos x$

$f(x)=\ln(x+2)$
$f(x)=x^2\cos x+x\tan x$

Answer: $f'(x) = x\sec^2 x+2x\cos x+\tan x−x^2\sin x$

$f(x)=\sqrt{3x^2+2}$
$f(x)=\dfrac{x}{4}\sin^{−1}(x)$

Answer: $f'(x) = \frac{1}{4}\left(\frac{x}{\sqrt{1−x^2}}+\sin^{−1} x\right)$

$x^2y=(y+2)+xy\sin x$

Exercises 15 - 17

Find the indicated derivatives of various orders.

First derivative of $y=x(\ln x)\cos x$

Answer: $\dfrac{dy}{dx} = \cos x⋅(\ln x+1)−x(\ln x)\sin x$

Third derivative of $y=(3x+2)^2$
Second derivative of $y=4^x+x^2\sin x$

Answer: $\dfrac{d^2y}{dx^2} = 4^x(\ln 4)^2+2\sin x+4x\cos x−x^2\sin x$

Exercises 18 - 19

Find the equation of the tangent line to the following equations at the specified point.

$y=\cos^{−1}(x)+x$ at $x=0$
$y=x+e^x−\dfrac{1}{x}$ at $x=1$

Answer: $y = (2+e)x−2$

Exercises 20 - 21

Draw the derivative of the functions with the given graphs.

f goes from (-3,1) to (-2,-2) to (0,2) to (2,6) to (3,3) — Figure $\PageIndex{1}$ : A smooth curve represents a function that oscillates, showing both local minimum and maximum points. The function starts at about $(-3,0)$ , then dips to a local minimum around x=−2, rises to a near flat graph at $x=0$ , and then continues to rise further before turning around to curve downward at about $x=2.2$ .

f goes from (-1,4) to (0,0) to (2,4) — Figure $\PageIndex{2}$ : The given graph is continuous and has 2 section: For $x \le 0$ , the function is a line with slope of about $-4$ . For $x \ge 0$ , the graph is shaped like a standard parabola.

Answer: $f goes from (-2,-4) to (0,-4) open. f goes from (0,0) open to (2,4)$
Figure $\PageIndex{3}$ : A piecewise function is plotted with two distinct segments: For negative $x$ -values, the function is a horizontal line at $y=−4$ , extending to the left. There is an open circle at $(0,−4)$ . For positive $x$ -values, the function is a diagonal line with a positive slope, starting from the origin (0,0), though there is an open hole there, and increasing as $x$ increases.

Exercises 22 and 23

These exercises concern the water level in Ocean City, New Jersey, in January, which can be approximated by $w(t)=1.9+2.9\cos(\frac{π}{6}t),$ where $t$ is measured in hours after midnight, and the height is measured in feet.

Find and graph the derivative. What is the physical meaning?
Find $w′(3).$ What is the physical meaning of this value?

Answer: $w′(3)=−\frac{2.9π}{6}$ . At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

Exercises 24 and 25

Consider the wind speeds of Hurricane Katrina, which affected New Orleans, Louisiana, in August 2005. The data are displayed in a table.

Table $\PageIndex{1}$ : Wind Speeds
Hours after Midnight, August 26	Wind Speed (mph)
1	45
5	75
11	100
29	115
49	145
58	175
73	155
81	125
85	95
107	35

Wind Speeds of Hurricane KatrinaSource: news.nationalgeographic.com/n..._timeline.html.

Using the table, estimate the derivative of the wind speed at hour 39. What is the physical meaning?
Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

Answer: $−7.5.$ The wind speed is decreasing at a rate of 7.5 mph/hr