# 3R: Chapter 3 Review Exercises

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## Chapter Review Exercises

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

1) Every function has a derivative.

False

2) A continuous function has a continuous derivative.

3) A continuous function has a derivative.

False

4) If a function is differentiable, it is continuous.

In exercises 5 and 6, use the limit definition of the derivative to exactly evaluate the derivative.

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

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

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

In exercises 7 - 15, find the derivatives of the given functions.

7) $$f(x)=3x^3−\dfrac{4}{x^2}$$

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

9) $$f(x)=(4−x^2)^3$$

10) $$f(x)=e^{\sin x}$$

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

11) $$f(x)=\ln(x+2)$$

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

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

13) $$f(x)=\sqrt{3x^2+2}$$

14) $$f(x)=\dfrac{x}{4}\sin^{−1}(x)$$

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

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

In exercises 16 - 18, find the indicated derivatives of various orders.

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

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

17) Third derivative of $$y=(3x+2)^2$$

18) Second derivative of $$y=4^x+x^2\sin x$$

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

In exercises 19 and 20, find the equation of the tangent line to the following equations at the specified point.

19) $$y=\cos^{−1}(x)+x$$ at $$x=0$$

20) $$y=x+e^x−\dfrac{1}{x}$$ at $$x=1$$

$$y = (2+e)x−2$$

In exercises 21 and 22, draw the derivative of the functions with the given graphs.

21)

22)

Questions 22 and 23 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.

22) Find and graph the derivative. What is the physical meaning?

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

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

Questions 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.

 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.

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

25) Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

$$−7.5.$$ The wind speed is decreasing at a rate of 7.5 mph/hr
The text passage is a set of review exercises for a calculus chapter, containing multiple questions and their answers, primarily focused on derivatives and their applications.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
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system
# 3R: Chapter 3 Review Exercises

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## Chapter Review Exercises

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

1) Every function has a derivative.

False

2) A continuous function has a continuous derivative.

3) A continuous function has a derivative.

False

4) If a function is differentiable, it is continuous.

In exercises 5 and 6, use the limit definition of the derivative to exactly evaluate the derivative.

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

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

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

In exercises 7 - 15, find the derivatives of the given functions.

7) $$f(x)=3x^3−\dfrac{4}{x^2}$$

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

9) $$f(x)=(4−x^2)^3$$

10) $$f(x)=e^{\sin x}$$

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

11) $$f(x)=\ln(x+2)$$

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

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

13) $$f(x)=\sqrt{3x^2+2}$$

14) $$f(x)=\dfrac{x}{4}\sin^{−1}(x)$$

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

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

In exercises 16 - 18, find the indicated derivatives of various orders.

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

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

17) Third derivative of $$y=(3x+2)^2$$

18) Second derivative of $$y=4^x+x^2\sin x$$

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

In exercises 19 and 20, find the equation of the tangent line to the following equations at the specified point.

19) $$y=\cos^{−1}(x)+x$$ at $$x=0$$

20) $$y=x+e^x−\dfrac{1}{x}$$ at $$x=1$$

$$y = (2+e)x−2$$

In exercises 21 and 22, draw the derivative of the functions with the given graphs.

21)

22)

Questions 22 and 23 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.

22) Find and graph the derivative. What is the physical meaning?

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

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

Questions 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.

 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.

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

25) Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

$$−7.5.$$ The wind speed is decreasing at a rate of 7.5 mph/hr
user
The text passage is a set of review exercises for a calculus chapter, containing multiple questions and their answers, primarily focused on derivatives and their applications.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS