Chapter 2: Limits and The Derivative

# Chapter 2 Review Exercises: The Derivative

*From Calculus, Volume 1, by Strang and Herman, OpenStax (Web), licensed under a **CC BY-NC-SA 4.0 License.*

True or false?

** 1. ** Every function has a derivative.

## Solution

False. Counterexample: [latex]f(x) = |x|[/latex]

** 2. ** A continuous function has a continuous derivative.

** 3. ** A continuous function has a derivative.

## Solution

False. Counterexample: [latex]f(x) = |x|[/latex]

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

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

** 5. ** [latex]f(x)=\sqrt{x+4}[/latex]

## Solution

[latex]f'(x) =\frac{1}{2\sqrt{x+4}}[/latex]

** 6. ** [latex]f(x)=\frac{3}{x}[/latex]

** 7. ** [latex]f(x)=\frac{9}{x^2}[/latex]

## Solution

[latex]f'(x) = -\frac{18}{x^3}[/latex]

** 8. ** [latex]f(x)=\sqrt{3x+1}[/latex]

Find the first derivative of the following functions.

** 9. ** [latex]f(x)=3x^3-\frac{4}{x^2} + \pi^6[/latex]

## Solution

[latex]f'(x) = 9x^2+\frac{8}{x^3}[/latex]

** 10. ** [latex]f(x)=(4-x^2)^3[/latex]

** 11. ** [latex]f(x)=\ln(3x^8+2)[/latex]

** 12. ** [latex]f(x)=(\sqrt{3x^2+2})(e^{3x} - \ln |x|)[/latex]

** 13. ** [latex]v(t)= t^2 - \frac{1}{\sqrt[5]{t}^6}[/latex]

## Solution

[latex]v'(t) = 2t + \frac{6}{5} t^{-\frac{11}{5}}[/latex]

** 14. ** [latex]f(x) = \sqrt{2x + \sqrt{3x}}[/latex]

**15. ** [latex]f(x)= \log_7 (2x +e^{-x})[/latex]

**16. ** [latex]f(x)= \ln |9x^3 - e^{2x} + 5^{x}|[/latex]

## Solution

[latex]f'(x) = \frac{27x^2 -2e^{2x} + 5^x \ln 5}{9x^3 - e^{2x} + 5^{x}}[/latex]

**17. ** [latex]g(x)= \log_2 [\ln (4x^6 -3x + 1][/latex]

**18. ** [latex]2xy = y^2 + 6[/latex]

## Solution

[latex]y' = \frac{y}{y-x}[/latex]

**19. ** [latex]f(x) = x^{2x}[/latex]

## Solution

[latex]f'(x) = 2x^{2x}(\ln x +1)[/latex]

Find the following derivatives of various orders.

**20. ** Third derivative of [latex]y=(3x+2)^2[/latex]

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

**21. ** [latex]y=x+e^x-\frac{1}{x}[/latex] at [latex]x=1[/latex]

## Solution

[latex]y=(2+e)x-2[/latex]

**22. ** [latex]x^2y^3 - x^3 = 3y^2-7[/latex] at the point [latex](2,1)[/latex]

**23. ** [latex]f(x) = (x+2)^{x^2}[/latex] at [latex]x=-1[/latex]

## Solution

[latex]y = x+ 2[/latex]

Draw the derivative for the following graphs.

**24.**

**25.**

## Solution

Answer the following questions.

**26. ** A particle moves on a vertical line so that its coordinate at time [latex]t[/latex] is [latex]y= t^3 -18t+7, t \ge 0[/latex].

a) Find the velocity function, [latex]v(t)[/latex].

b) Find the acceleration function, [latex]a(t)[/latex].

c) When is the particle moving upward? When is the particle moving downward?

**27. ** A ball is thrown vertically in the air with an upward velocity of 80 ft per second. Its height after [latex]t[/latex] seconds is [latex]h = 80t - 10t^2[/latex].

a) What is the maximum height reached by the ball? How many seconds does it take for the ball to reach its maximum height?

b) Find the velocity function, [latex]v(t)[/latex].

c) What is the velocity of the ball at [latex]t = 3[/latex]?

d) What is the velocity of the ball [latex]t = 5[/latex]?

e) How long does it take for the ball to reach the ground?

## Solution

a) 160 ft; 4 seconds

b) [latex]v(t) = 80 - 20t[/latex]

c) [latex]v(3) = 80 - 20 (3) = 20[/latex] ft/s (upward)

d) [latex]v(5) = 80 - 20 (5) = -20[/latex] ft/s (downward)

e) [latex]0 = 80t - 10t^2[/latex] so [latex]t=8[/latex] seconds

The following questions 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 |

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

**29. ** Estimate the derivative of the wind speed at hour 83. What is the physical meaning?

## Solution

-7.5. The wind speed is decreasing at a rate of 7.5 mph/hr