# Chapter 3 Review Exercises

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

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

Answer:

False.

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

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

Answer:

False

**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]

Answer:

[latex]\frac{1}{2\sqrt{x+4}}[/latex]

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

Find the derivatives of the following functions.

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

Answer:

[latex]9x^2+\frac{8}{x^3}[/latex]

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

**9. **[latex]f(x)=e^{\sin x}[/latex]

Answer:

[latex]e^{\sin x} \cos x[/latex]

**10. **[latex]f(x)=\ln(x+2)[/latex]

**11. **[latex]f(x)=x^2 \cos x+x \tan x[/latex]

Answer:

[latex]x \sec^2 x+2x \cos x+ \tan x-x^2 \sin x[/latex]

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

**13. **[latex]f(x)=\frac{x}{4} \sin^{-1} x[/latex]

Answer:

[latex]\frac{1}{4}(\frac{x}{\sqrt{1-x^2}}+ \sin^{-1} x)[/latex]

**14. **[latex]x^2 y=(y+2)+xy \sin (x)[/latex]

Find the following derivatives of various orders.

**15. **First derivative of [latex]y=x \ln x \cos x[/latex]

Answer:

[latex] \cos x (\ln x+1) -x \ln x \sin x[/latex]

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

**17. **Second derivative of [latex]y=4^x+x^2 \sin x[/latex]

Answer:

[latex]4^x(\ln 4)^2+2 \sin x+4x \cos x-x^2 \sin x[/latex]

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

**18. **[latex]y= \cos^{-1} x+x[/latex] at [latex]x=0[/latex]

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

Answer:

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

Draw the derivative for the following graphs.

**20.**

**21.**

Answer:

The following questions concern the water level in Ocean City, New Jersey, in January, which can be approximated by [latex]w(t)=1.9+2.9 \cos (\frac{\pi}{6}t)[/latex], where [latex]t[/latex] 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 [latex]w^{\prime}(3)[/latex]. What is the physical meaning of this value?

Answer:

[latex]w^{\prime}(3)=-\frac{2.9\pi}{6}[/latex]. At 3 a.m. the tide is decreasing at a rate of 1.514 ft/hr.

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 |

**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?

Answer:

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

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