Stefan Waner and Steven R. Costenoble

for

Section 3: Derivatives of Trigonometric Functions

2. The Six Trigonometric Functions |
3. Derivatives of Trigonometric Functions |
4. Integrals of Trigonometric Functions |
Trigonometric Functions Main Page |
"RealWorld" Page |
Everything for Calculus |
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Answers to Odd-Numbered Exercises |

Find the derivatives of the following functions

1. $f(x) = \sin x - \cos x$ | 2. $f(x) = \tan x -\sin x$ | 3. $g(x) = (\sin x)(\tan x)$ |

4. $g(x) = (\cos x)(cotan x)$ | 5. $h(x) = 2cosec x - \sec x + 3x$ | 6. $h(x) = 2\sec x + 3\tan x + 3x$ |

7. $r(x) = x \cos x + x^2 + 1$ | 8. $r(x) = 2x \sin x - x^2$ | 9. $s(x) = (x^2-x+1)\tan x$ |

10. $s(x) = \frac{\tan x}{x^2-1}$ |
11. $t(x) = \frac{cotan x}{1 + \sec x}$ | 12. $t(x) = (1+\sec x)(1-\cos x)$ |

13. $k(x) = \cos^2x$ | 14. $k(x) = \tan^2x$ | 15. $j(x) = \sec^2x$ |

16. $j(x) = cosec^2x$ | 17. $u(x) = \cos(x^2-x)$ | 18. $u(x) = \sin(3x^2+x-1)$ |

19. $v(x) = \sec(x^{2.2}+1.2x-1)$ | 20. $v(x) = \tan(x^{2.2}+1.2x-1)$ | 21. $w(x) = (\sec x)(\tan(x^2-1))$ |

22. $w(x) = (\cos x)(\sec(x^2-1))$ | 23. $y(x) = \cos(e^x) + e^x \cos x$ | 24. $y(x) = \sec(e^x)$ |

25. $z(x) = \ln \|\sec x + \tan x\|$ | 26. $z(x) = \ln \|cosec x + cotan x\|$ |

Obtain the following derivatives

27. |
$\frac{d}{dx} \tan x = \sec^2x$ | 28. |
$\frac{d}{dx} cotan x = - cosec^2x$ |

29. |
$\frac{d}{dx} \sec x = \sec^x \tan x$ | 30. |
$\frac{d}{dx} \ln \|\cos x\| = - \tan x$ |

Calculate the following

31. |
$\frac{d}{dx}[e^{-2x}\sin(3x)]$ | 32. |
$\frac{d}{dx}[e^{5x} \sin(-4x)]$ | 33. |
$\frac{d}{dx}[\sin(3x)]^{0.5}$ |

34. |
$\frac{d}{dx} \cos \left(\frac{x^2}{x - 1}\right)$ | 35. |
$\frac{d}{dx} \sec \left(\frac{x^3}{x^2 - 1} \right)$ | 36. |
$\frac{d}{dx} \left(\frac{\tan x}{2 + e^x} \right)^2$ |

37. |
$\frac{d}{dx}([\ln \|x\|][cotan(2x-1)])$ | 38. |
$\frac{d}{dx} \ln \|\sin x-2xe^{-x}\|$ |

**Applications**

**39. Cost **

The cost of Dig-It brand snow shovels is given by

$c(t) = 3.5\sin[2π(t-0.75)],$

where t is time in years since January 1, 1997. How fast, in dollars per week, is the cost increasing each September 1?

**40. Sales **

Daily sales of Doggy brand cookies can be modeled by

$s(t) = 400\cos[2π(t-2/7)]$

cartons per week, where t is time in weeks since Monday morning. How fast are sales going up on Thursday morning?

The depth of water at my favorite surfing spot varies from $5$ft to $15$ft, depending on the time. Last Sunday, high tide occurred at 5:00 am. and the next high tide occurred at 6:30 pm.

**(a)** Obtain a cosine model describing the depth of water as a function of time t in hours since 5:00 am. on Sunday morning.

**(b)** How fast was the tide rising (or falling) at noon on Sunday?

Repeat Exercise 41 using data from the depth of water at my other favorite surfing spot, where the tide last Sunday varied from a low of $6$ft at 4:00 am to a high of $10$ft. at noon.

Taking a $3.5%$ rate of inflation into account, the cost of Dig-In brand snow shovels is given by

$c(t) = 1.035^t[0.8\sin(2πt) + 10.2],$

How fast, in dollars per week, is the cost of Dig In shovels increasing on January 1, 1998?

**44. Deflation**

Sales (in bottles per day) of my exclusive 1997 vintage Chateau Petit Mont Blanc follow the function

$s(t) = 4.5e^{-0.2t}\sin(2πt),$

where t is time in years since January 1, 1997. How fast were sales rising or falling one year later?

**Communication and Reasoning Exercises**

2. The Six Trigonometric Functions |
3. Derivatives of Trigonometric Functions |
4. Integrals of Trigonometric Functions |
Trigonometric Functions Main Page |
"RealWorld" Page |
Everything for Calculus |

Answers to Odd-Numbered Exercises |

Stefan Waner (matszw@hofstra.edu) | Steven R. Costenoble (matsrc@hofstra.edu) |