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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 | Español |
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. $h(x) = 2\cosec x - \sec x + 3x$ | 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. $\frac{d}{dx} \ln \|\sin x-2x\ e^{-x}\|$ |
Applications
39. Cost
The cost of Dig-It brand snow shovels is given by
$c(t) = 3.5\sin[2\pi(t-0.75)],$
40. Sales
Daily sales of Doggy brand cookies can be modeled by
$s(t) = 400\cos[2\pi(t-2/7)]$
cartons per week, where t is time in weeks since Monday morning. How fast are sales going up on Thursday morning?
41. Tides(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?
42. Tides$c(t) = 1.035^t[0.8\sin(2\pit) + 10.2],$
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
$s(t) = 4.5e^{-0.2t}\sin(2\pit),$
where t is time in years since January 1, 1997. How fast were sales rising or falling one year later?
Communication and Reasoning Exercises
45. Give two examples of a function $f(x)$ with the property that $f'(x) = -f(x).$ 46. Give three examples of a function $f(x)$ with the property that $f^{(4)}(x) = f(x).$ 47. By referring to the graph of $f(x) = \cos x,$ explain why $f'(x) = - \sin x,$ rather than $\sin x.$ 48. At what angle does the graph of $f(x) = \sin x$ depart from the origin?
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) |