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# solnshw5 - 274 Chapter 8 Trigonometric‘ Functions...

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Unformatted text preview: 274 Chapter 8 Trigonometric‘ Functions Relative maxima: (4.01. —38.5). (0.70. w5.8) Relative minimum: (1.95. —7.8) ) (4.01. —38.5) 84. f(x) = sin ./32 Relative maximum: (2.47. l) 88. False. the maximum value of y is about 3.6. 82. f(.r) = lnx sinx Relative minima: (0.35. «0.36). (4.84. - 1.56) Relative maximum: (2.13. 0.64) 2 (11.35.41.361 (2.13.1101) 86. False.f’(x) = 2 sin(2x) cos(2x)(2) Section 8.5 Integrals of Trigonometric Functions 3 2. [(tz—sint)dt=%+cost+C 6. f(secytany- seczy)dy= secy ~ tan_v + C l 10. [Jsinxzdr=%f2xsinx2dx= ~5cosx3 + C l 14. fechxcothdx=§J2csc22xcot2xdx = —%csc2.x+ C 18. flan 5x (1'); = %J 5 tan Sxdx = —%ln|cos 5x) + C sin x 1 sin x 22. , dx = - dx cos- x cos x cos x = [secxtanxdx = secx + C 3 4. [(02+sec36)d0=%—+tan0+C 8. fcos6xdx =éf6cos6xdx = ésin6x + C M _ 1 2i __ 5 12.fcsc 2dx—2f2csc 2dx— 2c0t2+C 16. f ./c0tx csczxdx = —f (cot x)‘/3(—c303 x) dr = —%(cotx)3/3 + C -_v 2. l a 20.jsec2dr 2J2sec2dx =21n r seci + tan'— 2 2 +C 24. f—C‘Ldzz lnll + sintI + C 1+ stnt Section 8.5 Integrals ofTrigonometriC Functions W 26- [Siigdx = 21W“ ﬁ)<2\l/E)dx : -2 cos J; + C 30. It)" tan e“ (Ix = —f (~e“) tan e“ dx = lnlcos e‘-"| + C l—cosé) 8. h 0: «f + 2 f0~sin0d ln|0 Sm 0| C 32. few“ “ sec x tan x dx = e‘“ " + C 34. [(csc 20 — cot 20)2 d0 = [(csc2 20 — 2 csc 20 cot 20 + cot: 20) d0 = f(2csc320— 2csc20cot20—1)d0 = -cot20+csc20— 0+ Czcsc20~cot20— 0+C 36. Using integration by parts, we let u = x and (Iv = sin x dx. Then du = dx and v = —cos x. fxsinxdx= -xcosx + Icosxdx = ".xcosx + sinx + C I 17/2 17/2 40. f sin 2x dx = -; cos 2x] 1) (1 .. 1+1 2 2 = I 17/8 11/8 44. I sin 2.x cos 2x dx = %f sin 2r(2 cos 2x) (1x 0 0 = [l _ sin2 211”” = l 2 2 0 8 17/4 48. Area = f tan xdx 1) 11/4 = —ln|c0s 1|] 411;) ﬁ In J2 z 0.3466 square unit 0 ll 52. From Exercise 36, we have 11 Area =f x sinx dx 0 1? = [‘xcosx + sinx] 0 = 11 square units 38. Using integration by pans. we let u = 0 and dv = sec 0tan 0:10. Then du = d0 and v = sec0. J0 sec 0tan 0:10 = 0360 0 - [sec 0:10 = 0sec0— lnlsec0+ tan 0| + C 71/2 x: 7/2 42. I (x + cos x) dx = [-— + sin x] o 2 0 113 = —— + 8 1 3 + = 77 8 = 2.2337 8 ”/4 71/4 46. f sec x tan x dx = sec x] 2 J2 — l 0 0 m 0.4142 50. Area = I (2 sin x + sin 2x) dx 0 = [— 2cosx — 1c0521] 2 0 = 4 square units 511/6 54. V = 11] csczxdx 17/6 511/6 = _ ”NCO! .1] 11/6 __ /' — Av 371 cubic units 276 Chapter 8 Trigonomerric' Functions 56. Trapezoidal Rule: §l[f(0) + 1G) + 24%) + 2f(3 E) +1111] m 0.8958 Simpson's Rule: é[f(0) + 4f<i> + 2/6) + 4/62) +f(1)] :2 0.9045 Calculator: 0.9045 ” 262.5 1 b 58. (74.50 + 43.75 sin 171M = [74.501 — cos 1] b—aa 6 b-a 7r 6., F) 3 (a) [74.501 — 26-5 cos 33] 2 102.352 thousand units = 102.352 units 3 ‘ 0 7T 6 () .5 “ (b) *1 [74.501 - 262 cos 7—71] -~ 102.352 thousand units 2 102,352 units 6 7- 3 7T 6 0 .5 ‘3 . . (c) [74.501 — 262 cos 31] = 74.5 thousand units = 74,500 units 12 — 0 7T 6 1) 60. P = 1.07 sin(0.59t + 3.94) + 1.52 [’7 7 , ‘ . 3 (a) P (t) = 0.6313 cos(0.59t + 3.94) = 0 (b) Average = i P(t1dr ~ E2— 2 1.41 inches/month 12 7, 12 cos(0.59t + 3.94) = 0 ll 71' (C) Total = I P(t) dt 17- 16.93 inches 0.59! + 3.94 = j)- + n7r “ t 2 1.309, 6.634. 11.959 Maximum: 2.59 when 1 ~ 6.634 (June) Minimum: 0.45 when I '~* 1.309 and 11959 (January and December) 62. The volume is given by the following. 3 ' 3 3 . V: 0.9 sinﬂdt = ——(0.9)cos7T—t] = —27(-1 — 1) -~ 1.7189 liters 0 3 7T 3 0 7T l 200 277“ _ 60) ‘65 277“ _ 60) 200 4. . + ' ——— 2 5 — ‘——‘ *— x 6 200 _ of) 100 000[1 sm 365 Jdt 00[t 27 cos 365 1) 136.4951b 7/2 2 66. f cos xdx‘ 2 1.0057 68. I (4 + x + sin 77x) dx = 10 0 0 70. True d . . E(-cos 2x- + C) = 2 sm 2x = 43m .x' cos x Section 8.6 L’Hﬁpital’s Rule 00 2. Yes; 4_ N0.%—>0 6. Yes,— ...
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solnshw5 - 274 Chapter 8 Trigonometric‘ Functions...

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