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Chapter 3 25

# Chapter 3 25 - SECTION 3.4 DERIVATIVES 0F TRIGONOMETRIC...

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Unformatted text preview: SECTION 3.4 DERIVATIVES 0F TRIGONOMETRIC FUNCTIONS 175 25. (a) y = xcosa: => 3/ = a:(— sinm) + cosm(1) = cosz — xsinw. (b) So the slope of the tangent at the point (mg—71') is —1 gm 5 cos7r — 7rsin7r : —1 — 7r(0) = *1. and an equation is y+7r:—(m—7r)ory:—:c. 26. (a)y:sec:v—2cosm => y’zsecartanm—I-2sina: : the slope of the tangent line at (I 1) is sec—tan—'+25in—=2 ﬁ+2~— 23:3\/§andanequationis y~1:3\/§(\$—§)Ory=3x/§m+l*7r\/§. 27. (a) f(m) : 23v + cotm => f’(a:) : 2 — csc2 a: (b) 6 Notice that f’(;z:) : 0 when f has a horizontal tangent. f’ is positive when f is increasing and f’ is negative when f is decreasing. Also, f’(m) is large negative when the graph 74 ¥ - 1 ﬂ . 1 _1/2 _ sins; 28. (a) f(a:) — ﬁsmm :> f (at) ~ ﬁcosarl— (s1n:c)(§w )— ﬁcosm—I— 2f :c (b) 3 Notice that J“ (an) = 0 when f has a horizontal tangent. f' is positive when f is increasing and f’ is negative when f r ‘ is decreasing. 0 A v V 277 —3 29. f(x)2 : a: + 251nm has a horizontal tangent when f’(m) — 0 4:) 1 ' 2cos:c — 0 < cosm — —% :c— g 1+ 27m or4 "+ 27m where n is an integer. Note that4— 3and 24* are i— 7‘ units from 7r. This allows us to write the solutions in 3the more compact equivalent form (271 + 1)7r :: 3 , 72 an integer. cosa: 30. : “ :> y 2 + sinz 3,1" (2+sinx)(—sinx)—cosxcosm ~2sinm—sin2w—C052w —2sinar—1 ‘0 h (2—I—sinx)2 (2«I—sin:c)2 _ (2—I~sin\$)2 _ W en —2\$inx— 1 :0 4:) sinaz: —% 4:) :v : “—7” +27morw: 7?" +27rn,naninteger. Soy: % or H 117r 77r 1 y— —7 and the points on the curve with horizontal tangents are: (6 + 27m «5) ( 6 + 27m7 V5) n an integer. \ ...
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