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File0036 - Section 3.4 Derivatives of Trigonometric...

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Unformatted text preview: Section 3.4 Derivatives of Trigonometric Functions 12] 28. 2tanx 2> y’2sec2x 2> 510 eoftan entatx:« y P g wt: is sec2 (- g) 2 4; slope of tangent at x 2 O is see2 (0) 2 l; and slope of tangent at x— — 73’ is sec2 (g )2 4. The tangent at(—%.tan(~§))=(-— ~f)isy+f=4 (x+ —); the tangent at (0,0) is y 2 x; and the tangent at (3, tan (9) 2(§,\/§)isy— 324(x—g). 29. y 2 sec x 2> y’ 2 sec x tan x 2> slope of tangent at ’ y=secx x 2 — g is sec (—133) tan(— 3) 2 —2\/—; slope of tangent : at x 2 g is sec (’3') tn (g )2 \/2 The tangent at the point (—1sw<—;»=t~,2ww—2=4¢‘e+a; g the tangent at the po1nt(sec(4)) 2 (f, «2) is y — \/2 l =fih%l y=;2‘/3x—i3+2 y=\fix- [31+‘E 30. y 2 l + cos x 2> y’ 2 ~sin x 2> slope of tangent at x— — — 3 is —sin(— g) 2 g ; slope oftangentatx 2 321' is ~sin (3—2") 2 l. The tangent at the point elwmem=efl> isy——2 lg—Hx §;) the tangent atthe point (32” ,21+cos(?"))2 (3—2”,1)isy—12x—377r 31. Yes, y 2 x + sin x 2> y’ 2 1 + cos x; horizontal tangent occurs where l + cos x 2 O 2 cos x 2 —l 2 x 2 7r 32. No, y : 2x + sin x 2> y’ 2 2 + cos x; horizontal tangent occurs where 2 + cos x 2 O 2 cos x 2 —2. But there are no x-Values for which cos x 2 —2. 33. No, y 2 x — cot x 2> y’ 2 l + csc2 x; horizontal tangent occurs where 1 + csc2 x 2 0 2 csc2 x 2 —1. But there are no x—values for which csc2 x 2 —l. 34. Yes, y 2 x + 2 cos x 2> y’ 2 1 — 2 sin x; horizontal tangent occurs where l — 2 sin x 2 0 2> 1 2 2 sin x 12' —£ -51 2 2_s1nx=>x_60rx— 6 35. We want all points on the curve where the tangent line has slope 2. Thus, y 2 tan x 2 y’ 2 sec2 x so thaty’ 22 2> SCC2X=2 2> secx: :t \/2 2 x 2 i 1. Then the tangent line at (1,1) has equation y — 1— — 2 (x — —); the tangent line at (*4 — ,-1) has equation y +l— — 2 (x + 27‘.) ...
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