Pre-Calc Homework Solutions 89

Pre-Calc Homework Solutions 89 - Section 3.5 d dx d dx d d...

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Unformatted text preview: Section 3.5 d dx d dx d d 89 12. y (x) d tan x dx x x (tan x) x2 tan x (x) d sec x (b) dx x sec2 x tan x x2 ( 2)2 y 4 1 8 2 2 2 2 16 (cos x) (1) (1) (cos x) d 1 dx dx (cos x)2 dx cos x (cos x)(0) (1)( sin x) cos 2 x sin x sec x tan x cos2 x d cos x dx sin x (sin x) (cos x) d dx 4 4 16. (a) 8 The normal line has slope tan d cot x dx 8 4 16 16 (cos x) (sin x) d dx and passes through 4 , 4 2 4 , 4 4 . (sin x)2 (sin x)( sin x) (cos x)(cos x) sin 2 x Its equation is y 2 16 3 8 x 4 4 , or (sin2 x cos2 x) sin2 x 1 sin2 x y 16 8 x 64 32 . csc 2 x Using decimals, this equation is approximately (b) y 13. y (x) 1.081x 2.122. x 2 (sin x) d dx d csc x dx d 1 dx sin x (sin x) (1) d 2 (x sin x) dx d dx (1) (sin x) d dx (sin x) (x 2) d dx (sin x)2 (sin x)(0) (1)(cos x) sin 2 x cos x sin2 x x 2 cos x y (3) 2x sin x 6 sin 3 9 cos 3 The tangent line has slope 9 cos 3 6 sin 3 and passes through (3, 9 sin 3). Its equation is y (9 cos 3 6 sin 3)(x 3) 9 sin 3, or y (9 cos 3 6 sin 3)x 27 cos 3 9 sin 3. Using decimals, this equation is approximately y 8.063x 25.460. 14. lim h0 csc x cot x 0, so the slope of 0, 17. d sec x dx sec x tan x which is 0 at x d the tangent line is 0. cos x dx sin x which is 0 at x so the slope of the tangent line is 0. 18. d tan x dx d cot x dx cos (x h) h cos x (cos x cos h cos x(cos h cos h h sin x sin h) h 1) h 1 1 cos x sec 2 x lim h0 1 , which is never 0. cos2 x 1 csc 2 x , which is never 0. sin2 x lim h0 sin x sin h sin h h sin h h0 h 19. y (x) y 4 d ( dx 2 cos x) 2 sin 4 2 sin x 2 1 2 lim (cos x) h0 (sin x) 1 (cos x) lim (cos x)(0) cos h h h0 The tangent line has slope 4 1 and passes through (sin x) lim sin x , 2 cos 1 x 4 4 4 , 1 , so its equation is x 4 (sin x)(1) y d (sin x) (cos x) dx 1, or y 1. 4 15. (a) d d sin x tan x dx dx cos x (cos x)(cos x) (sin x)( sin x) cos 2 x cos2 x sin2 x cos2 x 1 cos2 x d (cos x) (sin x) dx The normal line has slope 1 and passes through its equation is y 1 x 4 , 1 , so 4 (cos x)2 1, or y x 1 . sec 2 x ...
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This note was uploaded on 10/05/2011 for the course MAC 1147 taught by Professor German during the Fall '08 term at University of Florida.

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