Pre-Calc Homework Solutions 120

Pre-Calc Homework Solutions 120 - 120 dy dx Chapter 3...

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Unformatted text preview: 120 dy dx Chapter 3 Review d csc 1 (sec x) dx 1 sec x 1 sec x tan 2 x sec x 2 29. 32. Since y d (sec x) 1 dx dy dx sin x x cos x is defined for all real x and (cos x)(1) x sin x, the cos x (x)( sin x) function is differentiable for all real x. sec x tan x 33. Since y dy dx 1 1 2 1 x x2 sec x tan x sec x tan x 1 cos x 1 cos x sin x cos x sin x cos x 1 1 x is defined for all x x2 (1 x 2)( 1) (1 (1 x 2)2 1 x 2)3/2 x)(2x) 1 and sin x sin x 3 2 x2 2 1 2x x (1 , which is defined only for x 1, sign (sin x), x Alternate method: On the domain 0 x 2 , , the function is differentiable for all x 2 ,x 2 1. ,x 3 , we may 2 rewrite the function as follows: y csc 2 2 2 2 1 34. Since y x (2x 7) (2x 1 (x 5) (x 7)2 x 2x (sec x) 1 1 7 dy and 2 dx 7)(1) (2x 5 is defined for all 7 5)(2) 17 , the (2x 7)2 7 2 sec cos x, ( x, 2 (sec x) (cos x) 0 x), 0 x x x x , , x function is differentiable for all x 35. Use implicit differentiation. 2 3 2 . 2 , x x xy 2x 3y 1 d (1) dx 2 x, 2 , x 2 3 2 1, 0 dy Therefore, dx x x , x 2 2 d d d (xy) (2x) (3y) dx dx dx dy dy x (y)(1) 2 3 dx dx dy (x 3) dx dy dx 0 (y y x 2) 2 3 1, 2 , x 36. Use implicit differentiation. 5x 4/5 d (5x 4/5) dx Note that the derivative exists at 0 and 2 only because these are the endpoints of the given domain; the two-sided derivative of y csc 1(sec x) does not exist at these points. 30. dr d d 1 d 1 sin cos sin c os sin c os sin c os 2 2 10y 6/5 15 d (15) dx 4x 1/5 d (10y 6/5) dx dy 12y 1/5 dx dy dx 0 4x 1/5 12y 1/5 1 3(xy)1/5 2 1 1 (1 cos cos (1 cos )(cos ) (1 sin )(sin ) (1 cos )2 cos2 sin (1 cos )2 sin 1 cos )2 sin2 37. Use implicit differentiation. xy d dx 1 2 xy 1 d (1) dx 1 2 1 xy 2 31. Since y dy dx 1 1 [x dy dx (y)(1)] x dy dx 0 0 y x 1 . x y dy dx ln x is defined for all x 2x x2 0 and 1 d 2 (x ) x 2 dx 2 , the function is differentiable for x Alternate method: Since xy dy dx all x 0. 1, we have xy 1 . x2 1 and y Therefore, ...
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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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