Unformatted text preview: Section 3.8
dy ds d (s ds 113 13. 1
1 s2) d (cos 1s) ds 18. s2)(1)
1 1 1 s
2 dy dx d [sin 1(2x)] 1 dx (s)
2 s2 1 s
2 1
2 s 2 ( 2s) 1 s2 ( 1 [sin [sin 1 (2x)] (2x)]
2 2 d dx
2 sin
1 1 4x 2 1 (2x) (2) 1 4x 2 s (1 1
2 1 s ) s2
2 s 2 1 [sin 1 (2x)]2 1 2s 1 dy ds d ds 1 2 s2 ss s dy 15. dx s2 19. (a) Since
s2 d sec 1 s ds 1 s s2 1 dy dx sec 2 x, the slope at 4 , 1 is sec 2
4 4 2. The tangent line is given by y 1 y 2x
dy dx 2 2 x 1, or 14. s2 1.
1 1 x2 1 1 + 12 4 1 . 2 (2s)
1 1 1 (b) Since , the slope at 1, 4 is The tangent line is given by y y
2 1 (x 2 1) , or 1 x 2 1 2 4 . 5x 4 6x 2 1. Thus f (1) 3 and d (tan 1 dx 1 1 1 x2 2 1 x x2 1 ( x2 1 x2 x 1) d (csc 1 x) dx 20. (a) Note that f (x) f (1) 12.
1 d ( 1)2 dx x2
1 x x2 1)
x 1 (b) Since the graph of y
1 f(x) includes the point (1, 3) and x 2 the slope of the graph is 12 at this point, the graph of y f
1 (2x)
1 1 x x2 1 (x) will include (3, 1) and the slope will be 1 and (f
1 1 . Thus, f 1(3) 12 ) (3) 1 . (We have 12 assumed that f x 1 (x) is defined and differentiable at 0 Note that the condition x 1 is required in the last step. 3. This is true by Theorem 3, because 5x 4 6x 2 1, which is never zero.) f (x) 16. dy dx 1 d cot 1 x dx 1 1
1 x2 d (tan 1 x) dx d 1 dx x 1 1 x2 21. (a) Note that f (x) sin x 3, which is always between 2 and 4. Thus f is differentiable at every point on the interval( , ) and f (x) is never zero on this interval, so f has a differentiable inverse by Theorem 3. (b) f (0) cos 0 3(0) f (0) sin 0 3 (c) Since the graph of y 1; 3 f (x) includes the point (0, 1) (
x2 1 1 1 1
1 x2 )
1 1 1 x2 1 1 x2 and the slope of the graph is 3 at this point, the graph of y f
1 (x) will include (1, 0) and the slope will be 0 and (f
1 x2 1 . Thus, f 1(1) 3 ) (1) 1 . 3 0, x 0 22. The condition x 0 is required because the original function was undefined when x 0. 17.
dy dx d (x sin 1 x) dx 1 1 x2
1 d ( dx 1
1 x 2)
1 2 1 x2 [ 2 , 2 ] by [ 4, 4] (x) (sin x)(1) ( 2x) (a) All reals (b)
2 2 sin x , ...
View
Full Document
 Fall '08
 GERMAN
 Calculus, Derivative, Mathematical analysis, dx

Click to edit the document details