5128_Ch03_pp098-184
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Section 3.8 Derivatives of Inverse Trigonometric Functions 171<br />
Standardized Test Questions<br />
You may use a graphing calculator to solve the following<br />
problems.<br />
35. True or False The domain of y sin 1 x is 1 x 1.<br />
Justify your answer. True. By definition of the function.<br />
36. True or False The domain of y tan 1 x is 1 x 1.<br />
Justify your answer. False. The domain is all real numbers.<br />
d<br />
37. Multiple Choice Which of the following is sin 1<br />
d x x <br />
2 ?<br />
(A) 4 <br />
2 x 2 (B) 1<br />
4 x 2 (C) 2<br />
<br />
4 x 2<br />
(D) 4 <br />
2 x 2 (E) 1<br />
4 x 2<br />
d<br />
38. Multiple Choice Which of the following is tan 1 (3x)?<br />
d x<br />
(A) 1 <br />
39x 2 (B) 1 <br />
19x 2 (C) <br />
1 <br />
19x 2<br />
E<br />
D<br />
Extending the Ideas<br />
47. Identities Confirm the following identities for x 0.<br />
(a) cos 1 x sin 1 x p2<br />
(b) tan 1 x cot 1 x p2<br />
(c) sec 1 x csc 1 x p2<br />
48. Proof Without Words The figure gives a proof without<br />
words that tan 1 1 tan 1 2 tan 1 3 p. Explain what is<br />
going on.<br />
3<br />
(D) 1 <br />
39x 2 (E) 1 9x<br />
2 d<br />
39. Multiple Choice Which of the following is sec 1 (x 2 )?<br />
d x<br />
2<br />
(A) xx 4<br />
1<br />
(B) 2<br />
xx 2<br />
1<br />
(C) 2<br />
x1 x 4<br />
(D) x1 <br />
2 x 2 (E) 2x<br />
1 <br />
x 4<br />
40. Multiple Choice Which of the following is the slope of the<br />
tangent line to y tan 1 (2x) at x 1? C<br />
(A) 25 (B) 15 (C) 25 (D) 52 (E) 5<br />
Explorations<br />
In Exercises 41–46, find (a) the right end behavior model,<br />
(b) the left end behavior model, and (c) any horizontal tangents for<br />
the function if they exist.<br />
41. y tan 1 x 42. y cot 1 x<br />
43. y sec 1 x 44. y csc 1 x<br />
45. y sin 1 x 46. y cos 1 x<br />
A<br />
49. (Continuation of Exercise 48) Here is a way to construct<br />
tan 1 1, tan 1 2, and tan 1 3 by folding a square of paper. Try<br />
it and explain what is going on.<br />
Fold 2<br />
Fold 1<br />
Fold 3<br />
tan –1 3<br />
tan –1 2<br />
tan –1 1<br />
1<br />
13. .<br />
⏐2s 1⏐s <br />
2 s<br />
2<br />
15. <br />
(x 2 1)x <br />
2 2<br />
1<br />
17. 1 t<br />
2<br />
1<br />
19. <br />
2t t<br />
1<br />
d<br />
32. cos 1 d<br />
(x) d x<br />
d x 2 sin1 x <br />
d<br />
0 sin 1 x d x<br />
1<br />
1 x<br />
2<br />
d<br />
33. cot 1 d<br />
x d x d x 2 tan1 x <br />
d<br />
0 tan 1 x d x<br />
1<br />
<br />
1 x 2<br />
d<br />
34. csc 1 d<br />
(x) d x<br />
d x<br />
d<br />
0 sec 1 x d x<br />
1<br />
<br />
⏐x⏐x <br />
2 1<br />
<br />
2 sec1 x<br />
<br />
41. (a) y 2 (b) y 2 (c) None<br />
42. (a) y 0 (b) y (c) None<br />
43. (a) y 2 (b) y 2 (c) None<br />
44. (a) y 0 (b) y 0 (c) None<br />
45. (a) None (b) None (c) None<br />
46. (a) None (b) None (c) None