5128_Ch03_pp098-184
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164 Chapter 3 Derivatives<br />
56. The line that is normal to the curve x 2 2xy 3y 2 0 at<br />
1, 1 intersects the curve at what other point? (3, 1)<br />
57. Find the normals to the curve xy 2x y 0 that are parallel<br />
to the line 2x y 0. At (1, 1): y 2x 3;at (3, 3): y 2x 3<br />
58. Show that if it is possible to draw these three normals from the<br />
point a, 0 to the parabola x y 2 shown here, then a must be<br />
greater than 12. One of the normals is the x-axis. For what<br />
value of a are the other two normals perpendicular?<br />
y<br />
x y 2<br />
63. Multiple Choice Which of the following is equal to dydx if<br />
y x 3/4 ? E<br />
(A) 3x 13<br />
(B) 4x 14<br />
(C) 3x 14<br />
4 3<br />
(D) (E) <br />
4 3 4 3x<br />
1/4 4x<br />
1/4<br />
64. Multiple Choice Which of the following is equal to the slope<br />
of the tangent to y 2 x 2 1 at (1,2)? C<br />
1<br />
1<br />
(A) (B) 2 (C) (D) 2 (E) 0<br />
2<br />
2<br />
Extending the Ideas<br />
65. Finding Tangents<br />
(a) Show that the tangent to the ellipse<br />
x<br />
0<br />
(a, 0)<br />
The normal at the<br />
point (b 2 , b) is:<br />
y 2bx 2b 3 b.<br />
This line intersects the x-axis at x b 2 1 2 , which must be greater than 1 if b 0.<br />
2<br />
The two normals are perpendicular when a 3/4.<br />
Standardized Test Questions<br />
You should solve the following problems without using a<br />
graphing calculator.<br />
59. True or False The slope of xy 2 x 1 at (12, 1) is 2.<br />
Justify your answer. False. It is equal to 2.<br />
60. True or False The derivative of y 3 1<br />
x is . Justify your<br />
3x<br />
2/3<br />
answer. True. By the power rule.<br />
In Exercises 61 and 62, use the curve x 2 xy y 2 1.<br />
61. Multiple Choice Which of the following is equal to dydx? A<br />
(A) y <br />
<br />
2x<br />
(B) y <br />
<br />
2x<br />
2y x<br />
2y x<br />
2x<br />
(C) <br />
(D) 2 x y<br />
<br />
x 2y<br />
x 2y<br />
(E) y 2x<br />
<br />
x<br />
62. Multiple Choice Which of the following is equal to d 2y<br />
dx2? A<br />
6<br />
10y<br />
(A) (2y x) 3 (B)<br />
2 10x 2 10xy<br />
<br />
(2y x) 3<br />
(C) 8x 2 4xy<br />
8y <br />
(<br />
2<br />
x 2y) (D) 10 x 2 10y <br />
3 (x <br />
2<br />
2y) 3<br />
(E) 2 x <br />
7. 1 x cos2 (xy) y x dy x 2<br />
12. ,34<br />
dx y 3<br />
dy 2xy y 2<br />
13. , defined at every point except where x 0 or y x2<br />
dx 2xy x<br />
2<br />
dy 1<br />
14. , defined at every point except where y k, k any integer<br />
dx sin y<br />
dy 3x 2 y<br />
15. , defined at every point except where y 2 x3<br />
dx x 3y<br />
2<br />
dy 3 2x 4y<br />
1<br />
16. , defined at every point except where y x<br />
dx 4x 8y<br />
2<br />
x<br />
2 a<br />
2 y 2 b<br />
2 1<br />
at the point x 1 , y 1 has equation<br />
x 1<br />
a<br />
x<br />
2<br />
y 1y<br />
b 2<br />
1.<br />
(b) Find an equation for the tangent to the hyperbola<br />
x<br />
2 a<br />
2 y 2 b<br />
2 1<br />
at the point x 1 , y 1 .<br />
66. End Behavior Model Consider the hyperbola<br />
Show that<br />
(a) y b a x2 a 2 .<br />
x<br />
2 a<br />
2 y 2 b<br />
2 1.<br />
(b) gx bax is an end behavior model for<br />
f x bax 2 a .<br />
2<br />
(c) gx bax is an end behavior model for<br />
21. (a) y 6 7 x 6 7 <br />
(b) y 7 6 x 7 6 <br />
27. d y<br />
x dx<br />
y <br />
d 2y<br />
dx2 (x2 y 2 ) 1<br />
<br />
y 3 y 3<br />
28. d y<br />
<br />
dx<br />
<br />
y x 13<br />
d 2y<br />
dx2 x 2/ 3<br />
<br />
3x4<br />
/3 y2<br />
/3<br />
y1/<br />
3<br />
1<br />
3x 4/ 3 <br />
y 1/3<br />
39. 1 4 (1 x1/2 ) 1/2 x 1/2<br />
40. x 32 (2x 12 1) 43<br />
41. 9 2 (csc x)32 cot x<br />
42. 5 4 [sin (x 5)]1/4 cos (x 5)<br />
f x bax 2 a 2 .<br />
19. (a) y 3x 6 (b) y 1 3 x 8 3 <br />
23. (a) y 2 x <br />
2 2 <br />
(b) y x 2 <br />
29. d y<br />
x 1<br />
<br />
dx<br />
y<br />
d 2y<br />
dx2 y2 x 1) (y<br />
2 1<br />
<br />
3 y 3<br />
30. d y 1<br />
<br />
dx<br />
y 1<br />
2 y<br />
d 1<br />
dx2 (y 1) 3