5128_Ch03_pp098-184

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