5128_Ch03_pp098-184

Section 3.7 Implicit Differentiation 163
43. Which of the following could be true if f x x 13 ? (b), (c), and (d) 47. (a) Confirm that 1, 1 is on the curve defined by
3
(a) f x x 23 9
x 3 y 2 cos py.
3 (b) f x x 2 1 0
(1)
7
(1) 2 cos () is true since both sides equal: 1.
(b) Use part (a) to find the slope of the line tangent to the curve
1
(c) f x x
43
3
(d) f x x 23 at 1, 1. The slope is 3/2.
6
3 2 48. Grouping Activity
44. Which of the following could be true if gt 1t 34 ? (a) and (c)
(a) Show that the relation There are three values: 1, 1 5
(a) gt 4 4 t 4
(b) gt 4 4 2
t
y 3 xy 1
x
y
(b) One way is to graph the
–3
3 at (3, 2): 2 7
;
(a) Tangent: y 2x 1
y 2 (2 x) x 3
8
normal: y 1 equations
(–3, –2)
(3, –2)
2 x 3 2
x3
at (3, 2): 2 7
y
–2
2 x
8
(c) gt t 7 165t 54 (d) gt 14t 14
cannot be a function of x by showing that there is more than one
45. The Eight Curve (a) Find the slopes of the figure-eightshaped
possible y-value when x 2.
f(2) 1,
curve
(b) On a small enough square with center 2, 1, the part f (2) 4
y 4 y 2 x 2
at the two points shown on the graph that follows.
(b) Use parametric mode and the two pairs of parametric
equations
of the graph of the relation within the square will define a
function y f x. For this function, find f 2 and f 2.
49. Find the two points where the curve x 2 xy y 2 7 crosses
the x-axis, and show that the tangents to the curve at these
x 1 t t 2 t , 4 y 1 t t,
points are parallel. What is the common slope of these tangents?
x 2 t t 2 t , 4 y 2 t t,
50. Find points on the curve x 2 xy y 2 7 (a) where the tangent
is parallel to the x-axis and (b) where the tangent is parallel to the
to graph the curve. Specify a window and a parameter interval.
y-axis. (In the latter case, dydx is not defined, but dxdy is.
y
What value does dxdy have at these points?)
51. Orthogonal Curves Two curves are orthogonal at a point of
1 ⎛√3
⎯
— —
3
⎝
,
⎯√
intersection if their tangents at that point cross at right angles.
4 2
Show that the curves 2x 2 3y 2 5 and y 2 x 3 are
orthogonal at 1, 1 and 1, 1. Use parametric mode
⎛⎯√
—
3 1
⎝
, –2
to draw the curves and to show the tangent lines.
4
y 4 y 2 x 2
52. The position of a body moving along a coordinate line at time t is
x
s 4 6t 32 , with s in meters and t in seconds. Find the
0
body’s velocity and acceleration when t 2 sec.
53. The velocity of a falling body is v 8s t 1 feet
per second at the instant tsec the body has fallen s feet from its
starting point. Show that the body’s acceleration is 32 ftsec 2 .
54. The Devil’s Curve (Gabriel Cramer [the Cramer
–1
of Cramer’s Rule], 1750) Find the slopes of the devil’s
curve y 4 4y 2 x 4 9x 2 at the four indicated points.
46. The Cissoid of Diocles (dates from about 200 B.c.)
(a) Find equations for the tangent and normal to the cissoid of
Diocles,
y y 4 4y 2 x 4 9x 2
y 2 2 x x 3 ,
At (3, 2): 2 7 ;
at the point 1, 1 as pictured below.
(–3, 2)
2
8
(3, 2)
at (3, 2): 2 7
;
(b) Explain how to reproduce the graph on a grapher.
8
⎛
⎝
⎛
⎝
1
0
1
(1, 1)
x
55. The Folium of Descartes (See Figure 3.47 on page 157)
(a) Find the slope of the folium of Descartes, x 3 y 3 9xy 0
at the points 4, 2 and 2, 4.
5
(a) At (4, 2):
4 ; at (2, 4): 4 5
(b) At what point other than the origin does the folium have a
horizontal tangent? At (3 3 2, 3 3 4) (3.780, 4.762)
(c) Find the coordinates of the point A in Figure 3.47, where the
folium has a vertical tangent. At (3 3 4, 3 3 2) (4.762, 3.780)