5128_Ch03_pp098-184
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Section 3.7 Implicit Differentiation 163<br />
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<br />
3<br />
(a) f x x 23 9<br />
x 3 y 2 cos py.<br />
3 (b) f x x 2 1 0<br />
(1)<br />
7<br />
(1) 2 cos () is true since both sides equal: 1.<br />
(b) Use part (a) to find the slope of the line tangent to the curve<br />
1<br />
(c) f x x<br />
43<br />
3<br />
(d) f x x 23 at 1, 1. The slope is 3/2.<br />
6<br />
3 2 48. Grouping Activity<br />
44. Which of the following could be true if gt 1t 34 ? (a) and (c)<br />
(a) Show that the relation There are three values: 1, 1 5<br />
<br />
(a) gt 4 4 t 4<br />
(b) gt 4 4 2<br />
t<br />
y 3 xy 1<br />
x<br />
y<br />
(b) One way is to graph the<br />
–3<br />
3 at (3, 2): 2 7<br />
;<br />
(a) Tangent: y 2x 1<br />
y 2 (2 x) x 3<br />
8<br />
normal: y 1 equations<br />
(–3, –2)<br />
(3, –2)<br />
2 x 3 2 <br />
x3<br />
at (3, 2): 2 7<br />
<br />
y <br />
–2<br />
2 x<br />
8<br />
(c) gt t 7 165t 54 (d) gt 14t 14<br />
cannot be a function of x by showing that there is more than one<br />
45. The Eight Curve (a) Find the slopes of the figure-eightshaped<br />
possible y-value when x 2.<br />
f(2) 1,<br />
curve<br />
(b) On a small enough square with center 2, 1, the part f (2) 4<br />
y 4 y 2 x 2<br />
at the two points shown on the graph that follows.<br />
(b) Use parametric mode and the two pairs of parametric<br />
equations<br />
of the graph of the relation within the square will define a<br />
function y f x. For this function, find f 2 and f 2.<br />
49. Find the two points where the curve x 2 xy y 2 7 crosses<br />
the x-axis, and show that the tangents to the curve at these<br />
x 1 t t 2 t , 4 y 1 t t,<br />
points are parallel. What is the common slope of these tangents?<br />
x 2 t t 2 t , 4 y 2 t t,<br />
50. Find points on the curve x 2 xy y 2 7 (a) where the tangent<br />
is parallel to the x-axis and (b) where the tangent is parallel to the<br />
to graph the curve. Specify a window and a parameter interval.<br />
y-axis. (In the latter case, dydx is not defined, but dxdy is.<br />
y<br />
What value does dxdy have at these points?)<br />
51. Orthogonal Curves Two curves are orthogonal at a point of<br />
1 ⎛√3<br />
⎯<br />
— —<br />
3<br />
⎝<br />
,<br />
⎯√<br />
intersection if their tangents at that point cross at right angles.<br />
4 2<br />
Show that the curves 2x 2 3y 2 5 and y 2 x 3 are<br />
orthogonal at 1, 1 and 1, 1. Use parametric mode<br />
⎛⎯√<br />
—<br />
3 1<br />
⎝<br />
, –2<br />
to draw the curves and to show the tangent lines.<br />
4<br />
y 4 y 2 x 2<br />
52. The position of a body moving along a coordinate line at time t is<br />
x<br />
s 4 6t 32 , with s in meters and t in seconds. Find the<br />
0<br />
body’s velocity and acceleration when t 2 sec.<br />
53. The velocity of a falling body is v 8s t 1 feet<br />
per second at the instant tsec the body has fallen s feet from its<br />
starting point. Show that the body’s acceleration is 32 ftsec 2 .<br />
54. The Devil’s Curve (Gabriel Cramer [the Cramer<br />
–1<br />
of Cramer’s Rule], 1750) Find the slopes of the devil’s<br />
curve y 4 4y 2 x 4 9x 2 at the four indicated points.<br />
46. The Cissoid of Diocles (dates from about 200 B.c.)<br />
(a) Find equations for the tangent and normal to the cissoid of<br />
Diocles,<br />
y y 4 4y 2 x 4 9x 2<br />
y 2 2 x x 3 ,<br />
At (3, 2): 2 7 ;<br />
at the point 1, 1 as pictured below.<br />
(–3, 2)<br />
2<br />
8<br />
(3, 2)<br />
at (3, 2): 2 7<br />
;<br />
(b) Explain how to reproduce the graph on a grapher.<br />
8<br />
⎛<br />
⎝<br />
⎛<br />
⎝<br />
1<br />
0<br />
1<br />
(1, 1)<br />
x<br />
55. The Folium of Descartes (See Figure 3.47 on page 157)<br />
(a) Find the slope of the folium of Descartes, x 3 y 3 9xy 0<br />
at the points 4, 2 and 2, 4.<br />
5<br />
(a) At (4, 2): <br />
4 ; at (2, 4): 4 5 <br />
(b) At what point other than the origin does the folium have a<br />
horizontal tangent? At (3 3 2, 3 3 4) (3.780, 4.762)<br />
(c) Find the coordinates of the point A in Figure 3.47, where the<br />
folium has a vertical tangent. At (3 3 4, 3 3 2) (4.762, 3.780)