James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Solving the equations ln a − ca 2 and 1ya − 2ca, we get
y
ln a − ca 2 − c ?
1
2c − 1 2
y=ln x
Thus a − e 1y2 and
0
FIGURE 4
x
c − ln a
a 2
1y2
ln e
− − 1 e 2e
For negative values of c we have the situation illustrated in Figure 4: All parabolas
y − cx 2 with negative values of c intersect y − ln x exactly once. And let’s not forget
about c − 0: The curve y − 0x 2 − 0 is just the x-axis, which intersects y − ln x
exactly once.
To summarize, the required values of c are c − 1ys2ed and c < 0.
■
Problems
1. Find points P and Q on the parabola y − 1 2 x 2 so that the triangle ABC formed by the
x-axis and the tangent lines at P and Q is an equilateral triangle. (See the figure.)
y
A
P
Q
B
0
C
x
;
2. Find the point where the curves y − x 3 2 3x 1 4 and y − 3sx 2 2 xd are tangent to each
other, that is, have a common tangent line. Illustrate by sketching both curves and the
common tangent.
3. Show that the tangent lines to the parabola y − ax 2 1 bx 1 c at any two points with
x-coordinates p and q must intersect at a point whose x-coordinate is halfway between p
and q.
4. Show that
d
dxS sin2 x
1 1 cot x 1 cos2 x
− 2cos 2x
1 1 tan xD
sec t 2 sec x
5. If f sxd − lim
, find the value of f 9sy4d.
t l x t 2 x
6. Find the values of the constants a and b such that
y
lim
x l 0
7. Show that sin 21 stanh xd − tan 21 ssinh xd.
s 3 ax 1 b 2 2
x
− 5 12
x
8. A car is traveling at night along a highway shaped like a parabola with its vertex at the
origin (see the figure). The car starts at a point 100 m west and 100 m north of the origin
and travels in an easterly direction. There is a statue located 100 m east and 50 m north of
the origin. At what point on the highway will the car’s headlights illuminate the statue?
FIGURE FOR PROBLEM 8
9. Prove that d n
dx n ssin4 x 1 cos 4 xd − 4 n21 coss4x 1 ny2d.
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