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

18. Evaluate lim
x l
e sin x 2 1
x 2 .
19. Let T and N be the tangent and normal lines to the ellipse x 2 y9 1 y 2 y4 − 1 at any point
P on the ellipse in the first quadrant. Let x T and y T be the x- and y-intercepts of T and x N
and y N be the intercepts of N. As P moves along the ellipse in the first quadrant (but not
on the axes), what values can x T, y T, x N, and y N take on? First try to guess the answers just
by looking at the figure. Then use calculus to solve the problem and see how good your
intuition is.
y
y T
T
2
P
x N
0
3
y N
N
x T
x
sins3 1 xd 2 2 sin 9
20. Evaluate lim
.
x l 0 x
21. (a) Use the identity for tansx 2 yd (see Equation 14b in Appendix D) to show that if two
lines L 1 and L 2 intersect at an angle , then
tan −
m2 2 m1
1 1 m 1m 2
where m 1 and m 2 are the slopes of L 1 and L 2, respectively.
(b) The angle between the curves C 1 and C 2 at a point of intersection P is defined to be
the angle between the tangent lines to C 1 and C 2 at P (if these tangent lines exist). Use
part (a) to find, correct to the nearest degree, the angle between each pair of curves at
each point of intersection.
(i) y − x 2 and y − sx 2 2d 2
(ii) x 2 2 y 2 − 3 and x 2 2 4x 1 y 2 1 3 − 0
22. Let Psx 1, y 1d be a point on the parabola y 2 − 4px with focus Fsp, 0d. Let be the angle
between the parabola and the line segment FP, and let be the angle between the
horizontal line y − y 1 and the parabola as in the figure. Prove that − . (Thus, by a
prin ciple of geometrical optics, light from a source placed at F will be reflected along a
line parallel to the x-axis. This explains why paraboloids, the surfaces obtained by rotating
parabolas about their axes, are used as the shape of some automobile headlights and
mirrors for telescopes.)
y
∫ y=›
P(⁄, ›)
å
0 F(p, 0)
x
¥=4px
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