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

y
O
y
O
P{t, sin(t@)}
y=sin{≈}
A(t)
t x
P{t, sin(t@)}
B(t)
t x
FIGURE for problem 8
C
B
A
FIGURE for problem 16
3. If y 4 4 0 esx22d dx − k, find the value of y 4 4 0 xesx22d dx.
; 4. (a) Graph several members of the family of functions f sxd − s2cx 2 x 2 dyc 3 for c . 0 and
look at the regions enclosed by these curves and the x-axis. Make a conjecture about
how the areas of these regions are related.
(b) Prove your conjecture in part (a).
(c) Take another look at the graphs in part (a) and use them to sketch the curve traced out
by the vertices (highest points) of the family of functions. Can you guess what kind of
curve this is?
(d) Find an equation of the curve you sketched in part (c).
5. If f sxd − y tsxd 1
dt, where tsxd − y cos x
f1 1 sinst 2 dg dt, find f 9sy2d.
0
s1 1 t 3 0
6. If f sxd − y x 0 x 2 sinst 2 d dt, find f 9sxd.
1
7. Evaluate lim
x l 0 x yx s1 2 tan 2td 1yt dt.
0
8. The figure shows two regions in the first quadrant: Astd is the area under the curve
y − sinsx 2 d from 0 to t, and Bstd is the area of the triangle with vertices O, P, and st, 0d.
Find lim AstdyBstd.
t l 0 1
9. Find the interval fa, bg for which the value of the integral y b s2 1 x 2 x 2 a
d dx is a maximum.
10. Use an integral to estimate the sum 10000
o si .
i−1
11. (a) Evaluate y n 0
v xb dx, where n is a positive integer.
(b) Evaluate y b a
v xb dx, where a and b are real numbers with 0 < a , b.
12. Find d 2
y x
dx 2
Sy sin t
0 1
s1 1 u 4 duDdt.
13. Suppose the coefficients of the cubic polynomial Psxd − a 1 bx 1 cx 2 1 dx 3 satisfy the
equation
a 1 b 2 1 c 3 1 d 4 − 0
Show that the equation Psxd − 0 has a root between 0 and 1. Can you generalize this result
for an nth-degree polynomial?
14. A circular disk of radius r is used in an evaporator and is rotated in a vertical plane. If it is
to be partially submerged in the liquid so as to maximize the exposed wetted area of the
disk, show that the center of the disk should be positioned at a height rys1 1 2 above
the surface of the liquid.
15. Prove that if f is continuous, then y x
f sudsx 2 ud du − y x
0
0
Sy u
0
f std dtD du.
2
2
2
16. The figure shows a parabolic segment, that is, a portion of a parabola cut off by a chord
AB. It also shows a point C on the parabola with the property that the tangent line at C is
parallel to the chord AB. Archimedes proved that the area of the parabolic segment is 4 3
times the area of the inscribed triangle ABC. Verify Archimedes’ result for the parabola
y − 4 2 x 2 and the line y − x 1 2.
17. Given the point sa,bd in the first quadrant, find the downward-opening parabola that passes
through the point sa,bd and the origin such that the area under the parabola is a minimum.
18. The figure shows a region consisting of all points inside a square that are closer to the
center than to the sides of the square. Find the area of the region.
FIGURE for problem 18
426
2
19. Evaluate lim
n l `S
1
sn sn 1 1 1 1
sn sn 1 2 1 ∙ ∙ ∙ 1 1
sn sn 1 nD.
20. For any number c, we let f c sxd be the smaller of the two numbers sx 2 cd 2 and
sx 2 c 2 2d 2 . Then we define tscd − y 1 0 f csxd dx. Find the maximum and minimum values
of tscd if 22 < c < 2.
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