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

y
y=| 2x |
0 x
FIGURE for problem 15
;
(b) Make a guess as to the value of the limit
lim
n l ` y1 0
f sxd sin nx dx
based on graphs of the integrand.
(c) Using integration by parts, confirm the guess that you made in part (b). [Use the fact
that, since f 9 is continuous, there is a constant M such that | f 9sxd | < M for
0 < x < 1.]
11. If 0 , a , b, find
lim
t l 0Hy 1
0
fbx 1 as1 2 xdg t dxJ1yt
12. Graph f sxd − sinse x d and use the graph to estimate the value of t such that y t11
t
f sxd dx is
a maximum. Then find the exact value of t that maximizes this integral.
13. Evaluate y`
21S x 4
6D
2
1 1 x
14. Evaluate y stan x dx.
dx.
15. The circle with radius 1 shown in the figure touches the curve y − | 2x | twice. Find the
area of the region that lies between the two curves.
16. A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second.
Let v − vstd be the velocity of the rocket at time t and suppose that the velocity u of the
exhaust gas is constant. Let M − Mstd be the mass of the rocket at time t and note that
M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s
Second Law that
F − M dv
dt 2 ub
where the force F − 2Mt. Thus
1
M dv
dt
2 ub − 2Mt
Let M 1 be the mass of the rocket without fuel, M 2 the initial mass of the fuel,
and M 0 − M 1 1 M 2. Then, until the fuel runs out at time t − M 2yb, the mass is
M − M 0 2 bt.
(a) Substitute M − M 0 2 bt into Equation 1 and solve the resulting equation for v. Use
the initial condition vs0d − 0 to evaluate the constant.
(b) Determine the velocity of the rocket at time t − M 2yb. This is called the burnout
velocity.
(c) Determine the height of the rocket y − ystd at the burnout time.
(d) Find the height of the rocket at any time t.
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