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

282 Chapter 4 Applications of Differentiation
maximum and minimum points (or using a built-in maximum or minimum feature),
but instead let’s use calculus.
(b) The function f sxd − x 2 2 sin x is continuous on f0, 2g. Since
f 9sxd − 1 2 2 cos x, we have f 9sxd − 0 when cos x − 1 2 and this occurs when
x − y3 or 5y3. The values of f at these critical numbers are
f sy3d − 3 2 2 sin 3 − 3 2 s3 < 20.684853
and f s5y3d − 5 3 2 2 sin 5 3 − 5 3 1 s3 < 6.968039
The values of f at the endpoints are
f s0d − 0 and f s2d − 2 < 6.28
Comparing these four numbers and using the Closed Interval Method, we see that the
absolute minimum value is f sy3d − y3 2 s3 and the absolute maximum value is
f s5y3d − 5y3 1 s3 . The values from part (a) serve as a check on our work. n
Example 10 The Hubble Space Telescope was deployed on April 24, 1990, by the
space shuttle Discovery. A model for the velocity of the shuttle during this mission,
from liftoff at t − 0 until the solid rocket boosters were jettisoned at t − 126 seconds,
is given by
vstd − 0.001302t 3 2 0.09029t 2 1 23.61t 2 3.083
NASA
(in feet per second). Using this model, estimate the absolute maximum and minimum
values of the acceleration of the shuttle between liftoff and the jettisoning of the
boosters.
SOLUtion We are asked for the extreme values not of the given velocity function,
but rather of the acceleration function. So we first need to differentiate to find the
acceleration:
astd − v9std − d dt s0.001302t 3 2 0.09029t 2 1 23.61t 2 3.083d
− 0.003906t 2 2 0.18058t 1 23.61
We now apply the Closed Interval Method to the continuous function a on the interval
0 < t < 126. Its derivative is
a9std − 0.007812t 2 0.18058
The only critical number occurs when a9std − 0:
t 1 − 0.18058
0.007812 < 23.12
Evaluating astd at the critical number and at the endpoints, we have
as0d − 23.61 ast 1 d < 21.52 as126d < 62.87
So the maximum acceleration is about 62.87 ftys 2 and the minimum acceleration is
about 21.52 ftys 2 .
n
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