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

Section 3.4 The Chain Rule 207
; 90. The table gives the US population from 1790 to 1860.
CAS
CAS
Year Population Year Population
1790 3,929,000 1830 12,861,000
1800 5,308,000 1840 17,063,000
1810 7,240,000 1850 23,192,000
1820 9,639,000 1860 31,443,000
(a) Use a graphing calculator or computer to fit an exponential
function to the data. Graph the data points and
the exponential model. How good is the fit?
(b) Estimate the rates of population growth in 1800 and
1850 by averaging slopes of secant lines.
(c) Use the exponential model in part (a) to estimate the
rates of growth in 1800 and 1850. Compare these estimates
with the ones in part (b).
(d) Use the exponential model to predict the population
in 1870. Compare with the actual population of
38,558,000. Can you explain the discrepancy?
91. Computer algebra systems have commands that differentiate
functions, but the form of the answer may not be convenient
and so further commands may be necessary to simplify the
answer.
(a) Use a CAS to find the derivative in Example 5 and
compare with the answer in that example. Then use the
simplify command and compare again.
(b) Use a CAS to find the derivative in Example 6. What
happens if you use the simplify command? What happens
if you use the factor command? Which form of the
answer would be best for locating horizontal tangents?
92. (a) Use a CAS to differentiate the function
f sxd −Î x 4 2 x 1 1
x 4 1 x 1 1
and to simplify the result.
(b) Where does the graph of f have horizontal tangents?
(c) Graph f and f 9 on the same screen. Are the graphs
consistent with your answer to part (b)?
93. Use the Chain Rule to prove the following.
(a) The derivative of an even function is an odd function.
(b) The derivative of an odd function is an even function.
94. Use the Chain Rule and the Product Rule to give an
alternative proof of the Quotient Rule.
[Hint: Write f sxdytsxd − f sxdftsxdg 21 .]
95. (a) If n is a positive integer, prove that
d
dx ssinn x cos nxd − n sin n21 x cossn 1 1dx
(b) Find a formula for the derivative of y − cos n x cos nx
that is similar to the one in part (a).
96. Suppose y − f sxd is a curve that always lies above the
x-axis and never has a horizontal tangent, where f is
dif ferentiable everywhere. For what value of y is the rate
of change of y 5 with respect to x eighty times the rate of
change of y with respect to x?
97. Use the Chain Rule to show that if is measured in
degrees, then
d
ssin d −
d
180 cos
(This gives one reason for the convention that radian
measure is always used when dealing with trigonometric
functions in calculus: the differentiation formulas would
not be as simple if we used degree measure.)
98. (a) Write | x | − sx 2 and use the Chain Rule to show that
d
dx | x | −
x
| x |
(b) If f sxd − | sin x | , find f 9sxd and sketch the graphs of f
and f 9. Where is f not differentiable?
, find t9sxd and sketch the graphs of t
and t9. Where is t not differentiable?
(c) If tsxd − sin | x |
99. If y − f sud and u − tsxd, where f and t are twice differen
tiable functions, show that
d 2 y
dx − d 2 2
y du
2 du 2S 1
dxD
dy d 2 u
du dx 2
100. If y − f sud and u − tsxd, where f and t possess third
derivatives, find a formula for d 3 yydx 3 similar to the one
given in Exercise 99.
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