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

266 Chapter 3 Differentiation Rules
57. At what point of the curve y − cosh x does the tangent have
slope 1?
; 58. Investigate the family of functions
f nsxd − tanhsn sin xd
where n is a positive integer. Describe what happens to the
graph of f n when n becomes large.
59. Show that if a ± 0 and b ± 0, then there exist numbers
and such that ae x 1 be 2x equals either
sinhsx 1 d or coshsx 1 d
In other words, almost every function of the form
f sxd − ae x 1 be 2x is a shifted and stretched hyperbolic sine
or cosine function.
3 Review
CONCEPT CHECK
1. State each differentiation rule both in symbols and in words.
(a) The Power Rule (b) The Constant Multiple Rule
(c) The Sum Rule
(d) The Difference Rule
(e) The Product Rule (f) The Quotient Rule
(g) The Chain Rule
2. State the derivative of each function.
(a) y − x n (b) y − e x (c) y − b x
(d) y − ln x (e) y − log b x (f) y − sin x
(g) y − cos x (h) y − tan x (i) y − csc x
( j) y − sec x (k) y − cot x (l) y − sin 21 x
(m) y − cos 21 x (n) y − tan 21 x (o) y − sinh x
(p) y − cosh x (q) y − tanh x (r) y − sinh 21 x
(s) y − cosh 21 x (t) y − tanh 21 x
3. (a) How is the number e defined?
(b) Express e as a limit.
(c) Why is the natural exponential function y − e x used more
often in calculus than the other exponential functions
y − b x ?
Answers to the Concept Check can be found on the back endpapers.
(d) Why is the natural logarithmic function y − ln x used more
often in calculus than the other logarithmic functions
y − log b x?
4. (a) Explain how implicit differentiation works.
(b) Explain how logarithmic differentiation works.
5. Give several examples of how the derivative can be interpreted
as a rate of change in physics, chemistry, biology, economics,
or other sciences.
6. (a) Write a differential equation that expresses the law of
natural growth.
(b) Under what circumstances is this an appropriate model for
population growth?
(c) What are the solutions of this equation?
7. (a) Write an expression for the linearization of f at a.
(b) If y − f sxd, write an expression for the differential dy.
(c) If dx − Dx, draw a picture showing the geometric meanings
of Dy and dy.
TRUE-FALSE Quiz
Determine whether the statement is true or false. If it is true, explain
why. If it is false, explain why or give an example that disproves the
statement.
1. If f and t are differentiable, then
d
f f sxd 1 tsxdg − f 9sxd 1 t9sxd
dx
2. If f and t are differentiable, then
d
f f sxdtsxdg − f 9sxdt9sxd
dx
3. If f and t are differentiable, then
d
dx f f stsxddg − f 9stsxddt9sxd
4. If f is differentiable, then d sf sxd −
f dx
5. If f is differentiable, then d dx
6. If y − e 2 , then y9 − 2e.
7. d dx s10 x d − x10 x21 8.
9. d dx stan2 xd − d dx ssec2 xd
10.
d
dx | x 2 1 x | − | 2x 1 1 |
9sxd
2sf sxd .
f ssx d −
f 9sxd
2sx .
d
dx sln 10d − 1 10
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