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

CHAPTER 2 Review 165
64. The left-hand and right-hand derivatives of f at a are
defined by
f sa 1 hd 2 f sad
f 9 2sad − lim
h l0 2 h
and
f 9 1sad − lim
h l0 1
f sa 1 hd 2 f sad
h
if these limits exist. Then f 9sad exists if and only if these
one-sided derivatives exist and are equal.
(a) Find f 9 2s4d and f 9 1s4d for the function
f sxd −
0
5 2 x
1
5 2 x
(b) Sketch the graph of f.
(c) Where is f discontinuous?
(d) Where is f not differentiable?
if x < 0
if 0 , x , 4
if x > 4
65. Nick starts jogging and runs faster and faster for 3 mintues,
then he walks for 5 minutes. He stops at an intersection for 2
minutes, runs fairly quickly for 5 minutes, then walks for 4
minutes.
(a) Sketch a possible graph of the distance s Nick has covered
after t minutes.
(b) Sketch a graph of dsydt.
66. When you turn on a hot-water faucet, the temperature T of
the water depends on how long the water has been running.
(a) Sketch a possible graph of T as a function of the time t
that has elapsed since the faucet was turned on.
(b) Describe how the rate of change of T with respect to t
varies as t increases.
(c) Sketch a graph of the derivative of T.
67. Let be the tangent line to the parabola y − x 2 at the point
s1, 1d. The angle of inclination of is the angle that
makes with the positive direction of the x-axis. Calculate
correct to the nearest degree.
2 Review
CONCEPT CHECK
1. Explain what each of the following means and illustrate with
a sketch.
(a) lim f sxd − L
x la
(b) lim x la 1 (c) lim x la 2
(d) lim f sxd − `
x la
(e) lim f sxd − L
xl`
2. Describe several ways in which a limit can fail to exist. Illustrate
with sketches.
3. State the following Limit Laws.
(a) Sum Law
(b) Difference Law
(c) Constant Multiple Law
(d) Product Law
(e) Quotient Law
(f) Power Law
(g) Root Law
4. What does the Squeeze Theorem say?
5. (a) What does it mean to say that the line x − a is a vertical
asymptote of the curve y − f sxd? Draw curves to illustrate
the various possibilities.
(b) What does it mean to say that the line y − L is a horizontal
asymptote of the curve y − f sxd? Draw curves to
illustrate the various possibilities.
6. Which of the following curves have vertical asymptotes?
Which have horizontal asymptotes?
(a) y − x 4 (b) y − sin x (c) y − tan x
(d) y − tan 21 x (e) y − e x (f) y − ln x
(g) y − 1yx (h) y − sx
Answers to the Concept Check can be found on the back endpapers.
7. (a) What does it mean for f to be continuous at a?
(b) What does it mean for f to be continuous on the
interval s2`, `d? What can you say about the graph
of such a function?
8. (a) Give examples of functions that are continuous on
f21, 1g.
(b) Give an example of a function that is not continuous
on f0, 1g.
9. What does the Intermediate Value Theorem say?
10. Write an expression for the slope of the tangent line to
the curve y − f sxd at the point sa, f sadd.
11. Suppose an object moves along a straight line with position
f std at time t. Write an expression for the instantaneous
velocity of the object at time t − a. How can you
interpret this velocity in terms of the graph of f ?
12. If y − f sxd and x changes from x 1 to x 2, write expressions
for the following.
(a) The average rate of change of y with respect to x
over the interval fx 1, x 2g.
(b) The instantaneous rate of change of y with respect to
x at x − x 1.
13. Define the derivative f 9sad. Discuss two ways of interpreting
this number.
14. Define the second derivative of f. If f std is the position
function of a particle, how can you interpret the second
derivative?
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.