CHAPTER 1
Functions, Graphs,
and Limits
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Figure 1.1 Interpretations of the function f(x).
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Figure 1.2 The composition f(g(x)) as an
assembly line.
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Figure 1.3 (a) A production function.
(b) Bounded population growth.
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Figure 1.4 (a) The graph of y = x2. (b) Other graphs through the points in Example
2.1.
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Figure 1.5 The graph of f(x) =
2 x
2
x

3
0  x  1

1  x  4


x  4.
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Figure 1.6 The graph of f(x) = –x2 + x + 2.
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Figure 1.7 The graph of the function
y = x3 – x2 – 6x.
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Figure 1.8 The graph of the parabola y = Ax2 + Bx + C. (a) If A > 0, the parabola opens up.
(b) If A < 0, the parabola opens down.
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Figure 1.9 A revenue function.
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Figure 1.10 The graphs of y = f(x) and y = g(x) intersect at P and Q.
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Figure 1.11 The intersection of the graphs of
f(x) = 3x + 2 and g(x) = x2.
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Figure 1.12 Three polynomials of degree 3.
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Figure 1.13 Graphs of three rational functions.
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Figure 1.14 The vertical line test.
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Figure 1.15 The cost function C(x) = 50x + 200.
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Figure 1.16 Slope 
y 2  y1
y

.
x 2  x1
x
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Figure 1.17 The line joining (–2, 5) and (3, –1).
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Figure 1.18 The direction and steepness of a line.
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Figure 1.19 Horizontal and vertical lines.
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Figure 1.20 The slope and y intercept of the line
y = mx + b.
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Figure 1.21 The line 3y + 2x = 6.
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Figure 1.22 The line
y  1x  3 .
2
2
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Figure 1.23 The line y = –4x + 10.
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Figure 1.24 The rising price of bread: y = 2x + 136.
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Figure 1.25 Growth of federal civilian employment in the United States (1950–
1989).
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Figure 1.26
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Figure 1.27 Lines parallel and perpendicular
to a given line L.
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Figure 1.28 Rectangular picnic area.
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Figure 1.29 The length of fencing:
F ( x )  x  10 ,000 .
x
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Figure 1.30 Cylindrical can for Example 4.2.
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Figure 1.31 The cost function:
C( r )  6 r 2  96  .
r
r
C(r)
0.5
608
1.0
320
1.5
243
2.0
226
2.5
238
3.0
270
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Figure 1.32 The cost of water in Marin County.
x
C(x)
0
0
12
14.64
24
134.64
30
434.64
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Figure 1.33 The rate of bounded population growth: R(p) = kp(b – p).
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Figure 1.34 The profit function
P(x) = (6,000 – 400x)(x – 2).
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Figure 1.35 Market equilibrium: the intersection of supply and demand.
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Figure 1.36 The supply and demand curves
for Example 4.6.
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Figure 1.37 Geometric interpretation of the limit.
(a) If
limthef height
( x ) ofL,
x c
the graph of f approaches L as
x approaches c.
(b) Geometric interpretation of the limit statement
x
lim
x 1
2
 x  2
 3
x  1
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Figure 1.38 Three functions for which
lim f ( x )  L.
x c
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Figure 1.39 Two functions for which
does not exist.
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lim f ( x )
x c
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Figure 1.40 Limits of two linear functions.
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Figure 1.41 The graph of
f (x )  x  1 .
x 2
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Figure 1.42 The graph of
2
x
1
f (x )  2
.
x  3x  2
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Figure 1.43 Just in time inventory.
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Figure 1.44 The graph of
1  x 2
f (x )  
2 x  1
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if x  2
.
if x  2
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Figure 1.45 A continuous graph.
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Figure 1.46 Three functions with discontinuities of x = c.
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Figure 1.47 Functions for Example 6.3.
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Figure 1.48 The graph of
f (x )  x  2 .
x 3
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Figure 1.49 The intermediate value property.
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Figure 1.50 The graph of
y  x2  x  1 
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1 .
x 1
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