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Copyright © 2000 by the McGraw-Hill Companies, Inc.
C H A P T E R 3
Additional Applicationsof the Derivative
Copyright © 2000 by the McGraw-Hill Companies, Inc.
Figure 3.1 Military expenditure of former Soviet bloc countries as a percentage of GDP.
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Figure 3.2 Increasing and decreasing functions.
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Figure 3.3 The graph of f(x) = 2x3 + 3x2 – 12x - 7.
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Figure 3.4 The graph of .2
)(2
xxxf
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Figure 3.5 A graph with various kindsof “peaks” and “valleys.”
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Figure 3.6 Three critical points where f’(x) = 0: (a) relative maximum, (b) relative minimum, and
(c) not a relative extremum.
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Figure 3.7 Three critical points where f’(x) is undefined: (a) relative maximum, (b) relative minimum, and (c) not a relative extremum.
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Figure 3.8 The graph of f(x) = x4 + 8x3 + 18x2 – 8.
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Figure 3.9 The graph of g(t) = .23 2tt
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Figure 3.10 The graph of R(x) =
for 0 x 63.63
632
2
xxx
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Figure 3.11 The output of a factory worker.
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Figure 3.12 Concavity and the slopeof the tangent.
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Figure 3.13 Possible combinations of increase, decrease, and concavity.
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Figure 3.14 The graph of f(x) = x4.
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Figure 3.15 The graph off(x) = 3x4 – 2x3 – 12x2 + 18x + 15.
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Figure 3.16 A possible graph of f(x).
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Figure 3.17 The second derivative test.
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Figure 3.18 The graph of f(x) = 2x3 – 3x2 – 12x – 7.
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Figure 3.19 Three functions whose first and second derivatives are zero at x = 0.
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Figure 3.20 The production of an average worker.
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Figure 3.21 A graphical illustrationof limits involving infinity.
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Figure 3.22 The graph of f(x) = .21
xx
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Figure 3.23 The graph of f(x) = .39
2
2
xxx
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Figure 3.24 The graph of f(x) = .152
32
2
xxx
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Figure 3.25 The graph of f(x) = .)1( 2x
x
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Figure 3.26 The graph of f(x) = .152
32
2
xxx
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Figure 3.27 The average cost .7553)(q
qqA
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Figure 3.28 Absolute extrema.
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Figure 3.29 Absolute extrema of a continuous function on a closed interval: (a) the absolute maximum coincides with a relative maximum,
(b) the absolute maximum occurs at an endpoint,(c) the absolute minimum coincides with a relative minimum,
and (d) the absolute minimum occurs at an endpoint.
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Figure 3.30 The absolute extrema ofy = 2x3 + 3x2 – 12x – 7 on –3 x 0.
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Figure 3.31 Traffic speedS(t) = t3 – 10.5t
2 + 30t + 20.
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Figure 3.32 The speed of air during a coughS(r) = ar
2(r0 – r).
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Figure 3.33 Extrema of functions on unbounded intervals: (a) no absolute maximum for x > 0 and
(b) no absolute minimum for x 0.
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Figure 3.34 The function f(x) = x2 +
on the interval x > 0.x
16
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Figure 3.35 The relative minimum is notthe absolute minimum because of the
effect of another critical point.
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Figure 3.36 Graphs of profit, average cost, and marginal cost for Example 4.5.
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Figure 3.37 Rectangular picnic area.
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Figure 3.38 The graph of F(x) = x +
For x > 0.x000,10
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Figure 3.39 A cylinder of radius r and height hhas lateral (curved) area A = 2rh
and volume V = r2h.
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Figure 3.40 The cost function
for r > 0. tVrrC 46)( 2
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3 3/ V
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Figure 3.41 The profit functionP(x) = 400(15 – x)(x – 2).
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Figure 3.42 Relative positions of factory, river, and power plant.
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Figure 3.43 Two choices for the variable x.
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Figure 3.44 The revenue functionR(x) = (35 + x)(60 – x).
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Figure 3.45 Inventory graphs: (a) actual inventory
graph and (b) constant inventory of tires.2x
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Figure 3.46 Total cost
C(x) = 0.48x + .500,34000,120 x
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Figure 3.47 Elasticity in relationto a revenue curve.
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