copyright © 2000 by the mcgraw-hill companies, inc. c h a p t e r 2 differentiation: basic concepts

Copyright © 2000 by the McGraw-Hill Companies, Inc.

C H A P T E R 2

Differentiation:Basic Concepts


Figure 2.1 The graph of s = 16t2.

(a) The secant line through P(2, 64) and Q(2 + h, 16(2 + h)2).

(b) As h0, the secant line PQ tends toward the tangent line at P.

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Figure 2.2 Secant lines approximatinga tangent line.

(a) The graph of f(x) with a secant line through points P(x, f(x)) and Q(x + h, f(x + h)).

(b) As h0 the secant lines tend toward the tangent line at P.

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Figure 2.3 Inflation as a functionof unemployment

Source: Adapted from Robert Eisner, The Misunderstood Economy: What Counts and How to Count It, Boston, MA: Harvard Business School Press, 1994, page 173.

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Figure 2.4 The graph of y = x3.

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Figure 2.5 The graph of R(x) = 0.5x2 + 3x – 2,for x 0.

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Figure 2.6 Three functions that are not differentiable at (0, 0). (a) The graph has a gap at x = 0. (b) There is a sharp

“corner” at (0, 0). (c) There is a “cusp” at (0, 0).

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Figure 2.7 The graph of f(x) = c.

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Figure 2.8 The motion of a ball thrown upward from the top of a building.

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Figure 2.9 Marginal cost MC(x0) approximatesC(x0 + 1) – C(x0).

(a) The marginal cost MC(x0) at x = x0 is C’(x0).

(b) The cost of producing the (x0 + 1)th unit is C(x0 + 1) – C(x0).

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Figure 2.10 The graph of the profit function

display function .98222411

)( 2 xxxP

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Figure 2.11 Approximation of y by the differential dy.

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Figure 2.12 The graph of the circle x2 + y2 = 25.

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Figure 2.13 The graph of the equationx2 – y2 = 2x + 2y.

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Figure 2.14 A ladder moving down a wall.

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copyright © 2000 by the mcgraw-hill companies, inc. c h a p t e r 2 differentiation: basic concepts

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