chapter 3et, slide 1 chapter 3 et. finney weir giordano, thomas’ calculus, tenth edition © 2001....
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Chapter 3ET, Slide 1Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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Chapter 3ET, Slide 2Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.1: How to classify maxima and minima.
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Chapter 3ET, Slide 3Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.4: Some possibilities for a continuous function’s maximum and minimum on a closed interval [a, b].
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Chapter 3ET, Slide 4Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
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Chapter 3ET, Slide 5Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.6: A curve with a local maximum value. The slope at c, simultaneously the limit of nonpositive numbers and nonnegative numbers, is zero.
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Chapter 3ET, Slide 6Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.13: Geometrically, the Mean Value Theorem says that somewhere between A and B the curve has at least one tangent parallel to chord AB.
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Chapter 3ET, Slide 7Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.14: The chord AB is the graph of the function g(x). The function h(x) = ƒ(x) – g(x) gives the vertical distance between the graphs of f and g at x.
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Chapter 3ET, Slide 8Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.21: The graph of ƒ(x) = x3 – 12x – 5. (Example 1)
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Chapter 3ET, Slide 9Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.24: The graph of f (x) = x3 is concave down on (–, 0) and concave up on (0, ).
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Chapter 3ET, Slide 10Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.30: The graph of f (x) = x4 – 4x3 + 10. (Example 10)
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Chapter 3ET, Slide 11Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
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Chapter 3ET, Slide 12Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.31: Graphical solutions from Example 2.
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Chapter 3ET, Slide 13Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.40: The completed phase line for logistic growth. (Equation 6)
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Chapter 3ET, Slide 14Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.41: Population curves in Example 5.
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Chapter 3ET, Slide 15Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.42: Logistic curve showing the growth of yeast in a culture. The dots indicate observed values. (Data from R. Pearl, “Growth of Population.” Quart. Rev. Biol. 2 (1927): 532-548.)
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Chapter 3ET, Slide 16Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.43: An open box made by cutting the corners from a square sheet of tin. (Example 1)
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Chapter 3ET, Slide 17Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.46: The graph of A = 2r 2 + 2000/r is concave up.
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Chapter 3ET, Slide 18Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.48: A light ray refracted (deflected from its path)as it passes from one medium to another. (Example 4)
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Chapter 3ET, Slide 19Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.51: The graph of a typical cost function starts concave down and later turns concave up. It crosses the revenue curve at the break-even point B. To the left of B, the company operates at a loss. To the right, the company operates at a profit, with the maximum profit occurring where c´(x) = r´(x). Farther to the right, cost exceeds revenue (perhaps because of a combination of rising labor and material costs and market saturation) and production levels become unprofitable again.
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Chapter 3ET, Slide 20Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.53: The average daily cost c(x) is the sum of a hyperbola and a linear function.
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Chapter 3ET, Slide 21Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.54: The more we magnify the graph of a function near a point where the function is differentiable, the flatter the graph becomes and the more it resembles its tangent.
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Chapter 3ET, Slide 22Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
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Chapter 3ET, Slide 23Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.59: Approximating the change in the function f by the change in the linearization of f.
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Chapter 3ET, Slide 24Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.61: Newton’s method starts with an initial guess x0
and (under favorable circumstances) improves the guess one step at a time.
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Chapter 3ET, Slide 25Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.62: The geometry of the successive steps of Newton’s method. From xn, we go up to the curve and follow the tangent line down to find xn–1.
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Chapter 3ET, Slide 26Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.68: Newton’s method fails to converge. You go from x0 to x1 and back to x0, never getting any closer to r.
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Chapter 3ET, Slide 27Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
x
Figure 3.69: If you start too far away, Newton’s method may miss the root you want.
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Chapter 3ET, Slide 28Chapter 3 ET. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.70: (a) Starting values in (–2/2), (–21/7, 21/7), and (2/2, lead respectively to roots A, B, and C. (b) The values x = ± 21/7 lead only to each other. (c) Between 21/7 and 2/2, there are infinitely many open intervals of points attracted to A alternating with open intervals of points attracted to C. This behavior is mirrored in the interval (–2/2, –21/7).