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Chapter 3, Slide 1Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Finney Weir GiordanoFinney Weir Giordano
Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Chapter 3, Slide 2Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.1: How to classify maxima and minima.
Chapter 3, Slide 3Chapter 3. 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].
Chapter 3, Slide 4Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 3, Slide 5Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.7: A curve with a local maximum value. The slope at c, simultaneously the limit of nonpositive numbers and nonnegative numbers, is zero.
Chapter 3, Slide 6Chapter 3. 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.
Chapter 3, Slide 7Chapter 3. 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.
Chapter 3, Slide 8Chapter 3. 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)
Chapter 3, Slide 9Chapter 3. 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, ).
Chapter 3, Slide 10Chapter 3. 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)
Chapter 3, Slide 11Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
AIT p.253
Chapter 3, Slide 12Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.31: Graphical solutions from Example 2.
Chapter 3, Slide 13Chapter 3. 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)
Chapter 3, Slide 14Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.41: Population curves in Example 5.
Chapter 3, Slide 15Chapter 3. 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.)
Chapter 3, Slide 16Chapter 3. 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)
Chapter 3, Slide 17Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Figure 3.46: The graph of A = 2 r 2 + 2000/r is concave up.
Chapter 3, Slide 18Chapter 3. 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)
Chapter 3, Slide 19Chapter 3. 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.
Chapter 3, Slide 20Chapter 3. 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.
Chapter 3, Slide 21Chapter 3. 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.
Chapter 3, Slide 22Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 2001. Addison Wesley Longman All rights reserved.
Continued.
Chapter 3, Slide 23Chapter 3. 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.
Chapter 3, Slide 24Chapter 3. 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.
Chapter 3, Slide 25Chapter 3. 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.
Chapter 3, Slide 26Chapter 3. 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.
Chapter 3, Slide 27Chapter 3. 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.
Chapter 3, Slide 28Chapter 3. 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).
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