Chapter 3, Slide 1 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Finney Weir Giordano Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved.

Chapter 3, Slide 2 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.1: How to classify maxima and minima.

Chapter 3, Slide 3 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 4 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Continued.

Chapter 3, Slide 5 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 6 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 7 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 8 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.21: The graph of ƒ(x) = x 3 – 12x – 5. (Example 1)

Chapter 3, Slide 9 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.24: The graph of f (x) = x 3 is concave down on (– , 0) and concave up on (0,  ).

Chapter 3, Slide 10 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.30: The graph of f (x) = x 4 – 4x (Example 10)

Chapter 3, Slide 11 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. AIT p.253

Chapter 3, Slide 12 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.31: Graphical solutions from Example 2.

Chapter 3, Slide 13 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.40: The completed phase line for logistic growth. (Equation 6)

Chapter 3, Slide 14 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.41: Population curves in Example 5.

Chapter 3, Slide 15 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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): )

Chapter 3, Slide 16 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 17 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.46: The graph of A = 2  r /r is concave up.

Chapter 3, Slide 18 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 19 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 20 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 21 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 22 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Continued.

Chapter 3, Slide 23 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 24 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.61: Newton’s method starts with an initial guess x 0 and (under favorable circumstances) improves the guess one step at a time.

Chapter 3, Slide 25 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.62: The geometry of the successive steps of Newton’s method. From x n, we go up to the curve and follow the tangent line down to find x n–1.

Chapter 3, Slide 26 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © Addison Wesley Longman All rights reserved. Figure 3.68: Newton’s method fails to converge. You go from x 0 to x 1 and back to x 0, never getting any closer to r.

Chapter 3, Slide 27 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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 28 Chapter 3. Finney Weir Giordano, Thomas’ Calculus, Tenth Edition © 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).