Chapter 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.

Figure 3.4: Some possibilities for a continuous function’s maximum and minimum on a closed interval [a, b].

Continued.

Figure 3. 6: A curve with a local maximum value 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.

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.

Figure 3. 14: The chord AB is the graph of the function g(x) 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.

Figure 3.21: The graph of ƒ(x) = x3 – 12x – 5. (Example 1)

Figure 3.24: The graph of f (x) = x3 is concave down on (–, 0) and concave up on (0, ).

Figure 3.30: The graph of f (x) = x4 – 4x3 + 10. (Example 10)

Figure 3.31: Graphical solutions from Example 2.

Figure 3. 40: The completed phase line for logistic growth Figure 3.40: The completed phase line for logistic growth. (Equation 6)

Figure 3.41: Population curves in Example 5.

Figure 3. 42: Logistic curve showing the growth of yeast in a culture 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.)

Figure 3.43: An open box made by cutting the corners from a square sheet of tin. (Example 1)

Figure 3.46: The graph of A = 2r 2 + 2000/r is concave up.

Figure 3.48: A light ray refracted (deflected from its path) as it passes from one medium to another. (Example 4)

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.

Figure 3.53: The average daily cost c(x) is the sum of a hyperbola and a linear function.

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.

Continued.

Figure 3.59: Approximating the change in the function f by the change in the linearization of f.

Figure 3.61: Newton’s method starts with an initial guess x0 and (under favorable circumstances) improves the guess one step at a time.

Figure 3. 62: The geometry of the successive steps of Newton’s method 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.

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.

Figure 3.69: If you start too far away, Newton’s method may miss the root you want. x

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).