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Homework Homework Assignment #25 Read Section 4.5 Page 243, Exercises: 1 – 57 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 1. Match the graphs in Figure 12 with the description. (b) (c) (a) (d) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 5. If Figure 14 is the graph of the derivative of f ′(x), where do the points of inflection of f (x) occur, and on which interval is f (x) concave down? Points of inflection occur at a. and b. the graph of f (x) is concave down on [d, f]. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Determine the intervals on which the function is concave up or down and find the points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Determine the intervals on which the function is concave up or down and find the points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 13.Continued The function is concave up on (–∞, –1) and (1, ∞) and concave down on (–1, 1). Points of inflection are at x = {–1, 1}. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Determine the intervals on which the function is concave up or down and find the points of inflection. The function is concave up on (–∞, –4.236) and (0.236, ∞) and concave down on (–4.236, 0.236). Points of inflection are at x = {–4.236, 0.236}. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 21. The growth of a sunflower during its first 100 days is modeled well by the logistic curve y = h (t) in Figure 15. Estimate the growth rate at the point of inflection and explain its significance. Then make a rough sketch of the first and second derivatives of h (t). The growth rate at the point of inflection appears to be about 7 cm/day. It is the greatest rate of growth as the second derivative goes from + to – at that point. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 29. Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the critical points of f (x) and use the Second Derivative Test to determine whether each corresponds to a local minimum or maximum. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the intervals on which f is concave up or down, the points of inflection, and the critical points and determine whether each critical point corresponds to a local maximum or minimum (or neither). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 37.Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the intervals on which f is concave up or down, the points of inflection, and the critical points and determine whether each critical point corresponds to a local maximum or minimum (or neither). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the intervals on which f is concave up or down, the points of inflection, and the critical points and determine whether each critical point corresponds to a local maximum or minimum (or neither). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 Find the intervals on which f is concave up or down, the points of inflection, and the critical points and determine whether each critical point corresponds to a local maximum or minimum (or neither). Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 49. Continued Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework, Page 243 52. Water is pumped into a sphere at a variable rate in such a way that the water level rises at a constant rate c. Let V (t) be the volume at time t. Sketch the graph of V (t). Where does the point of inflection occur? Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The point of inflection occurs at the time when the height of the water equals the radius of the tank.

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Homework, Page 243 Sketch the graph of a function satisfying the given condition. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Jon Rogawski Calculus, ET First Edition Chapter 4: Applications of the Derivative Section 4.5: Graph Sketching and Asymptotes

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Graphs of functions that are at least twice differentiable are made up of segments shown in Figure 1. The keys to hand sketching are finding the transition points and selecting the correct curve shape.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company The text uses solid dots to indicate local extrema and solid squares to indicate transition points, as shown in Figure 2.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Figure 4 shows the sign combinations of f ′ and f ″ for Another way to represent this information is:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Calculating the values of f at the critical points and point of inflection gives us sufficient information to sketch the graph.

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Sketch the graph of the function. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company Asymptotic behavior refers to the behavior of a function as either x or f (x) approaches ±∞. A horizontal line y = L is called a horizontal asymptote if either of the following exists: Similarly, a vertical line y = L is called a vertical asymptote if either of the following exists:

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Sketch the graph over the given interval. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Sketch the graph over the given interval. Indicate the transition points. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Calculate the following limits. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Calculate the limit. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Example, Page 256 Sketch the graph of the function. Indicate the asymptotes, local extrema, and points of inflection. Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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Homework Homework Assignment #26 Read Section 4.6 Page 256, Exercises: 1 – 89 (EOO) Rogawski Calculus Copyright © 2008 W. H. Freeman and Company

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