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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Section 9.3 Partial Derivatives

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Recall - Contours and Level Curves

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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example

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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Boxes 1 and 2 on page 361 Example

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Visualizing

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Tangent Plane

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Notation and Rules

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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example from Instructors manual from 9.3 RECALL

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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Example from Instructors manual from 9.3 EXTENSION TO FUNCTIONS OF TWO VARIABLES The following table gives the number of calories burned per minute B = f(s,w) for someone who is rollerblading, as a function of the person's weight, w, and speed, s. w \ s8 mph9 mph10 mph11 mph 120 lbs4.25.87.48.9 140 lbs5.16.78.39.9 160 lbs6.17.79.210.8 180 lbs7.08.610.211.7 200 lbs7.99.511.112.6 Estimate f w (160,10) and f s (160,10) and interpret your answers.

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Use the level curves of f (x, y) in Figure 9.4 to estimate (a) f x (2, 1) (b) f y (1, 2) PROBLEMS

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Use the level curves of f (x, y) in Figure 9.4 to estimate (a) f x (2, 1) (b) f y (1, 2) PROBLEMS

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Applied Calculus, 3/E by Deborah Hughes-Hallet Copyright 2006 by John Wiley & Sons. All rights reserved. Box on page 363 and Problem 17 Approximations

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Problems 1 2 concern the contour diagram for a function f (x, y) in Figure 9.3. At the point P, which of the following is true? (a) f x > 0, f y > 0 (b) f x > 0, f y < 0 (c) f x 0 (d) f x < 0, f y < 0 PROBLEMS

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(a) (1, 0.5) (b) (0.4, 1.5) (c) (1.5, 0.4) (d) (0.5, 1) PROBLEMS (a) At (1, 0.5), the level curve is approximately horizontal, so f x 0, but f y 0. (b) At (0.4, 1.5), neither are approximately zero. (c) At (1.5, 0.4), neither are approximately zero. (d) At (0.5, 1), the level curve is approximately vertical, so f y 0, but f x 0. Figure 9.5 shows level curves of f (x, y). At which of the following points is one or both of the partial derivatives, f x, f y, approximately zero? Which one(s)?

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Figure 9.6 is a contour diagram for f (x, y) with the x and y axes in the usual directions. Is f x (P) positive, negative, or zero? Is f xx (P) positive, negative, or zero? Is f y (P) positive, negative, or zero? PROBLEMS At P, the values of f are increasing at an increasing rate as we move in the positive x- direction, so f x (P) > 0, and f xx (P) > 0. At P, the values of f are not changing as we move in the positive y-direction, so f y (P) = 0.

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Figure 9.7 is a contour diagram for f (x, y) with the x and y axes in the usual directions. At the point P, if x increases, what is true of f x (P)? If y increases, what is true of f y (P)? (a) Have the same sign and both increase. (b) Have the same sign and both decrease. (c) Have opposite signs and both increase. (d) Have opposite signs and both decrease. (e) None of the above. PROBLEMS We have f x (P) > 0 and decreasing, and f y (P) < 0 and decreasing, so the answer is (d).