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HH_15_1

# HH_15_1 - f =-2x 3-3y 4 6xy 2 and use the Second-Derivative...

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(11/25/08) Math 10C. Lecture Examples. Section 15.1. Local extrema Example 1 Find the second-degree Taylor polynomial approximation y = P 2 ( x , y ) of f ( x , y ) = 1 - cos x cos y at x = 0 , y = 0 . (The graphs of the two functions are in Figures 1 and 2.) Answer: T 2 ( x,y ) = 1 2 x 2 + 1 2 y 2 x y z z = 1 - cos x cos y x y z z = P 2 ( x,y ) FIGURE 1 FIGURE 2 Example 2 Figure 3 shows the graph of f = - x 4 - y 4 - 4xy + 1 16 and Figure 4 shows its level curves. Find its critical points and use the Second-Derivative Test to classify them. FIGURE 3 FIGURE 4 Answer: f has a saddle point at (0 , 0) and local maxima at (1 , - 1) and at ( - 1 , 1). Lecture notes to accompany Section 15.1 of Calculus by Hughes-Hallett et al. 1

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Math 10C. Lecture Examples. (11/25/08) Section 15.1, p. 2 Example 3 Find the critical points of
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Unformatted text preview: f =-2x 3-3y 4 + 6xy 2 and use the Second-Derivative Test to classify them. Answer: The function has local maxima at (1 , 1) and (1 ,-1). • The Second-Derivative Test fails at (0 , 0). The graph of the function of Example 3 is in Figure 13 and its level curves are in Figure 14. FIGURE 13 FIGURE 14 Interactive Examples Work the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/: ‡ Section 15.2: Examples 1–3 ‡ The chapter and section numbers on Shenk’s web site refer to his calculus manuscript and not to the chapters and sections of the textbook for the course....
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HH_15_1 - f =-2x 3-3y 4 6xy 2 and use the Second-Derivative...

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