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UCSD MATH 10C - Lecture Examples

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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 = P2(x, y) off(x, y) = 1 − cos x cos y at x = 0, y = 0. (The graphs of the two functionsare in Figures 1 and 2.)Answer: T2(x, y) =12x2+12y2xyzz = 1 − cos x cos yxyzz = P2(x, y)FIGURE 1 FIGURE 2Example 2 Figure 3 shows the graph of f = −x4− y4− 4xy +116and Figure 4 shows itslevel curves. Find its critical points and use the Second-Derivative Test toclassify them.FIGURE 3 FIGURE 4Answer: 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.1Math 10C. Lecture Examples. (11/25/08) Section 15.1, p. 2Example 3 Find the critical points of f = −2x3− 3y4+ 6xy2and 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 14Interactive ExamplesWork 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 sectionsof the textbook for the


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UCSD MATH 10C - Lecture Examples

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