(11/11/08)Math 10C. Lecture Examples.Section 14.7. Second-order partial derivatives†Example 1 What are the second-order derivatives of f(x, y) = xy2+ x3y5?Answer: fxx= 6xy5• fxy= fyx= 2y + 15x2y4• fyy= 2x + 20x3y3Example 2 Find the second-degree Taylor polynomial approximation off(x, y) = 1 − cos x cos y at x = 0, y = 0.Answer: P2(x, y) =12x2+12y2(The graph of f(x, y) = 1 − cos x cos y is in Figure 1, and the graph of itsTaylor polynomial approximation is the circular paraboloid shown in Figure 2 with the graph of f.)xyzz = 1 − cos x cos yxyzz = P2(x, y)FIGURE 1 FIGURE 2Example 3 Figure 3 shows the graph of f(x, y) = 5 − x + y − (x − 1)4− (y − 1)4andFigure 25 shows the graph of f with the graph of its second-degree Taylorpolynomial P2(x, y) at x = 1, y = 1. Give a formula for P2(x, y). What typeof surface is its graph?xyzz = 5 − x + y − (x − 1)4− (y − 1)4xyzz = P2(x, y)FIGURE 3 FIGURE 4Answer: P2(x, y) = 5 − x + y • The graph of P2is a plane.†Lecture notes to accompany Section 14.7 of Calculus by Hughes-Hallett et al.1Math 10C. Lecture Examples. (11/11/08) Section 14.7, p. 2Example 4 Figure 5 shows the graph of f(x, y) = y2−112y4− x, and Figure 6 shows itsgraph with the graph of its second-degree Taylor polynomial z = P2(x, y) atx = 0, y = 0. Give a formula for P2(x, y).xyzz = y2−112y4− xxyzz = P2(x, y)FIGURE 5 FIGURE 6Answer: P2(x, y) = y2−
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