(11/11/08)Math 10C. Lecture Examples.Section 14.4. Directional derivatives and gradient vectors in the plane†Example 1 (a) Find the directional derivative of f (x, y) = x2+ y2at (1, −1) in thedirection of the unit vector u= h12√2, −12√2i (Figure 1).(b) Why is it plausible that the directional derivative is positive?x−1 2y−1−21u =12√2, −12√21sFIGURE 1Answer: (a) Duf(1, −1) = 2√2 (b) f(x, y) = x2+y2is increasing in the direction of u at (1, −1) in Figure 1because its graph is a circular paraboloid that opens upward.Example 2 What is the derivative of f(x, y) = x2y5at P = (2, 1) in the direction towardQ = (4, 0)?Answer: Duf(2, 1) = −2√5Example 3 What is the derivative of h(x, y) = exyat (2,3) in the direction at an angleof23π radians from the positive x-direction?Answer: Figure A3 • u = h−12,12√3i • Duh(2, 3) = (−32+√3)e6xy23π11212√3uFigure A3†Lecture notes to accompany Section 14.4 of Calculus by Hughes-Hallett et al.1Math 10C. Lecture Examples. (11/11/08) Section 14.4, p. 2Example 4 Figure 2 shows level curves of the temperature T = T(x, y) (deg rees Celsius)of the surface of the ocean off the west coast of the United States at onetime.(1)Find the approximate rate of change of the temperature toward thenortheast at point P in the drawing.FIGURE 2 Figure A4Answer: One answer: Figure A4 • DuT (P ) ≈ −0.005 degrees per mileExample 5 Draw ∇f (1, 1), ∇f(−1, 2), and ∇f (−2, −1) for f (x, y) = x2y. Use the scale onthe x- and y-axes to measure the lengths of the arrows.Answer: ∇f (1, 1) = h2, 1i • ∇f(−1, 2) = h−4, 1i • ∇f(−2, −1) = h4, 4i • Figure A5Figure A5Example 6 (a) What is the maximum directional derivative of g(x, y) = y2e2xat (2, −1)and in t he dire ction o f what unit vector does it occur?(b) What is the minimum directional derivative of g at (2, −1) and in thedirection of what unit vector does it occur?Answer: (a) The maximum directional derivative is√8 e4and occurs in the direction of u =h1, −1i√2.(b) The minimum directional derivative is = −√8 e4and occurs in the direction of u =h−1, 1i√2.(1)Data adapted from Zoogeography of the Sea by S. Elkman, London: Sidgwich and Jackson, 1953, p. 144.Section 14.4, p. 3 Math 10C. Lecture Examples. (11/11/08)Example 7 Give the two unit vectors u such that the f unction z = g(x, y) of Example 6has zero derivatives at (2, −1) in the direction of u.Answer: The directional derivative is zero in the directions of u =h−1, −1i√2and u =h1, 1i√2.Example 8 (a) Draw the gradient vector of f (x, y) = xy at ( 1,2) and the level curve off through that point.(b) Draw ∇f (−3, 1) and the level curve of f through (−3, 1). Use the scaleson the axes to measure the components.Answer: (a) ∇f(1, 2) = h2, 1i • The level curve is y =2x• Figure A8a(b) ∇f(−3, 1) = h1, −3i • The level curve is y =−3x. • Figure A8bx1y2xy = 2∇f(1, 2) = h2, 1ixy1xy = −3−3∇f(−3, 1) = h1, −3iFigure A8a Figure A8bInteractive ExamplesWork the following Interactive Examples on Shenk’s web page, http//www.math.ucsd.edu/˜ashenk/:‡Section 14.5: Examples 1 through 6‡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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