number 39

nates at the origin. nat origi- 37. The temperature in a neighborhood of the point (47, 0) is given by the function vlibest at noisups eufT JTyZ h the T ( x, y ) = V2e- cos x. Find the path followed by a heat-seeking particle that origi- hich nates at the center of the neighborhood. of f 38. Determine the path of steepest descent along the surface nich f z = A+x+2y - x2 - 3y2 from the point (0, 0, A). f in 39. Set f(x, y) = 3x2+ y. (a) Find "Ox audT lim f ( x ( 2 + h), y (2 + h)) - f(2, 4) h - 0 h given that x(t) = t and y(t) = t2. (These functions parametrizationola y = x2.) able. (b) Find venby sig- lim f ( x ( 4 + h ), y ( 4 + h)) - f(2, 4) to smoa thi monde h - 0 bau h no ballco ad liver dor 0) + given that x(t) = =(t + 4) and y(t) = t. (These func- tions parametrization = 4x - 4.) (c) Calculate the directional derivative of f at (2, 4) in the direction of i + 4j. (d) Observe that i + 4j is a direction vector for the line y = 4x - 4 and that this line is tangent to the parabola y = x2 at (2, 4). Explain then why the computations in (a), (b), (c) yield different values. 40. According to Newton's law of gravitation, the force exerted on a particle of mass m located at the point (x, y, z) by a particle of mass M located at the origin is given by GMm F(x, y , Z) = 1 3 where r = xi + yj +zk,r = lrll, and G is the universal find gravitational constant. Show that F is the gradient of the the function on; GMm