f(x, y)

18. The speed of sound traveling through ocean water is a functionof temperature, salinity, and pressure. It has beenmodeled by the functione = 1449.2 + 4.6T - 0.055T 2 + 0.00029T 3+ (1.34 - O.OIT)(S - 35) + 0.016Dwhere e is the speed of sound (in meters per second), T is thetemperature (in degrees Celsius), S is the salinity (the concentrationof salts in parts per thousand, which means the numberof grams of dissolved solids per 1000 g of water), and Dis the depth below the ocean surface (in meters). Computeae/aT, ae/as, and ae/aD when T = lOoC, S = 35 partsper thousand, and D = 100 m. Explain the physical significanceof these partial derivatives.19. f(x, y) = 4x 3 - xy221. f(x, y, z) = xkylzmaz az23. If z = xy + xe YI " show that x- + y- = xy + z.ax ayaz a 2 z az a 2 z---ax ax at at ax 225-29 Find equations of (a) the tangent plane and (b) the normalline to the given surface at the specified point.25. z = 3x 2 - y2 + 2x, (I, -2, I)26. z = eX cos y, (0, 0, I)27. x 2 + 2y2 - 3z 2 = 3, (2,-I, J)28. xy + yz + zx = 3, (1, I, I)29. sin(xyz) = x + 2y + 3z, (2, - 1,0)ffi 30. Use a computer to graph the surface z = x 2 + y4 and itstangent plane and normal line at (I, 1,2) on the same screen.Choose the domain and viewpoint so that you get a goodview of all three objects.31. Find the points on the hyperboloid x 2 + 4 y2 - Z2 = 4 wherethe tangent plane is parallel to the plane 2x + 2y + z = 5.35. If u = x 2 i + Z4, where x = p + 3p2, Y = peP, andz = p sin p, use the Chain Rule to find du/dp.36. If v = x 2 sin y + ye xy , where x = s + 2t and y = st, use theChain Rule to find av/as and avjat when s = 0 and t = 1.37. Suppose z = f(x, y), where x = g(s, t), Y = h(s, t),g(1,2) = 3, g,(1, 2) = -I, g,(l, 2) = 4, h(I, 2) = 6,h s (I,2) = -5, h,(1, 2) = 10,fx(3, 6) = 7, and Jy(3, 6) = 8.Find azjas and az/at when s = 1 and t = 2.38. Use a tree diagram to write out the Chain Rule for the casewhere w = f(t, u, v), t = t(p, q, r, s), u = u(p, q, r, s), andv = v(p, q, r, s) are all differentiable functions.39. If z = y + f(x 2 - y2), where f is differentiable, show thatazy-+x-=xax40. The length x of a side of a triangle is increasing at a rate of6 cm/s, the length y of another side is decreasing at a rate of4 cm/s, and the contained angle (}is increasing at a rate of0.05 radian/so How fast is the area of the triangle changingwhen x = 80 cm, y = 100 cm, and (}= n/6?41. If z = f(u, v), where u = xy, v = y/x, and f has continuoussecond partial derivatives, show thata 2 z a 2 z a 2 z azx 2 _- - y2 __ = -4uv-- + 2vax2 ay2 au av av4 ? 3 _ az az42. If yz + rz = e XY ', find - and -.ax ay43. Find the gradient of the function f(x, y, z) = z2exh.44. (a) When is the directional derivative of f a maximum?(b) When is it a minimum?(c) When is it O?(d) When is it half of its maximum value?45-46 Find the directional derivative of f at the given point inthe indicated direction.45. f(x, y) = 2JX - y2, (1,5),in the direction toward the point (4, I)46. f(x, y, z) = x 2 y + x~, (1,2,3),in the direction of v = 2i + j - 2kazay32. Find du if u = In(1 + se 2 ').33. Find the linear approximation of the functionf(x, y, z) = X\/y2 + Z2 at the point (2, 3, 4) and use itto estimate the number (1.98)3)(3.01)2 + (3.97)2.34. The two legs of a right triangle are measured as 5 m and12 m with a possible error in measurement of at most0.2 cm in each. Use differentials to estimate the maximumerror in the calculated value of (a) the area of the triangle and(b) the length of the hypotenuse.47. Find the maximum rate of change of f(x, y) = x 2 y + ,;;at the point (2, I). In which direction does it occur?48. Find the direction in which f(x, y, z) = ze xy increases mostrapidly at the point (0, 1,2). What is the maximum rate ofincrease?49. The contour map shows wind speed in knots during HurricaneAndrew on August 24, 1992. Use it to estimate the