Problem 1E: At the beginning of this section we discussed the function I = f(T, H), where I is the heat index, T... Problem 2E Problem 3E Problem 4E Problem 5E Problem 6E: A contour map is given for a function f. Use it to estimate fx(2, 1) and fy(2, 1). Problem 7E: If f(x, y) = 16 4x2 y2, find fx(1, 2) and fy(1, 2) and interpret these numbers as slopes.... Problem 8E: If f(x,y)=4x24y2, find fx(1,0) and fy(1, 0) and interpret these numbers as slopes. Illustrate with... Problem 9E Problem 10E Problem 11E Problem 12E Problem 13E Problem 14E Problem 15E Problem 16E Problem 17E Problem 18E Problem 19E Problem 20E Problem 21E Problem 22E Problem 23E Problem 24E Problem 25E Problem 26E Problem 27E Problem 28E Problem 29E Problem 30E Problem 31E Problem 32E Problem 33E Problem 34E Problem 35E Problem 36E Problem 37E Problem 38E: Find the indicated partial derivative. 42. f(x,y)=ysin1(x,y);fy(1,12) Problem 39E: Find the indicated partial derivative. 43. f(x,y,z)=ln1x2+y2+z21+x2+y2+z2;fy(1,2,2) Problem 40E: Find the indicated partial derivative. 44. f(x, y, z) = xyz; fz(e, 1, 0) Problem 41E: Use implicit differentiation to find z/x and z/y . 41. x2+2y2+3z2=1 Problem 42E: Use implicit differentiation to find z/x and z/y . 42. x2y2+z22z=4 Problem 43E: Use implicit differentiation to find z/x and z/y . 43. ez=xyz Problem 44E: Use implicit differentiation to find z/x and z/y . 44. yz+xlny=z2 Problem 45E: Find z/x and z/y. 51. (a) z = f(x) + g(y) (b) z = f(x + y) Problem 46E: Find z/x and z/y. 52. (a) z = f(x)g(y) (b) z = f(xy) (c) z = f(x/y) Problem 47E Problem 48E: Find all the second partial derivatives. 54. f(x, y) = ln(ax + by) Problem 49E: Find all the second partial derivatives. 55. z=y2z+3y Problem 50E Problem 51E Problem 52E Problem 53E: Verify that the conclusion of Clairauts Theorem holds, that is, uxy = uyx. 59. u = x4y3 y4 Problem 54E: Verify that the conclusion of Clairauts Theorem holds, that is, uxy = uyx. 60. u = exy sin y Problem 55E: Verify that the conclusion of Clairauts Theorem holds, that is, uxy = uyx. 61. u = cos(x2y) Problem 56E: Verify that the conclusion of Clairauts Theorem holds, that is, uxy = uyx. 62. u = ln(x + 2y) Problem 57E: Find the indicated partial derivative(s). 63. f(x, y) = x4y2 x3y; fxxx, fxyx Problem 58E: Find the indicated partial derivative(s). 64. f(x, y) = sin(2x + 5y); fyxy Problem 59E: Find the indicated partial derivative(s). 65. f(x,y,z)=exyz2;fxyz Problem 60E Problem 61E: Find the indicated partial derivative(s). 67. w=u+v2;3Wu2v Problem 62E Problem 63E: Find the indicated partial derivative(s). 69. w=xy+2z;3wzyx,3wx2y Problem 64E: Find the indicated partial derivative(s). 70. u = xaybzc; 6uxy2z3 Problem 65E: Use Definition 4 to find fx(x,y) and fy(x,y) . 65. f(x,y)=xy2x3y Problem 66E: Use Definition 4 to find fx(x,y) and fy(x,y) . 66. f(x,y)=xx+y2 Problem 67E: If f(x,y,z)=xy2z3+arcsin(xz), find fxzy. [Hint: Which order of differentiation is easiest?] Problem 68E: If g(x,y,z)=1+xz+1xy, find gxyz. [Hint: Which order of differentiation is easiest?] Problem 69E Problem 70E Problem 71E Problem 72E Problem 73E: Use the table of values of f(x, y) to estimate the values of fx(3, 2), fx(3, 2.2),and fxy(3, 2). Problem 74E Problem 75E Problem 76E: If u=ea1x1+a2x2++anxn, where a12+a22++an2=1, show that 2ux12+2ux22++2uxn2=u Problem 77E: Show that the function u=u(x,t) is a solution of the wave equation utt=a2uxx . (a) u=sin(kx)sin(akt)... Problem 78E Problem 79E: Verify that the function u=1/x2+y2+z2 is a solution of the three-dimensional Laplace equation uxx +... Problem 80E Problem 81E: The Diffusion Equation The diffusion equation ct=D2cx2 where D is a positive constant, describes the... Problem 82E Problem 83E Problem 84E Problem 85E: Show that the Cobb-Douglas production function satisfies P(L, K0) = C1(K0)L by solving the... Problem 86E Problem 87E Problem 88E Problem 89E: In the project following Section 4.7 we expressed the power needed by a bird during its flapping... Problem 90E Problem 91E Problem 92E Problem 93E: The ellipsoid 4x2+2y2+z2=16 intersects the plane y=2 in an ellipse. Find parametric equations for... Problem 94E Problem 95E Problem 96E: If a, b, c are the sides of a triangle and A, B, C are the opposite angles, find A/a, A/b, A /c by... Problem 97E Problem 98E Problem 99E: If f(x,y)=x(x2+y2)3/2esin(x2y) find fx(1,0). [Hint: Instead of finding fx(x, y) first, note that... Problem 100E: If f(x,y)=x3+y33 find fx(0, 0). Problem 101E Problem 1DP Problem 2DP Problem 3DP Problem 4DP Problem 5DP format_list_bulleted