Problem 1E: In Exercises 1–22, find and .
1.
Problem 2E: In Exercises 1–22, find and .
2.
Problem 3E: In Exercises 1–22, find and .
3.
Problem 4E: In Exercises 1–22, find and .
4.
Problem 5E: In Exercises 1–22, find and .
5.
Problem 6E: In Exercises 1–22, find and .
6.
Problem 7E: In Exercises 1–22, find and .
7.
Problem 8E: In Exercises 1–22, find and .
8.
Problem 9E: In Exercises 1–22, find and .
9.
Problem 10E: In Exercises 1–22, find and .
10.
Problem 11E: In Exercises 1–22, find and .
11.
Problem 12E: In Exercises 1–22, find and .
12.
Problem 13E: In Exercises 1–22, find and .
13.
Problem 14E: In Exercises 1–22, find and .
14.
Problem 15E: In Exercises 122, find f/x and f/y . 15. f(x,y)=ln(x+y) Problem 16E: In Exercises 1–22, find and .
16.
Problem 17E: In Exercises 1–22, find and .
17.
Problem 18E Problem 19E: In Exercises 1–22, find and .
19.
Problem 20E Problem 21E: In Exercises 1–22, find and .
21.
Problem 22E: In Exercises 1–22, find and .
22.
Problem 23E: In Exercises 23–34, find fx, fy, and fz.
23. f(x, y, z) = 1 + xy2 − 2z2
Problem 24E Problem 25E: In Exercises 23–34, find fx, fy, and fz.
25.
Problem 26E: In Exercises 23–34, find fx, fy, and fz.
26. f(x, y, z) = (x2 + y2 + z2)−1/2
Problem 27E: In Exercises 23–34, find fx, fy, and fz.
27. f(x, y, z) = sin−1 (xyz)
Problem 28E Problem 29E: In Exercises 23–34, find fx, fy, and fz.
29. f(x, y, z) = ln (x + 2y + 3z)
Problem 30E: In Exercises 23–34, find fx, fy, and fz.
30. f(x, y, z) = yz ln (xy)
Problem 31E: In Exercises 23–34, find fx, fy, and fz.
31.
Problem 32E: In Exercises 23–34, find fx, fy, and fz.
32. f(x, y, z) = e−xyz
Problem 33E: In Exercises 23–34, find fx, fy, and fz.
33. f(x, y, z) = tanh (x + 2y + 3z)
Problem 34E Problem 35E: In Exercises 35–40, find the partial derivative of the function with respect to each variable.
35.... Problem 36E: In Exercises 35–40, find the partial derivative of the function with respect to each variable.
36.... Problem 37E Problem 38E Problem 39E Problem 40E Problem 41E Problem 42E Problem 43E: Find all the second-order partial derivatives of the functions in Exercises 41–50.
43. g(x, y) = x2y... Problem 44E Problem 45E: Find all the second-order partial derivatives of the functions in Exercises 41–50.
45. r(x, y) = ln... Problem 46E Problem 47E Problem 48E: Find all the second-order partial derivatives of the functions in Exercises 4150. 48. w=yex2y Problem 49E: Find all the second-order partial derivatives of the functions in Exercises 41–50.
49.
Problem 50E Problem 51E: Find all the second-order partial derivatives of the functions in Exercises 41–50.
51.
Problem 52E Problem 53E Problem 54E: Find all the second-order partial derivatives of the functions in Exercises 41–50.
54.
Problem 55E: In Exercises 5560, verify that wxy=wyx . 55. w=ln(2x+3y) Problem 56E Problem 57E Problem 58E: In Exercises 55–60, verify that .
58.
Problem 59E: In Exercises 55–60, verify that .
59.
Problem 60E Problem 61E: Which order of differentiation enables one to calculate fxy. faster: x first or y first? Try to... Problem 62E Problem 63E Problem 64E Problem 65E Problem 66E Problem 67E Problem 68E Problem 69E Problem 70E Problem 71E: Exercises 71 and 72 are about the triangle shown here.
71. Express A implicitly as a function of a,... Problem 72E Problem 73E: Two dependent variables Express vx in terms of u and y if the equations x = v ln u and y = u ln v... Problem 74E Problem 75E: Let f(x, y) = 2x + 3y = 4. Find the slope of the line tangent to this surface at the point (2, −1)... Problem 76E Problem 77E: In Exercises 77-80, find a function z = f(x, y) whose partial derivatives are as given, or explain... Problem 78E: In Exercises 77-80, find a function z = f(x, y) whose partial derivatives are as given, or explain... Problem 79E: In Exercises 77-80, find a function z = f(x, y) whose partial derivatives are as given, or explain... Problem 80E: In Exercises 77-80, find a function z = f(x, y) whose partial derivatives are as given, or explain... Problem 81E: Let
Find fx, fy, fxy, and fyx, state the domain for each partial derivative.
Problem 82E: Let
Show that for all x, and for all y.
Show that
The three-dimensional Laplace equation
is... Problem 83E: Show that each function in Exercises 83-90 satisfies a Laplace equation.
83. f(x, y, z) = x2 + y2 ‒... Problem 84E: Show that each function in Exercises 83-90 satisfies a Laplace equation.
84. f(x, y, z) = 2z3 ‒ 3(x2... Problem 85E: Show that each function in Exercises 83-90 satisfies a Laplace equation.
85. f(x, y) = e−2y cos 2x
Problem 86E Problem 87E Problem 88E Problem 89E Problem 90E Problem 91E: Show that the functions in Exercises 91-97 are all solutions of the wave equation.
91. w = sin (x +... Problem 92E Problem 93E: Show that the functions in Exercises 91-97 are all solutions of the wave equation.
93. w = sin (x +... Problem 94E Problem 95E Problem 96E Problem 97E Problem 98E Problem 99E Problem 100E Problem 101E Problem 102E:
Show that fx(0, 0) and fy(0, 0) exist, but f is not differentiable at (0, 0).
Problem 103E Problem 104E format_list_bulleted