13.7. Maximum and Minimum Values
Ex. 1
[[x == 1, y == 3]] [[x == 1, y == 3]] |
2 0 2 4 2 0 2 4 |
4 4 |
Ex. 2
[[x == 0, y == 0]] [[x == 0, y == 0]] |
-2 0 2 -4 -2 0 2 -4 |
Sleeping...
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Ex. 3
[[x == I*(-1)^(1/4), y == -I*(-1)^(3/4)], [x == -(-1)^(1/4), y == -(-1)^(3/4)], [x == -I*(-1)^(1/4), y == I*(-1)^(3/4)], [x == (-1)^(1/4), y == (-1)^(3/4)], [x == -I, y == I], [x == I, y == -I], [x == -1, y == -1], [x == 1, y == 1], [x == 0, y == 0]] [[x == I*(-1)^(1/4), y == -I*(-1)^(3/4)], [x == -(-1)^(1/4), y == -(-1)^(3/4)], [x == -I*(-1)^(1/4), y == I*(-1)^(3/4)], [x == (-1)^(1/4), y == (-1)^(3/4)], [x == -I, y == I], [x == I, y == -I], [x == -1, y == -1], [x == 1, y == 1], [x == 0, y == 0]] |
0 -4 0 -16 0 -4 0 -16 |
12 -4 12 128 -1 12 -4 12 128 -1 |
12 -4 12 128 -1 12 -4 12 128 -1 |
Ex. 5
[z == -x - 2*y + 4] [z == -x - 2*y + 4] |
[[x == (11/6), y == (5/3)]] [[x == (11/6), y == (5/3)]] |
4 4 10 24 5/6*sqrt(6) 4 4 10 24 5/6*sqrt(6) |
Ex. 6
[z == -1/2*(x*y - 12)/(x + y)] [z == -1/2*(x*y - 12)/(x + y)] |
[[x == 0, y == 0], [x == 2*I*sqrt(3), y == -2*I*sqrt(3)], [x == -2*I*sqrt(3), y == 2*I*sqrt(3)], [x == -2, y == -2], [x == 2, y == 2]] [[x == 0, y == 0], [x == 2*I*sqrt(3), y == -2*I*sqrt(3)], [x == -2*I*sqrt(3), y == 2*I*sqrt(3)], [x == -2, y == -2], [x == 2, y == 2]] |
-1 -1/2 -1 3/4 4 -1 -1/2 -1 3/4 4 |
Ex. 7
[[x == 1, y == 1]] [[x == 1, y == 1]] |
x^2 -4*y + 9 x^2 - 4*x + 4 2*y x^2 -4*y + 9 x^2 - 4*x + 4 2*y |
1 0 9 1 0 4 1 0 9 1 0 4 |
Exercise 2
[[x == -2, y == 4]] [[x == -2, y == 4]] |
x^2 x^2 + 5*x - 5 y^2 - 9*y + 9 y^2 - 3*y + 9 x^2 x^2 + 5*x - 5 y^2 - 9*y + 9 y^2 - 3*y + 9 |
-12 9 9 0 -45/4 19 -45/4 -11 27/4 -12 9 9 0 -45/4 19 -45/4 -11 27/4 |
14.1. Double Integrals over Rectangles
Ex. 1
46.780 46.780 |
Ex. 3
by Marshall Hampton
Click to the left again to hide and once more to show the dynamic interactive window |
Ex. 4
27/2 27/2 |
27/2 27/2 |
Ex. 5
-12 -12 |
Ex. 6
0 0 |
-1/2*pi*Ei(2*I*pi) + 1/2*pi*Ei(I*pi) + 1/2*pi*Ei(-I*pi) - 1/2*pi*Ei(-2*I*pi) + 1/2*pi*gamma(-1, 2*I*pi) - 1/2*pi*gamma(-1, I*pi) - 1/2*pi*gamma(-1, -I*pi) + 1/2*pi*gamma(-1, -2*I*pi) -1/2*pi*Ei(2*I*pi) + 1/2*pi*Ei(I*pi) + 1/2*pi*Ei(-I*pi) - 1/2*pi*Ei(-2*I*pi) + 1/2*pi*gamma(-1, 2*I*pi) - 1/2*pi*gamma(-1, I*pi) - 1/2*pi*gamma(-1, -I*pi) + 1/2*pi*gamma(-1, -2*I*pi) |
Ex. 7
48 48 |
Ex. 8
1 1 1 1 |
14.2. Double Integrals over General Regions
Ex. 1
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[[x == -1, y == 2], [x == 1, y == 2]] [[x == -1, y == 2], [x == 1, y == 2]] |
32/15 32/15 |
Ex. 2
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[[x == 2, y == 4], [x == 0, y == 0]] [[x == 2, y == 4], [x == 0, y == 0]] |
216/35 216/35 |
Ex. 3
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[[x == -1, y == -2], [x == 5, y == 4]] [[x == -1, y == -2], [x == 5, y == 4]] |
[x == 1/2*y^2 - 3] [x == 1/2*y^2 - 3] |
[x == y + 1] [x == y + 1] |
36 36 |
Ex. 4
Sleeping...
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[[x == 1, y == (1/2)]] [[x == 1, y == (1/2)]] |
[y == -1/2*x + 1] [y == -1/2*x + 1] |
[z == -x - 2*y + 2] [z == -x - 2*y + 2] |
1/3 1/3 |
Ex. 5
-1/2*cos(1) + 1/2 -1/2*cos(1) + 1/2 |
Ex.
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[x == -1, x == 1, x == 0] [x == -1, x == 1, x == 0] |
-1/2 -1/2 |
Exercise 1
1/8 1/8 |
Exercise 2
Sleeping...
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[[y == 1, z == 3]] [[y == 1, z == 3]] |
[[x == -1, y == 1], [x == 1, y == 1]] [[x == -1, y == 1], [x == 1, y == 1]] |
16/15 16/15 |
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