Math241 D1 Quiz#6 July 7, 2011
Name: Answer Key
1.[10pts] Use Lagrange multipliers to find the maximum and minimum values of the
function subject to the given constraint.
f(x, y) = x2y;x2+ 2y2= 24
Sol) Let g(x, y) = x2+ 2y2.
∇f=λ∇g, (1)
<2xy, x2>=λ < 2x, 4y > (2)
2xy = 2λx (3)
x2= 4λy (4)
xy =λx
x2= 4λy
x2+ 2y2= 24
The first equation implies that x= 0 or y=λ. If x= 0 then by plugging in the third
equation y=±2√6. If x6= 0 the y=λ....
2.[10pts] Calculate the double integral
ZZR
√1−r2rdA, R : 0 ≤r≤1,0≤θ≤2π.
Sol)
ZZR
√1−r2rdA =Z2π
0Z1
0
√1−r2rdrdθ (5)
=Z2π
0·−1
3(1 −r2)3
2¸1
0
dθ (6)
=... (7)
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