Calculus II Quiz 2 May 19, 2010
Name: TA/研討課教室:
1. (10 pts) Given sin (xyz) =x+ 2y+ 3z, use implicit differentiation to find ∂z∂x.
2. (5 pts) Suppose thatg(t) =f(x(t), y(t)), wheref is a differentiable function ofx and y, and x=x(t) and y=y(t) both have first-order derivatives. Given that
x(1) = 1, y(1) = 2, x(2) = 3, y(2) = 4,
x′(1) = 5, y′(1) = 6, x′(2) = 7, y′(2) = 8, fx(1,2) = 7, fy(1,2) = 8, fx(3,4) = 1, fy(3,4) = 2, f(1,2) = 5, f(2,3) = 6, f(3,4) = 7, f(4,5) = 8, computeg′(2).
3. (10 pts) Given Y = uev, u = r+s, v = rs, use the Chain Rule to find the partial derivatives, ∂Y∂r, ∂Y∂s, when (r, s) = (1,0).
4. (15 pts) Givenf(x, y, z) =xe2yz,P(3,0,2) and⃗u=< 23,−23,13 >, (a) find the gradient of f,
(b) evaluate the gradient at the pointP,
(c) find the rate of change off atP in the direction of the vector⃗u.
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Name: Student ID number:
5. (a) (10 pts) Find the local maximum and minimum values and saddle points of the function,
f(x, y) = 2x2+ 3y2−8x−5.
(b) (10 pts) Use the Method of Lagrange Multiplier to find the absolute maxi- mum and minimum values of the function f(x, y) = 2x2+ 3y2−8x−5 subject to the constraintx2+y2 = 1.
(c) (10 pts) Find the absolute maximum and minimum values of the functionf(x, y) = 2x2+ 3y2−8x−5 subject to the constraint x2+y2≤1.
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Name: Student ID number:
6. (10 pts) Evaluate the iterated integral,
∫ 1
0
∫ x
0
1 + 2y dydx.
7. (10 pts) Evaluate the iterated integral by first changing the order of integration,∫ 1
0
∫ 1
y
ex2dxdy.
8. (10 pts) 求由兩曲面z=x2+y2 與z= 4−x2−y2 所圍之體積
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