MIDTERM 2 FOR CALCULUS
Time: 8:15–9:55 AM, Friday, June 1, 2001
No calculator is allowed. No credit will be given for an answer without reasoning.
1.
(i) [5%] Suppose that (0, 2) is a critical point of a function g with continuous second derivatives.
Suppose that gxx(0, 2) = −1, gxy(0, 2) = 2 and gyy(0, 2) = −8. Use second derivative test to classify the critical point (0, 2).
(ii) [5%] Find an equation of the tangent plane to the surface z = exln y at the point (3, 1, 0).
2. [10%] Let u = x + at and v = x − at. Then use chain rule to show that any differentiable function of the form
z = f (x + at) + g(x − at) is a solution of the wave equation
∂2z
∂t2 = a2∂2z
∂x2.
3. [10%] Find the directional derivative of the function g(x, y, z) = z3− x2y at the point (1, 6, 2) in the direction v = 3i + 4j + 12k.
4. [20%] Find the extreme values of the function f (x, y) = e−xy on the region x2+ 4y2≤ 1.
5. [10%] Evaluate
Z π/2
0
Z π/2
0
sin(x + y) dy, dx.
6. [10%] Find the area of the part of the paraboloid z = x2+ y2that lies under the plane z = 4.
7. [10%] Use triple integral to show that the volume of the solid bounded by a sphere of radius a is
4 3a3π.
8. [10%] The average value of a function f (x, y, z) over a solid region E is defined to be fave= 1
V (E) Z Z Z
E
f (x, y, z) dV
where V (E) is the volume of E. Find the average value of the function f (x, y, z) = x + y + z over the tetrahedron with vertices (0, 0, 0), (1, 0, 0), (0, 1, 0) and (0, 0, 1).
9. [10%] Evaluate the integral Z 1
0
Z 1
√y
px3+ 1 dx dy
by reversing the order of integration.
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