The Review Test IV- (Chapter 8- Sec 10.3)

The Review Test IV- (Chapter 8- Sec 10.3)

Deadline 2008/5/29

5 Problems. Total 80 points.

1. Given two vectors ~a =< 2, 1,−2 >,~b =< 1, 3, 0 >

(a) (5pts) Compute ~c = 4~a + 2~b.

(b) (5pts) Write ~c as the product of its magnitude and a unit vector.

2. (a) (5pts) Find the distance between the point q = (3,−2, 1) and the straight line through two points (2, 1,−1) and (1, 1, 1).

(b) (5pts) Find the intersection of the two planes L₁ : 3x + 4y = 1 and L₂ : x + y− z = 3

3. (a) (10pts) Given ~r(t) =< t ln t, e^3t, 3^t >, find lim

t→0~r(t) if it exists, and the derivative of ~r(t).

(b) (10pts) Given ~s(t) =< 4t, 2t, t² >, find the unit tangent vector to the curve determined by ~s(t) at the points t = −1 and t = 0.

4. Let f (x, y) =

{ x³−xy²

x²+y² for (x, y) 6= (0, 0),

c for (x, y) = (0, 0). , please answer the following problems.

(a) (10pts) Find c such that f (x, y) is continuous at the (0, 0).

(b) (10pts) Evaluate f_x(x, y) and f_x(0, 0).

(c) (10pts) Determine whether f_x(x, y) is continuous at (0, 0) or not.

5. (10pts) Find the corresponding contour plots of the surfaces a and b.