Find the length of the curve given by x 3t2 y 2t3 0 le t

Question

Midterm 1 Review - Math 53, section 213

1. Find the length of the curve given by x = 3t², y = 2t³, 0 ≤ t ≤ 2.

2. Compute the cross product of (1, 1, -1) and (2, 4, 6). What is the area of the parallelogram spanned by these vectors?

3. Find an equation for the plane passing through (1, 2, -2) that contains the line x = 2t, y = 3 - t, z = 1 + 3t.

4. Evaluate the following limit or show that it does not exist:

lim_{(x,y)→(0,0)}(x³ + y³/x² + y²).

What about the limit

lim_{(x,y)→(0,0)}(x² + y²/x² - y²)?

5. Find all second partial derivatives of f(x, y, z) = x^ky^lz^m.

6. Find equations of the tangent plane and the normal line to the surface xy + yz + zx = 3 at the point (1, 1, 1).

7. Find the maximum rate of change of f(x, y) = x²y + √y at the point (2, 1). In which direction does it occur?

8. Find the critical points (local maxima, minima, and saddle points) of the function f(x, y) = x³ - 6xy + 8y³.