Math 210 E1 S92 — Calculus III — Exam #2 (Ch. 12) — Hanson — page 1 of 2 Show work on this sheet in 50 minutes ((16 + 20 + 4 × 16 = 100 points).

Name  .  SocSecNo.  .

1. Find the Point on the Plane 3x − 2y + 4z = 5 that is closest to the Point (-4, 5, 6).

2. Find the Positions of Local Extrema and Saddle Points of f(x, y) = 2y^{3} + x^{2}y + 4y^{2} + 3x^{2} (identification of critical points is part of the problem).

3. Find the Direction of the Minimum Rate of Change of f(x, y, z) = \frac{x}{y} − z^{2} at (3, -1, 2).

Math 210 E1 S92 — Calculus III — Exam #2 — Hanson — page 2 of 2

4. If e^{x+2y+3z} = cos(3x + 2y + z), find the differential dz =

5. If z = sin(xy^{2}), x = r cos θ and y = r sin θ, find

(a) \frac{∂z}{∂r}

(b) \frac{∂z}{∂θ}

6. Find the Limit if it Exists, else show that it Does Not Exist:

lim_{(x,y)→(0,0)} \frac{2xy}{x^{2} + y^{2}} =