Math 210 S92 — Calculus III — Exam #2 (Ch. 12) — Berger — page 1 of 3 Show work on these sheets in 100 minutes ((12 + 16 + 6 × 12 = 100 points).

Name  .  SocSecNo.  .

1. Find the Point on the Plane x − y + z = 1 that is closest to the Point (1,1,1).

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

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

Math 210 S92 — Calculus III — Exam #2 — Berger — page 2 of 3

4. If x + y − cos(3x + 2y + z) = 0, find the differential dz =

5. If z = cos(xy), r = s 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{xy}{3x^{2} + 2y^{2}} =

Math 210 S92 — Calculus III — Exam #2 — Berger — page 3 of 3

7. Find the equation of the tangent plane to the 3-dimensional surface given by

xy^{2}z^{3} − 12 = 0 at (3, 2, 1) :

8. (a) State any one of Kepler’s Laws of Planetary Motion.

(b) Explain why Sir Isaac Newton’s deduction of Kepler’s laws using Newton’s second law of motion and the Law of Universal gravitation is one of the great accomplishments of Calculus.