12.2 Partial Derivatives and Directional Derivatives 1. Show that .

(Sol)

Let , so .

Thus  2. If , find .

http://matrix.skku.ac.kr/cal-lab/cal-12-2-2.html (Sol) (x, y, z) |--> (-x/(x^2 + y^2 + z^2)^(3/2), -y/(x^2 + y^2 +

z^2)^(3/2), -z/(x^2 + y^2 + z^2)^(3/2))

Therefore, .

3. If , find .

4. State and chat on the mean value Theorem for multi-variable functions.

5. Let with .

(a) Show that and exist everywhere.

(b) Are and continuous at the origin? Justify your answer.

http://matrix.skku.ac.kr/cal-lab/cal-12-2-5.html (Sol)

(a) If ,  .

Else -(x^2*y - y^3)/(x^4 + 2*x^2*y^2 + y^4)

(x^3 - x*y^2)/(x^4 + 2*x^2*y^2 + y^4)

Thus and exist everywhere.

(b) 0

Infinity is not continuous at the origin. Infinity

0 is not continuous at the origin.

6.(a) Take an example of the  solution for the .

(b) In (a), Can you find the examples other than polynomial?

(Sol)

(a) http://matrix.skku.ac.kr/cal-lab/cal-12-2-6.html  ([x == -y], )

(b)

7. Can you find and prove an analogue for Rolle's Theorem, for functions of two real variables?

(Sol)

Rolle's Theorem for a variable: If a real-valued function is continuous on a closed interval [a, b], differentiable on the open

interval (a, b), and , then there exists a c in the open interval (a, b) such that .