CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 3, Spring 2004

Exam 3 will cover sections 5.1,5.2, and 5.3 of Walker, and sections 12.6, 14.1, 14.2, 14.3, 14.4, 14.5, 14.6 of Stewart. Only those topics that I covered in lecture will be included on the exam - consult your class notes. I may also include a number of true/false questions again, so pleased be prepared for these. Below is a number of problems that may be similar to your exam problems. Please understand and be able to do these. Also, be prepared to do problems similar to HW or quiz problems. Monday April 5th will be a review day - please be ready to ask any questions that you may have. Good luck!

1. Find the absolute maximum and minimum values of $f(x)=xe^{-x}$ on the interval $[0,2]$.

2. Find all critical points of $g(x)=x^3-3x-2$. Use the first derivative test to classify these critical points as local maximum, local minimum, or neither.

3. Find all critical points of $h(x)=2x+\frac{8}{x}$, $x>0$. Use the second derivative test to classify these critical point(s) as local maximum or local minimum.

4. Let $x=y^2+z^2-2y-4z+5$. Identify the graph of this equation as either an ellipsoid, elliptic paraboloid, hyperbolic paraboloid, cone, hyperboloid of one sheet, hyperboloid of two sheets, or none of the above.

5. Show that $\lim_{(x,y)\rightarrow (0,0)} \frac{x^2y}{x^4+y^2}$ does not exist.

6. Let $f(x,y)=\frac{xy^2}{x^2+y}$. Verify that the conclusion of Clairaut's Theorem holds for this function.

7. Find the linear approximation $L(x,y)$ to the function $f(x,y)=\sqrt{x^2+y}$ at the point $(1,3)$. Use this to approximate $f(\frac{3}{2}, \frac{10}{3} )$.

8. Suppose $u=xy^2z^2$, $x=p^2q$, $y=pr^2$, and $z=q^2r$. Use the chain rule to find $\frac{\partial u}{\partial p}$ when $p=2$, $q=-1$, and $r=-2$.

9. Find the directional derivative of $f(x,y)=\ln(x^2+y^2)$ at the point $(1,3)$ in the direction $\vec{v}=\langle 2,-1 \rangle$.