CARNEGIE MELLON UNIVERSITY
DEPARTMENT OF MATHEMATICAL SCIENCES
21-256 Review Exam 2, Spring , 2006

• Your exam shall consist of problems similar to the homework and quiz problems. All homework problems are posted on the course web site.

• Wednesday, March 29th will be a review day. Please ask any questions you may have from the sections 14.1-14.3, 14.5-14.8(S), 5.5(W),5.6(W),6.2(W),6.3(W).

• The exam will be held on Friday March 31st in Porter Hall 100.

• Notes and texts will not be permitted during the exam.

• Calculators or other electronic devices are not permitted.

• You will have 50 minutes for the exam, and must stop when told to stop.

• Please consult the course web page for exam policies - these will be enforced. What follows is a practice exam. This is designed for you to take in 50 minutes. The actual exam will consist of different questions. To fully prepare please practice many homework problems.

1. Find the limit, or show it does not exist:
   
   $$\lim_{(x,y) \rightarrow (0,0)} \frac{x^2y}{x^4+2y^2}$$
   
   

2. Find $\frac{\partial w}{\partial t}$ when $t=0$, $s=1$, given that $w=x^2+2xyz+y^4+z^3$, $x=e^{-t}\cos(\pi s)$, $y=e^{-2t}\sin(\pi s)$, $z=st$.

3. 1. Find the directional derivative of $f(x,y)=\frac{xy^2}{x^2+y^2}$ at the point $(1,1)$ in the direction of $\vec{v}=\langle 3,-4 \rangle$.
   2. Find the maximum directional derivative of $f(x,y)=\frac{xy^2}{x^2+y^2}$ at the point $(1,1)$ and the direction in which this occurs.

4. Find all local maximum, minimum, and saddle points of $f(x,y,z)=4x^2+2y^2+4z^2+4x-14y-xy+2yz+16$.

5. Use Lagrange multipliers to determine the absolute maximum of $f(x,y,z)=3x+2y-5z$ subject to the constraints $x^2+4z^2=1$ and $2x+y=4$.

6. Find the Least Squares Approximation of the data:
   

   x	1	2	3	4
   y	5	3	2	0