Math 213 Exam 2 October 25, 2006
Read each problem carefully. Please show all your work for each problem! Use only those
methods discussed thus far in class. Always simplify when possible. No calculators!
1. (18 points)
(a) Use the Chain Rule to find the partial derivatives ∂w
∂u and ∂w
∂v , where
w= 3xy +y3−1, x = 2u+v2, y =u3−3v,
when u= 1 and v= 2.
(b) Integrate:
Z2
−1Z6
3
(2x2y−3x)dydx.
2. (10 points) Find the velocity and position of a particle that has the given acceleration
and the given initial velocity and position:
a(t) = 2ti−10 k,v(0) = 3 j,r(0) = 0.
3. (14 points) Find an equation of the tangent plane to the graph of the function
f(x, y) = (x+ 2y) cos(3xy)
at the point (0,1,2).
4. (14 points) Let g(x, y, z) = ez−xy2+zsin(πx).
(a) Find the directional derivative of gin the direction of the vector v=h1,1,1iat
the point P(1,−2,0).
(b) In which direction does ghave the maximum rate of change at that point? What
is its maximum rate of change there?
5. (14 points) The total resistance Rof two resistors with resistances R1and R2con-
nected in parallel is given by the following well-known formula:
1
R=1
R1
+1
R2
.
Suppose that R1and R2are measured to be 300 and 600 ohms, respectively, with an
error of ±15 ohms in each measurement. Use differentials to estimate the maximum
error in ohms in the calculated value of R.
6. (15 points) Find all the local minima, maxima, and saddle point points of the function
f(x, y) = 5x2−4xy +y2−4x+ 6y+ 10.
7. (15 points) Use the method of Lagrange Multipliers to find the minimum and the
maximum value of the function f(x,y , z) = xy +yz subject to the constraints xy = 1
and y2+z2= 1.
