MATH261 EXAM II SPRING 2015
NAME:
CSU ID:
Problem Points
1
16
2
14
3
20
4
18
5
14
6
18
Total
100
SECTION NUMBER:
You may NOT use calculators or
any references. Show work to receive full credit.
GOOD LUCK !!!
Score
1. (a) (8 pts) For a function f (r, s) with r = r(x, y, z) and s = s(x, y, z), what is
the general multivariate Chain Rule for computing ∂f /∂y. DO NOT actually
compute the partial derivatives involved until part (b).
(b) (8 pts) Given f = rs + s2 , r = xy 2 , and s = cos(yz), evaluate ∂f /∂y at the point
(x, y, z) = (e, π/2, 4). Be sure to clearly label all partial derivatives you compute.
2. Consider the surface given by
F (x, y, z) = xy 2 z + ln(xy) − 1 = 0
(a) (6 pts) Verify that the point (x, y, z) = (e, 1, 0) is on the surface.
(b) (8 pts) Find the equation of the tangent plane at (e, 1, 0). Please simplify the
equation so it is in the form Ax + By + Cz = D.
3. (a) (10 pts) Find the linearization of f (x, y) = x2 y + y at P0 (1, 2). Do NOT simplify
beyond the form L(x, y) = a + b(x − x0 ) + c(y − y0 ).
(b) (10 pts) Using the method from class, find the lowest possible upper bound for
|E|, the error in the approximation of f (x, y) with L(x, y) over the rectangle
|x − 1| ≤ 0.1, |y − 2| ≤ 0.2.
4. Consider the function f (x, y) = x3 + 8y 3 − 12xy.
(a) (8 pts) (0, 0) is one critical point. Classify it as a min, a max, or a saddle point.
Clearly identify the value of fxx fyy − (fxy )2 and the second value needed to distinguish between a max and a min (if needed).
(b) (10 pts) Find and classify all remaining critical points. Clearly identify the value
of fxx fyy − (fxy )2 and the second value needed to distinguish between a max and
a min (if needed).
5. (14 pts) Using Lagrange multipliers, find the point on the line x + 2y = 5 nearest the
origin.
6. Consider the integral
Z 2Z 4
0
ex/y dy dx
2x
(a) (8 pts) Sketch the region, with all corners clearly labeled.
(b) (10 pts) Rewrite the integral with the variables in the other order, i.e., dx dy.
DO NOT EVALUATE.