MATH 16C
Dr. Dad-del
Test 2 Practice Problems
1. Find the first and second partial derivatives fx, fy, fxx, fyy , fxy, fy x , for the following
functions;
a) f(x, y) = x4/3
−4xy−1/3b) f(x, y) = x2y
3y2−x
c) f(x, y) = x
yey2+xat (1,1). d) f(x, y) = Zy
x(t−3)dt
2. Find the slope of the surface f(x, y ) = x
x−yin the x-direction and in y-direction at the
point (2,1).
3. Find the relative extrema of f(x, y) = x3
−3xy + 6y2,using the Second Partials Test.
4. A rectangular open box is to hold 108 cubic inches. What dimensions will require the
least anount of material?
5. Use Lagrange Multipliers to find the minimum of the function f(x, y) = qx2+y2subject
to the constraint x+ 2y= 10.
6. Use Lagrange Multipliers to find the minimum of the function f(x, y, z) = xz +yz sub ject
to the constraint x−2y= 0 and 3x+z= 5
7. Evaluate a) Z1
0Zy
√y(6x2
−2y2
−y)dxdy b) Z∞
0Z1
0
(yey2−x)dxdy
8. Use double integral to find the area of the region bounded by y=x/2 and y=√x.
Change the order of the integration and evaluate.
9. Find the volume of the solid bounded above by the plane z= 3 + x−yand bounded
below by the region bounded by y=x, x = 4 and y= 0.
10. Find the average value of the f(x, y) = x+yover the region bounded by the graphs of
x=y2and x−2y= 3 .
11. Find the nT h term of the sequence 1,2,27
6,256
24 ,3125
120 ,· · ·
12. Determine the convergence or divergence of the sequence whose nth term is :
a) 2n+ 1
n−2b) 3n−5
3n+1 −1
13. Determine the convergence or divergence of the following series:
a) ∞
X
n=1
2−3n
5n+1 b) ∞
X
n=1
en
en+ 2