3.6 Practice Prelim Questions - Review Problems for Prelim...

Review Problems for Prelim 3
Partial Derivatives
Find ∂f/∂x and ∂f/∂y
1. x - y
2
2. (3 + 2 xy 2 ) 3
3. ln( x 2 + y 3 )
4. y
2
e
x 2 y
5.
x 2 y
x 2 + y 2
Find ∂ 2 f/∂x 2 and ∂ 2 f/∂y∂x
6. x 4 + x 2 y 2 + xy 3 + y 4
7. x
y
-
y
x
8. xe y/x
Definite Integrals (includes Improper Integrals)
Volume problems turn into these after you use V =
b
a
πf ( x ) 2 dx
9.
25
4
1
√
x
dx
10.
∞
0
x (1 + x )
- 5 / 2
dx
11.
3
2
8 x ( x 2 - 4) 3 dx
12.
2
1
2 x
2
+1
x
dx
13.
2
0
2 x
x 2 +1
dx
14.
2
0
6 x
3
√
2 x
2
+1
dx
15.
4
1
ln x
√
x
dx
16.
e
1
√
ln x
x
dx
17.
∞
1
e
- 2
√
x
√
x
dx
18.
∞
0
x 3 e - x
2
/ 2 dx
Probability Density Problems
19. A random variable X has density function 12 x 2 (1 - x ), Y has density function 12 x (1 - x ) 2 .
Compute their means and variances. If you do this correctly then the means will add to 1
and the variances will be the same, since X and 1 - Y have the same distribution.