Math 2B
Practice Final Exam
Z
1. Suppose that
Z 1
(a)
6f (x)dx
1
Z
−1
1
4
Z
1
[2f (x) − 3x2 ]dx
(b)
−1
Z
4
f (x)dx
(c)
−1
d
2. Evaluate
dx
Z
x2
sin(x)
t3 tan(t)dt
4
f (x)dx = −2, evaluate the following integrals:
f (x)dx = 6 and
3. Evaluate the following integrals:
Z
(a) 6x2 tan−1 (x)dx
Z
(b)
1
dx
x ln(3x)
Z
(c)
sin5 (θ) cos2 (θ)dθ
Z √
(d)
x2 − 25
dx
x
2
Z
(e)
0
Z
(f)
0
1
dx
(x − 2)2
∞
z2
1
dz
+ 3z + 2
h πi
4. Find the average value of f (x) = sec2 (x) on 0,
4
5. Find the Maclaurin series for f (x) = (1 − x)−2 and find its radius of convergence.
11. Find thelimit of each sequence below if it exists.
2 n
(a) an =
+3
3
(b) bn = n3 e−n
(c) cn = tan−1 (ln(n))
12.Let R be the region bounded by the curves y = x2 + 1 and y = 3 − x
(a) Find the area of R
(b) Find the volume when R is rotated around y = 5
(c) Find the volume of the solid with base R and square cross-sections perpendicular to the x-axis.
13. Determine whether the following series are convergent or divergent and state any test used.
∞
X
n
(a)
n3 + 1
n=1
∞
X
(−1)n
√
(b)
n+1
n=1
(c)
∞
X
2n
n=1
n2
(d)
∞
X
n2 e−n
n=1
(e)
∞
X
n=1
(f)
3n
1
−1
∞
X
cos(πn)
n=1
(2n)!