MATH 205 Final Examination Solved

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10] 1. (a) Sketch a graph of the function
f(x) = (


4 − x
2 −2 ≤ x ≤ 0
2 − 2|x − 2| 0 < x ≤ 4
on the interval −2 ≤ x ≤ 4 and calculate the definite integral R
4
−2
f(x) dx
in terms of area (do not antidifferentiate).
(b) Use the Fundamental Theorem of Calculus to calculate the derivative of
F(x) =
x
2
R
x
e
sin(π t) dt,
and determine whether F is increasing or decreasing at x = 1.
[6] 2. Find F(x) such that F
0
(x) = x
2 + 2x
x
2 + 4
and F(0) = 0.
[11] 3. Find the following indefinite integrals:
(a)
Z
13 − x
x
2 − x − 6
dx (b)
Z
x
3/2
ln2
(x) dx
[12] 4. Evaluate the following definite integrals (give the exact answers):
(a)
Z
1
0
2
x
4
x + 1
dx (b)
Z
2
1

4 − x
2 dx .
[8] 5. Evaluate the given improper integral or show that it diverges:
(a)
Z∞
e
dx
x ln(x
2
)
(b)
Z
1
0
dx
(1 − x)
3/4
MATH 205 Final Examination
[18] 6. (a) Sketch the curves y = x (3 − x
2
) and y = −x, and find the area enclosed by
the two curves. (HINT: find first the points of intersection of the curves.)
(b) Sketch the region enclosed by y = cos(2x) and the x-axis on the interval
[0,
π
2
], and find the volume of revolution of this region about the axis y = −1.
(c) Find the average value of the function f(x) = sec4
(x) on the interval

π
4
,
π
4

.
[9] 7. Find the limit of the sequence {an} as → ∞ or prove that it does not exist:
(a) an =
e
n − n
3
3
n
(b) an =
(−1)nn

1 + 4n2
(c) an = ln(n + 2n
2
) − ln(2n + n
2
).
[12] 8. Determine whether the series is divergent or convergent, and if convergent,
then whether absolutely or conditionally convergent:
(a)
X∞
n=1
n
2/3
1 + 2n
(b)
X∞
n=1
(−1)n+1 sin 
1
n

(c)
X∞
n=1
(−1)n+1 2
n+1
n!
[6] 9. Find (a) the radius and (b) the interval of convergence of the series P∞
n=1
(x + 2)3n
n2 8
n
[8] 10. (a) Derive the Maclaurin series of f(x) = x
2
e
3x
.
(HINT: start with the series for e
z and then let z = 3x).
(b) Find the values of x for which the following series converges
X∞
n=0
(x
2 + 1)n
2
n+1
and, for these values of x, find the sum of the series as a function of x.
Bonus question [5]. Find the values of p (if any) for which the series
P∞
n=5
1
n ln n(ln (ln n))p is convergent