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Math 113 – Fall 2005 — key
Departmental Final Exam
Part I: Short Answer and Multiple Choice Questions Do not show your work for problems in this part.
- Fill in the blanks with the correct answer.
(a) The integral
cos(x + 2) dx equals
sin(x + 2) + C
(b) The integral
sec x tan x dx equals sec(x) + C
(c) The integral
0
dx 1 + x^2
equals tan−^1 x
0 =^
π 4
(d) The integral
0
dx √ 1 − x^2
equals sin−^1 x
0 =^
π 2
(e) The integral
tan^2 x dx equals
sec^2 (x) − 1 dx = tan(x) − x + C
(f) The integral
0
√^ dx x equals 2
x
(g) The integral
0
dx x^3 equals divergent integral
(h) The integral
x √ 1 + x^2
dx equals
1 + x^2 + C
(i) Give the limit of the sequence
n
)n} as n → ∞ if it is convergent, otherwise write DIVERGENT. e−^1 (j) State the integration by parts formula: ∫ u(x)v′(x) dx = u(x)v(x) −
u′(x)v(x) dx
(k) Give a limit definition of the improper integral
0
sin x √ x dx
lim ≤→ 0
≤
sin √ x x dx
(l) State the (2m)-th term of the MacLaurin series for sin x x (−1)m (2m + 1)! x^2 m
(m) The integral
cot x dx equals ln(sin(x)) + C
- True/False: Write T if statement always holds, F otherwise.
Let
an =
n=1 an^ be an arbitrary series. (a) F : need an → 0 If {an} is a positive decreasing sequence then
(−1)nan converges. (b) T: Divergence test If
an converges then an → 0. (c) F : 1 − 1 + 1 − 1 ... If the partial sums of
an are bounded, then
an converges.
Problems 3 through 9 are multiple choice. Each multiple choice problem is worth 3 points. In the grid below fill in the square corresponding to each correct answer.
- The most appropriate first step to integrate
x^2 − 1 3 x^3 − x^2 dx would be
(a) Integration by parts (d) Other (non trigonometric) substitution
(b) Partial fractions (e) Differentiate the integrand
(c) Trigonometric Substitution (f) None of these
- The series x^2 + x^4 + x^6 2
x^8 6
∑^ ∞
n=
x(2n+2) n! converges to the function
(a) (^) 1+x^2 x 2 (e) x^2 (sin x^2 + cos x^2 )
(b) x^2 tan−^1 x (f) sin x^2 + cos x^2
(c) ex^2 +2^ (g) None of these
(d) x^2 ex^2
- The improper integral
∫^ ∞
0
xe−xdx converges to
(a) 0 (e) 2
(b) 1 /e (f) e
(c) 1 / 2 (g) None of these
(d) 1 (h) It doesn’t converge
Part II: Written Solutions
For problems 10 – 18, write your answers in the space provided. Neatly show your work for full credit.
- (a) Evaluate the integral
0
t^2 et^ dt.
Let u = t^2 , dv = et^ dt, then du = 2t dt, v = et, ∫ (^1)
0
t^2 et^ dt = t^2 et
0
2 tet^ dt
Let u = 2t, dv = et^ dt, then du = 2dt, v = et, ∫ (^1)
0
t^2 et^ dt = e −
2 tet
0
2 et^ dt
= e − (2e − 2(e − 1)) = e − 2
(b) Expand in partial fraction form x^2 + 3 x^2 − 1
x^2 + 3 x^2 − 1
x^2 − 1 + 4 x^2 − 1
(x − 1)(x + 1)
x − 1
1 + x
(c) Evaluate the integral
x^2 + 3 x^2 − 1
dx.
x + 2 ln
x − 1 x + 1
- Evaluate the integral
4 − 3 sin x
dx. Let z = tan(x/2), then
dx = 2 dz 1 + z^2 , sin x = 2 z 1 + z^2
∫ 1 4 − 3 sin x
dx =
4 − (^3) 1+^2 zz 2
2 dz 1 + z^2
=
4 z^2 − 6 z + 4 dz =
2 z^2 − 3 z + 2 dz
=
z − (^34)
7 4
) 2 dz
Let z −
tan t, then dz =
sec^2 t dt and ∫ 1 4 − 3 sin x dx =
dt
=
t + C
√^2
tan−^1
√^4
z −
+ C
tan−^1
tan
(x 2
+ C
tan−^1
4 tan
(x 2
+ C
- Find the centroid of the region bounded by the curves
y =
1 + x^2 , x = 1 and y = 1 + x.
Express you answer in terms of unevaluated integrals. (Note: You should simplify the inte- grands as much as possible.)
The curves y =
1 + x^2 and y = 1 + x intersects at x = 0. Area of region A =
0
1 + x −
1 + x^2 dx
Coordinates of centroid (¯x, ¯y) :
x¯ =
0 x(1 +^ x^ −
1 + x^2 ) dx A
y ¯ =
0
1 2
1 + x +
1 + x^2
1 + x −
1 + x^2
dx A
=
0
1 2 ((1 +^ x) (^2) − (1 + x (^2) )) dx A =
A
0
x dx ( =
A)
- If a region in the first quadrant, with area 10π and centroid at the point (1, 12), is revolved around the line x = −5, find the resulting volume of revolution.
By the first theorem of Pappus, V = 2π¯rA. Now A = 10π, r¯ = 1 − (−5) = 6, so V = 120π^2
- Determine whether each infinite series is absolutely convergent, conditionally convergent, or divergent. Give reasons for your conclusion.
(a)
∑^ ∞
n=
ln n 3 n + 7
Divergent: comparison with harmonic series ∑^ ∞
n=
ln n 3 n + 7
∑^ ∞
n=
3 n + 7
∑^ ∞
n=
6 n
∑^ ∞
n=
n
(b)
∑^ ∞
n=
(3−n^ − 5 −n)
Absolutely convergent: geometric series ∑^ ∞
n=
(3−n^ − 5 −n) =
∑^ ∞
n=
3 −n^ −
∑^ ∞
n=
5 −n) =
(c)
∑^ ∞
n=
(−1)n n ln n
Conditionally convergent: alternating series related to decreasing sequence of positive terms and comparison test Series is convergent since the sequence
n ln n
is a decreasing sequence of positive
terms, hence the alternating series
∑^ ∞
n=
(−1)n n ln n is convergent.
Series is not absolutely convergent since the integral
2
x ln x dx is divergent.
(d)
∑^ ∞
n=
(−1)nn ln(2n)
Divergent: divergence test
nlim→∞
(−1)nn ln(2n) = lim n→∞ (−1)n 2 /n
- Given the polar curve r = θ^2 , 0 ≤ θ ≤ 3 /2,
(a) sketch the curve; 3 2 radian is slightly less than^ π 2 radian or 90 ◦. ( note: 3 2 radian is about 86
0
1
2
0.1 0.2 0.3 0.4 0.
(b) find the area swept out by the curve;
Area =
0
r^2 dθ =
0
θ^4 dθ =
(c) find the arc length.
Arc length =
0
r^2 +
dr dθ
dθ
0
θ^4 + 4θ^2 dθ
0
θ
θ^2 + 4 dθ
=
t^2 + 4
3 / 2 0 =
—End—
