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Math 113 – Fall 2006 – Key
Departmental Final Exam
Part I: Short Answer and Multiple Choice Questions Do not show your work for problem 1.
- Fill in the blanks with the correct answer.
(a) Does the improper integral
0
dx ex^ + 1 converge (yes or no) yes
(b) The integral
cos x sin^3 x dx equals −
2 sin^2 x
+ C
(c) The integral
∫ (^) e 2
1
dx 2 x equals 1
(d) x^2 4
y^2 25 = 1 is the equation of a/an hyperbola
(e) The radius of convergence of
n=
3 nxn^ is
(f) If n > 1, the integral
1
dx xn^ equals
n − 1
(g) The series x^2 − x^4 3!
x^6 5!
x^8 7! +... is the MacLaurin series for the function x sin x
(h) The integral
x sin x dx equals −x cos x + sin x + C
(i) The series 2 −
+... converges to
Problems 2 through 8 are multiple choice. Each multiple choice problem is worth 3 points. In the grid below fill in the square corresponding to each correct answer.
- Which of the following integrals represents the surface area of the surface generated by re- volving the curve y = tan x, 0 ≤ x ≤ π/4, about the line y = −2? (a)
∫ (^) π/ 4 0 π(tan^ x^ + 2)
1 + sec^2 x dx (f)
∫ (^) π/ 4 0 2 π(tan^ x^ −^ 2)
1 + sec^2 x dx
(b)
∫ (^) π/ 4 0 2 π(tan^ x^ + 2)
1 + sec^2 x dx (g)
∫ (^) π/ 4 0 π(tan^ x^ −^ 2)
1 + sec^2 x dx
(c)
∫ (^) π/ 4 0 π(tan^ x^ + 2)
1 + sec^4 x dx (h)
∫ (^) π/ 4 0 2 π(tan^ x^ −^ 2)
1 + sec^4 x dx
(d)
∫ (^) π/ 4 0 2 π(tan^ x^ + 2)
1 + sec^4 x dx (i) None of the above
(e)
∫ (^) π/ 4 0 π(tan^ x^ −^ 2)
1 + sec^4 x dx
- Which of the following substitutions will best simplify the integral
3 + 2x − x^2 dx? (a) x = 1 − 2 sec u (e) x =
3 sin u
(b) x =
3 + 2 cosh u (f) x = 1 + 2 sin u
(c) x =
3 cos u (g) x = 2 sin u
(d) x =
3 − 2 cosh u
- Consider the region R that is the portion of the circle x^2 +y^2 = 1 that lies in the first quadrant. What is the volume of the solid generated by revolving R about the line x + y = 2?
(a) π 2
(d) π^2 2 (g) π^2
(b) π 2 (e) π^2 3
(h) π^2 2
(c) π
(f) π^2 4 (i) None of the above
- The series
∑^ ∞
n=
3 n n! converges to
(a) ln 3 (d) 3 n+ n + 1 (g) cos 3
(b) ln 2 (e) ∞ (h) e^3 − 4
(c) ln(3) − 1 (f) e^3 (i) 3 e
- The interval of convergence of the power series
∑^ ∞
n=
n^2 (7x − 3)n^ is
(a)
(d) (0, 1) (g) (0, ∞)
(b)
(e)
(h) (−∞, ∞)
(c) (− 1 , 1) (f)
(i) None of these
- Find the general solution, in the form y = f (x), to the differential equation
dy dx = (4 + y^2 )(4 + x^2 ).
Separating variables and integrating dy 4 + y^2
(4 + x^2 ) dx
y = 2 tan
8 x +
x^3 + C
- Find the length of the graph of y =
x^2 −
ln x, on the interval 1 ≤ x ≤ 2.
Length of graph is given by the arc length formula
1
1 + (y′)^2 dx
Now, as y′^ =
x −
x
, so the length of graph is
∫ (^2)
1
x^2 − 2 +
x^2
dx =
1
x +
x
dx =
ln 2
- Find the centroid of the region that lies within the first quadrant and is bounded above by y = 1 − x^2.
A =
0
(1 − x^2 ) dx =
my =
0
x(1 − x^2 ) dx =
mx =
0
(1 − x^2 )^2 dx =
¯x = my A
y¯ = mx A
- Find the area enclosed by the polar curves r = 2 − cos θ and r = 1.
Unit circle lies entirely inside the first curve, Area is (^) ∫ (^2) π
0
r^2 (θ) dθ − π (1)^2 =
∫ (^2) π
0
(2 − cos θ)^2 dθ − π =
π
- Use the first three non-zero terms of the MacLaurin series for e−x^2 to estimate the definite
integral
0
e−x^2 dx. Write your answer as a fraction, if possible.
e−x 2 = 1 − x^2 +
x^4 −
x^6 + ...
∫ (^2)
0
e−x 2 dx =
0
1 − x^2 +
x^4 −
x^6 + ...
dx
0
1 − x^2 +
x^4 dx
= x − x^3 3
x^5 10
2
0
- Find the mass of the circular region x^2 + y^2 ≤ 1, whose density at each point is twice the distance from the point to the origin.
Mass is given by (^) ∫ 1 0
2 r · 2 πr dr = 4 πr^3 3
1
0
4 π 3
- Find the sum of the power series
∑^ ∞
n=
nxn−^1 (as a rational function of x).
The power series
∑^ ∞
n=
nxn−^1 is obtained by differentiating f (x) =
∑^ ∞
n=
xn^ =
1 − x , so
∑^ ∞
n=
nxn−^1 = d dx
1 − x
(1 − x)^2
- Determine whether each of the following infinite series converges. State any convergence/divergence test you used.
(a)
∑^ ∞
n=
n^2 + 1
Converges by comparison with 1/n^2.
(b)
∑^ ∞
n=
en n^30 + 2n
Diverges by divergence test (terms not approaching 0)
(c)
∑^ ∞
n=
√^ (−1)n n + 1
Converges by alternate series test
