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MAC 2312 August 12, 2004 Final Exam Prof. S. Hudson
- (10pts) For each series, answer either Converges absolutely (CA) or Con- verges conditionally (CC) or Diverges (D). State which “test” you are using and show work.
a)
k=1(−1)
k(1 + k)− (^2) ln(k)
b)
k=
(−1)k 2 k+
(5pts) Set up and simplify the integral with which you would find the length of the curve y = 13 (x^2 +2)^3 /^2 from x = 0 to x = 3. Do not evaluate.
(5pts) Express 1 − 2 /3 + 4/ 5 − 8 /7 + 16/9 in sigma notation.
[5 pts each] Compute (or explain why no answer exists):
∫ tan^2 x sec^2 x dx ∫ x^2 ex^ dx ∫ (^) π 0 |^ cos(x)|^ dx^ = ∫ (^) +∞ e
dx x ln x
- (15pts) a) Sketch the rose r = sin 2 θ.
b) Write down, but do not evaluate, an integral for the area of region enclosed by the rose.
c) Use parametric equations to find dy/dx for this curve.
- (10pts) a) Find the Taylor Series for the function f (x) = sin( xx ) at x = 0.
b) Use the result in (a) to approximate the integral
0 sin(x)/x dx^ (include 3 nonzero terms).
- (5pts) Find the interval of convergence of the series.
∑^ ∞
k=
2 kxk k
- [10 pts] Choose ONE:
a) State and prove the integration formula for area in polar coordi- nates. Draw a picture showing α, β, etc., and use a
and a limit in your explanation.
b) Prove: if a power series converges to f on (x 0 − R, x 0 + R), then it is the Taylor series of f at x 0. (Prove that ck =.. ., as Taylor predicts).
- [20 pts] Answer True or False:
The shell method formula for volume is V =
∫ (^) b a 2 πxf^ (x)^ dx.
The formula for arc length in Ch. 6 is L =
∫ (^) b a
1 + (f ′(x))^2 dx.
The integral
1 1 /x
p (^) dx converges if and only if p > 1.
There are exactly 8 points on the graph of r = cos(2θ) where the tangent line is horizontal.
If 0 ≤ an ≤ bn and
bn converges, then
an converges.
The Mclaurin series for tan−^1 x is x − x^3 /3 + x^5 /5 +... but it doesn’t converge for all real x.
The polar equations r = sin 4θ and r = cos 4θ have the same graphs.
A bounded increasing sequence must converge.
The sequence 10
n n! (for^ n^ = 0,^1 ,^2.. .) is strictly decreasing.
The series 1 − 1 /2 + 1/ 3 − 1 /4 +... converges conditionally to ln 2.
