Math 143: Calculus III Practice Final Exam
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Math 143: Calculus III
Practice Final Exam
December 17th, 2017
Part A
1. (10 points) If a sequence below converges, find its limit, and justify by citing any theorems/rules you use. If a sequence below diverges, state whether it diverges because it
oscillates, diverges to +● , or diverges to _● .
(a) an = ln(6n7 + 5n + 3) _ ln(4n7 + 2n + 8)
(b) an = / 、n
(c) an = n4 sin(n)
(_1)n
(d) an =
2. (10 points) Determine whether the following series converge absolutely, converge only conditionally, or diverge, naming any tests you use, and justifying their use com- pletely.
(a)
2 n5n + 6
(4.9)n _ n
n=1
(b)
2 e1/n
n2
n=1
(c)
2
1
n ln(n)
n=2
3. (10 points) Determine whether the following series converge absolutely, converge only conditionally, or diverge, naming any tests you use, and justifying their use com- pletely.
(a)
2 arctan(n)
0.9
(b)
2 / 、n
4. (10 points) Find the radius and interval of convergence of the power series below.
2 10n (x _ 3)2n+1
n(2n + 1)!
2 (_1)nn!(3x + 2)n
n=1 4n ^n + 2
2 (_5)n (10x _ 3)n
4n ^n
5. (10 points)
(a) Find a power series expansion of the function f (x) = about x = 0, write out the
first five nonzero terms, and express the series in sigma notation.
(b) What are the radius and interval of convergence of the series you found in (a)?
(c) Write out the first five nonzero terms, and express in sigma notation a power series
expansion for f (x) = dx about x = 0, assuming f (0) = 0.
(d) What are the radius and interval of convergence of the series you found in (a)?
6. (10 points) Consider the function f (x) = ex/2 .
(a) Write out the first five nonzero terms, and express in sigma notation the Taylor series expansion for f (x) about x = _2.
(b) What are the radius and interval of convergence of the series you found in (a)?
7. (10 points) Find the sum of the following convergent series. You do not need to justify that they converge.
2
1
n(n + 1)
n=1
2 (_6)n
7nn
(c) 90 + 30 + 10 + + + . . .
2 (_1)n 112n
32n+1(2n + 1)!
n=0
8. (10 points) Consider the function f (x) = 18
(a) Find the first five nonzero terms of the Taylor series expansion of f (x) about x = 0.
(b) What is the value of f(4) (0)?
(c) What is the value of lim f (x)?
x…0
(d) What is the Taylor polynomial of degree 5 of f (x) at x = 0?
Part B
9. (15 points) Consider the parametric curve defined by
x = t2
y = t3 _ 3t.
dy
(a) Calculate
(b) Find a point at which the curve has two different tangent lines.
(c) Find these tangent lines.
d2y
(d) Calculate
(e) Determine intervals of t-values for which the parametric curve is concave up and intervals
for which it is concave down.
10. (15 points) Consider the parametric curve defined by
x = cos3 (t)
y = sin3 (t).
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(a) Sketch this curve on the graph above, indicating the direction of increasing t. (b) Fill in the area under the curve from t = to t = on your sketch above.
(c) Find the area under this curve from t = to t = .
(d) Write down but do not evaluate an integral that would give the arc length of this curve from t = to t = .
11. (15 points) Consider the polar curve defined by r = 1 + ^2 sin(θ).
π/2
π
0
3π/2
(a) Draw a clear sketch of the curve above.
(b) At which angles does the curve cross itself?
(c) Write down but do not evaluate an integral that would give the arc length of the curve.
(d) Find the area inside the larger loop, but outside the smaller loop of this curve.
12. (15 points) Consider the polar curves defined by r = 2 + 2 sin(θ) and r = 2 sin(θ).
(a) Find the area inside the outer curve, but outside the inner curve.
π/2
3π/2
(b) Write down but do not evaluate an integral that would give the arc length of the inner loop.
2022-12-14