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Answers: APPM 1360 Final Spring 2012
- (a) (7 pts) Find the area of the region that lies inside r = 2 cos(2θ) and outside r = 1.
(b) (7 pts) Find the length of the polar curve r = eθ/^2 , 0 ≤ θ ≤ 2 π. Answers: (a) A =
8 π 12 +^
2 π 3 +^
(b) L =
5(eπ^ − 1)
- (a) (7 pts) Find the Cartesian equation of the curve given by the parametric equations
x(t) = 3 cos t − 1 , y(t) = sin t + 2, 0 ≤ t ≤ π
(b) (7 pts) Sketch the curve in part (a). Make sure to show the direction in which the curve is traced. (c) (7 pts) Find all of the points where the parametric curve x(r) = r + r^3 , y(r) = er^2 has a horizontal tangent. Give your answer in the Cartesian coordinates (Your answer should be in the form (x, y)). Answers:
(a) y =
1 − (x^ + 1)
2 9
2 9
-4 -3.2 -2.4 -1.6 -0.8 0 0.8 1.6 2.
4
(-4,2) (^) (2,2)
(c) (0, 1)
- (a) (7 pts) Determine whether
12
x − 1 x^3 + ex^ dx^ converges or diverges.
(b) (7 pts) Use the Trapezoidal Rule to approximate
∫ (^) π
0
cos^4 (x) dx using n = 4 subintervals.
(c) (7 pts) Given the following information about the second derivative of cos^4 (x):
d^2 dx^2
[
cos^4 (x)
]
= −4 cos^2 (x) [−1 + 2 cos(2x)] , and − 4 ≤ d
2 dx^2
[
cos^4 (x)
]
< 3 for 0 ≤ x ≤ π
what is the Trapezoidal Rule error bound of the approximation that you found in part (b)? Answers: (a) The integral is convergent by the Direct Comparison Test for integrals.
(b)
∫ (^) π
0
cos^4 (x) dx ≈
3 π 8
(c) |ET | ≤
π^3 48
- Determine if the given series are absolutely convergent, conditionally convergent or divergent. Justify your answer and find the sum when possible.
(a)(9 pts)
∑^ ∞
m=
m^2 + 5m + 6 (b)(9 pts)
∑^ ∞
k=
(−1)kπ−kek−^1 (c)(9 pts)
∑^ ∞
b=
be−b
Answers: (a) absolutely convergent Telescoping series that converges to 1/3. (b) absolutely convergent Geometric Series that converges to (^) e(π π+ e).
(c) absolutely convergent by the Integral Test.
- (a) (7 pts) Find the mass of a thin metal plate given by the region between y = ln x, the x-axis, and the line x = 4, with density ρ = 2. (b) (7 pts) Set up, but do not evaluate an integral to find the moment about the x-axis of the plate from part (a).
(c) (7 pts) Solve the following differential equation for y: e−x
2 dy dx =^ x^ cos
(^2) (y + 1).
Answers: (a) 8 ln 4 − 6
(b)
1
ln^2 (x) dx
(c) y = tan−^1
ex^2 2 +^ C
- (a) (7 pts) Find the integral
x cos(x^3 ) dx by finding a Maclaurin series for the integrand and inte- grating the series. (b) (7 pts) Consider the function f (x) =
3 (x^ −^ 1)
3 / (^2). Find the second order Taylor polynomial, T 2 (x), centered at a = 2. (c) (7 pts) Now use Taylor’s Formula to find a bound on the error of the approximation f (x) ≈ T 2 (x) found in part (b) on the interval 1. 2 ≤ x ≤ 3 .2. Answers:
(a) C +
∑^ ∞
n=
(−1)nx^6 n+ (2n)!(6n + 2)
(b)^2 3
(x − 2)^2
(c) |R 2 | <
4(0.2)^3 /^2 3!
(1.2)^3
- (5 pts ea.) Answer either Always True or False. Do NOT justify your answer.
