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Answers: APPM 1360 Final Fall 2011
- (a) (8 pts) Find the Cartesian equation of the curve x = 2 cos(θ), y = 1 + sin(θ)
(b) (8 pts) Sketch the parametric curve x = 2 cos(θ), y = 1 + sin(θ), 0 ≤ θ ≤ 2 π. (c) (8 pts) Find the slope of the tangent line to the curve x = 2 cos(θ), y = 1 + sin(θ) when θ = π/4. Answers: (a)
( (^) x 2
-2 -1.6 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 1.6 2
1
2
(c) the slope is − 1 /2.
- (a) (9 pts) Graph the functions r = sin (4θ) and r = 12 on the same axis, clearly label your graph.
(b) (9 pts) Find the area of the region in the first quadrant inside r = sin (4θ) and outside r = 12. (c) (9 pts) Set up, but do not solve, an integral to find the entire length of all the petals inside the circle. Answers: (a)
-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1
-0.
1
0 0.25^ 0.5^ 0.75^1 1.25^ 1.
(b) A =
π 24 +
(c) L = 16
∫ 24 π
0
sin^2 (4θ) + 16 cos^2 (4θ) dθ
- Evaluate the integral or show that it is divergent:
(a)(9 pts)
0
ln(x) x^2 dx^ (b)(9 pts)
1
4 x^2 − 7 x − 12 x^3 − x^2 − 6 x dx^ (c)(9 pts)
0
√^ x^2 4 − x^2
dx
Answers: (a) the integral is divergent.
(b)^9 5
ln
(c)
π 3 −
- (a) (8 pts) Set up, but do not solve, an integral to find the volume of the object created when the region bounded by y = x^3 + 1, y = 1 and x = 1 is rotated around the line x = 2 using the disk/washer method. (b) (8 pts) Set up, but do not solve, an integral to find the volume of the object created when the region bounded by y = x^3 + 1, y = 1 and x = 1 is rotated around the line x = 2 using the method of cylindrical shells. (c) (8 pts) Set up but do not solve an integral to find the surface area if we revolve the curve y = x^3 + 1, 0 ≤ x ≤ 1 around the line y = 0. Answers: (a) V = π
1
[(
y − 1
− (1)^2
]
dy
(b) V = 2π
0
(2x^3 − x^4 )dx
(c) SA = 2π
0
(x^3 + 1)
1 + 9x^4 dx
- Determine if the following converge or diverge:
(a)(9 pts)
e−nn!
n=0 (b)(9 pts)
1
dx x − ln (x) (c)(9 pts)
∑^ ∞
n=
2 n^2 + 7 n^3 − 2 n^2 − n − 1 Answers: (a) diverges (b) diverges (DCT) (c) diverges (LCT)
- (8 pts) For what values of x is the series
∑^ ∞
n=
n(x − 4)n n^3 + 1
is absolutely convergent? conditionally convergent?
