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The solutions to test 3 of math 112. It includes answers to various math problems covering integrals, comparisons of functions, and calculus. The document also includes geometric shapes and their areas.
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Test 3 Math 112 November 14, 2006 Answers
1a) -1/2 1b) e−^2 1c) 0
I is made up of 3 integrals, only one of which converges, so I diverges.
π/ 3
Compare to f (x) = 1/ 3 x^2 which is bigger. Show f (x) ≥ g(x) ≥ 0 and
∫ (^) ∞
and then invoke the theorem.^1 f^ (x)^ dx^ converges
(x, y) = (1/ 2 , 1 /2) and (x, y) = (0, 0)
Area = 2
∫ (^) π/ 6 0
sin^2 θ 2 dθ^ + 2
∫ (^) π/ 2 π/ 6
(1 − sin θ)^2 2 dθ
∣∣ ∣∣ ∣t=0^ =
d^2 dx^2
∣∣ ∣∣ ∣t=0^ =
8a)A = 1 8b) The curve starts at (1,0) and spirals around as r decreases until it crosses the origin at θ = π. Then it makes a small loop above the x–axis and across the y-axis returning to the origin along the same angle. The y-interccepts are at 2/π and 2/ 3 π. 9a) A line through the origin with slope 5. 9b) A parabola opening left turning around at (x, y) = (1/ 2 , 0) and crossing the y-axis at y = ±1.
√ 5 4 ·^ 2 (e
π (^) − 1)