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Math 112 Test 3 Solutions, Exams of Calculus

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.

Typology: Exams

Pre 2010

Uploaded on 08/16/2009

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koofers-user-tc2 🇺🇸

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Test 3 Math 112 November 14, 2006
Answers
1a) -1/2 1b) e21c) 0
2) Iis made up of 3 integrals, only one of which converges, so Idiverges.
3) π/3
4) Compare to f(x) = 1/3x2which is bigger. Show f(x)g(x)0 and Z
1
f(x)dx converges
and then invoke the theorem.
5) (x, y) = (1/2,1/2) and (x, y) = (0,0)
6) Area = 2 Zπ/6
0
sin2θ
2 + 2 Zπ/2
π/6
(1 sin θ)2
2
7) dy
dxt=0
=1
2
d2
dx2t=0
=3
8
8a)A= 1
8b) The curve starts at (1,0) and spirals around as rdecreases 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 and 2/3π.
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.
10) s5
4·2 (eπ1)

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Test 3 Math 112 November 14, 2006 Answers

1a) -1/2 1b) e−^2 1c) 0

  1. I is made up of 3 integrals, only one of which converges, so I diverges.

  2. π/ 3

  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

  1. (x, y) = (1/ 2 , 1 /2) and (x, y) = (0, 0)

  2. Area = 2

∫ (^) π/ 6 0

sin^2 θ 2 dθ^ + 2

∫ (^) π/ 2 π/ 6

(1 − sin θ)^2 2 dθ

  1. dy dx

∣∣ ∣∣ ∣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)