Test 3 Math 112 November 14, 2006
Answers
1a) -1/2 1b) e−21c) 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
2dθ + 2 Zπ/2
π/6
(1 −sin θ)2
2dθ
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/π 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 (eπ−1)