Directions:
Exam #3
Math 181 Wednesday 5/19/2021 Name:___________ Score:___________
– Be sure to read each question carefully.
– Do not necessarily work each problem in order.
– Do the easy problems first.
– Show all work clearly to receive full credit.
– No graphing calculators are allowed.
x = 2t2 10t+ 1. Determine the area bounded by the curve:
for 0 ≤ t ≤ 3 (3 pts)
(2t + 3)3/ 2
2. Find the exact arc-length of x = 3 0 ≤ t ≤ 4 (5pts)
2
y= t +5t 1+
t2 y=t +
2
3. Sketch the graph of the following functions in polar coordinates: (10pts) 3a) r2 = −4sin(2θ)
3b) r = 5 cos (3θ )
3c) r=2 5−sinθ
4. Find equation of tangent line (in rectangular coordinates) to the curve in polar coordinate: r = sin (3θ ) at θ = π6 (6pts)
5. Find the arc – length the following function in polar coordinate: r = 1 +cos (2θ ) for 0 ≤ θ ≤ π 2 (6pts)
6. Sketch the following curves:
the following: (12pts)
6a) Inside r1 / outside r2 6b) Inside r2 / outside r1 6c) Inside both r1 and r2.
r = −3 sin (θ ) and r = 3 + 3 sin (θ ) . Then SETUP the integrals of the region of 12
7. Evaluate the limits of the following sequences: (9 pts)
7a)
7b)
2n+3 b = 2n − 3
7c)
c =2n+1! ( )
{a }= 1 n2 2
3n +n− 3n +1
{n} 2n+1
{n} (2n+3)!
8. Test for convergence / divergence. Find the limit if convergence. (9 pts)
8a)
n=3
∞ 4n−2 ∑23n−4
∞1 1 8b) ∑e n+4 −e n+5
n=0
8c) ∑∞ ln(2n−1)−ln(n+2) n=1
9. Test for convergence / divergence (Indicate the theorem that you use)
9a)
(32 pts)
9b)
2 ∑ 2n +1
∞ 7n2 +2n+4 ∑73
n=1 4n +3n +1
∞
n
n=1 e
∞ 4n2 +5n+1n ∑ 3n2 +n+4
9c)
n=0
∞ (−1)n (7n+1)
9d) ∑ 8 3
n=1 5n +n +4
9e)
∞ 7n−1n2 ∑7n+3
n=1
∞
1⋅3⋅5⋅⋅⋅(2n−1)
∑2⋅4⋅6⋅⋅⋅(2n)3n+2 +1
9f)
n=1
9g)
∑()
9h)
∑64
n=1 7n −4n +2
7
n=1 n2+n
∞
∞ 5n2 −n+3
10. Determine number of terms are needed to ensure the approximation of the following sum accurate within 0.00001. (8 pts)
10a)
∞ ∑3
cos(nπ) n=1 3n +n+4
∞
10b)
n
2 ∑(2 )
n=1 2n +1