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Math 202 Skills Check 3
Directions: Complete each problem and write a solution that is mathematically sound and logical, explaining the steps taken to reach your answer to each problem. For this Skills Check, you may use the
course textbook, the instructor lecture notes, your class notes, and your class notes from Calculus I. You are
allowed to work together but you must write up solutions separately. Each question is worth 5 points. This
is due at the beginning of class on Friday, April 17th. Good luck!
1. Solve the initial-value problem
y 0 − xey = 2ey , y(0) = 0
2. A cell of the bacterium E. coli divides into two cells every 20 minutes when placed in a nutrient
culture. Let y = y(t) be the number of cells that are present t minutes after a single cell is placed in the
culture. Assume that the growth of the bacteria is approximated by an exponential growth model.
(a) Find an initial-value problem whose solution is y(t).
(b) Find a formula for y(t).
(c) How many cells are present after 2 hours?
(d) How long does it take for the number of cells to reach 1,000,000?
3. Consider the initial-value problem
√
y
dy
=
, y(0) = 1.
dx
2
(a) Use Euler’s Method with step sizes of h = 0.2 and h = 0.1 to obtain two approximations of y(1).
(b) Find y(1) exactly.
4. For what positive values of b does the sequence b, 0, b2 , 0, b3 , 0, b4 , · · · converge? Justify your answer.
5. Consider the sequence
√
a1 = p6
√
a2 = q6 + 6
p
√
a3 = r6 + 6 + 6
q
p
√
a4 = 6 + 6 + 6 + 6
..
.
(a) Find a recursion formula for an+1 .
(b) Assuming that the sequence converges, find the limit.
1
6. Consider the sequence {an }, where
an =
2
n
1
+ 2 + ··· + 2
n2
n
n
(a) Find a1 , a2 , a3 , and a4 .
(b) Use numerical evidence to make a conjecture about the limit of the sequence.
(c) Confirm your conjecture by expressing an in closed form and calculating the limit.
7. Determine whether the series converges, and if so find its sum.
∞ X
1
1
−
2k
2k+1
k=1
8. Determine whether the series converges, and if so find its sum.
∞
X
53k 71−k
k=1
9. Determine whether the series converges, and if so, find its sum.
∞
X
k=1
1
9k 2 + 3k − 2
10. Determine whether the series converges, and if so, find its sum.
∞
X
7−k 3k+1 −
k=1
2
2k+1
5k