Math 105/206 - Practice problems for Quiz 4 Problem 1

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Math 105/206 - Practice problems for Quiz 4
Problem 1
Compute Simpson’s approximation for the integral
Z π/2
sin(2x) dx
0
with n = 4 subintervals, and estimate the absolute error. How big do you need to take n to have an error of at
1
?
most 1000
Problem 2
Find the general solution of the following separable first-order differential equations.
dy
dx
= y(x2 + 1) where y > 0
√
• x2 dw
= w(3x + 1).
dx
•
Problem 3
Find the solution to the following first-order initial value problem.



dz
z2


= 2
dx
x +1



 z(0) = 16
Problem 4
You have a colony of bacteria that, if left undisturbed, grows proportionally to the population, with a factor
0.008 (and they never die......). Let’s say we use years as unit of time, and you harvest a quantity h of the
bacteria every year, and say the initial population is P0 .
• if P0 = 2000, what’s the value of h that gives you a constant population?
• if h = 200, what’s the value of P0 that gives you a constant population?
• if P0 = 100 and h = 50, find an expression for the population at a generic time t.
Problem 5
Are the following functions CDFs for a continuous random variable?
1. F (x) = π1 (arctan(x) + π2 )
2. F (x) = 1 + e−x
3. F (x) =
1
1+log(x2 +1)

if x < 1
 0
2x − 2 if 1 ≤ x < 32
4. F (x) =

1
if x ≥ 32

if x < 1
 0
x − 1 if 1 ≤ x < 32
5. F (x) =

1
if x ≥ 23
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