Name:
Advanced Integration and Differential Equations Study Guide
1.
ò
¥
2.
ò
3
xe- x dx
2
-¥
1
dx
3- x
0
3. Find the general solution to the differential equation
dy y2 (x - 3)
=
dx
x3
4. The volume of a cube is increasing at a rate proportional to its volume at any
time, t. If the volume is 8 ft3 originally, and 12 ft3 after 5 seconds, what is the volume
at t=12 seconds?
5. lim+ xln x
x®0
6.
ò cos(2x)e
7.
ò x(x
2x
dx
2x - 3
dx
2
+1)
8. Use Euler’s Method with a step size of h=0.1 to find approximate values of the
solution at t=0.3 if y(0)=1.
9.
The number of moose in a national park is modeled by M (t ) that satisfies the
dM
M 

 2 M 1 
logistic differential equation
 , where the initial
dt
 1000 
population M (0)  300 and
t is the time in years.
a)
At what rate is the moose population changing when there are 400
moose in the park?
b)
Find lim M (t )
t 
c)
Find the equation of M (t )
d)
How many moose are in the park at t = 1?
e)
At what time t is the moose population growing the fastest?
10.
Find the particular solution of the differential equation dP  3( P  40)dt  0
that satisfies the initial condition P (0)  60 . Express you answer as a
function P (t ) .
11.
On the graph below, sketch the graphs of three logistic growth functions, all
dy
of which are solutions to the same differential equation
with a carrying
dt
capacity of 100. Create an appropriate scale on the y-axis and place a tick
mark at each interval. Clearly indicate any points of inflection.
a.
Sketch the graph of the solution curve whose initial condition is
y (0)  25 .
b. Using a thicker line than before, sketch the graph of the solution curve
whose initial condition is y (0)  75 .
c. Using a dotted line, sketch the graph of the solution curve whose initial
condition is y (0)  150
**Also look at panther example from 6-3 PP Day 2