BC Calculus Techniques of Integration Test Review
I. Differential Equations
 Find both general and particular solutions to differential equations
 Review problems:
dy 3 x  1
.

dx
y
1.
Find the general solution of the differential equation
2.
Identify f(x) given f ' ' ( x)  2 x  cos x , f ' ( )  0 , and f (0) 

2
.
II. Slope Fields
 Be able to draw a slope field from a given differential equation.
 Review problems:
1.
Draw the slope field for the differential equation
dy x  y
.

dx
x
III. Euler’s Method
 Be able to approximate given values from differential equations.
 Review Problems:
1.
Use Euler’s method with two steps of width x 
1
dy x  y
to approximate y(4) if
and the point (3, 0)

2
dx
x
belongs to the graph of the specific solution of the differential equation.
2.
Use Euler’s method with three steps of width x 
1
 7  dy
to approximate y   if
 x y and the point (1, 0)
4
dx
4
 
belongs to the graph of the solution of the differential equation.
IV. Integration By Parts and Tabular Integration
 Find antiderivatives and definite integrals using by parts or tabular.
 Know proof of integration by parts formula.
 Review Problems:
1.
2.
3.
Evaluate
x
2
sin xdx both by parts and tabular.
 e sin xdx
 ln xdx
x
2
4.
x
2
ln xdx
1
V. Exponential Growth & Decay

Know what “a population grows at a rate proportional to the amount present.”

Solve this differential equation to get the exponential growth/decay model.

Newton’s Law of Cooling:

 dP

 kt 

 dt

T  T A  (T0  T A )e kt
1. T is an object’s temperature
2. TA is the temperature of the surrounding environment
3. T0 is the initial temperature of the object
Review Problems:
1. The radioactive isotope Carbon-14 has a half-life of 5730 years. Given N grams of C-14, in approximately how
many years will the mass decay to one third of its original measure?
2. The housekeeping staff in a hotel discovers a corpse, the apparent victim of a fatal overnight heart attack, and alerts
the police. Investigators arrive at 11:00 AM and note that the body’s temperature is 81F. By 12:30 PM, the
temperature has dropped to 77F. Noting that the thermostat in the room is set to maintain a constant temperature of
67F, and assuming the victim’s temperature was 98.6F when he died, at what time (to the nearest minute) did the
heart attack occur?
3.
At the start of a science experiment, a scientist observes 125 bacterial colonies growing in the agar of a Petri dish.
Exactly six hours later, the number of colonies has grown to 190. Assuming that resources are not a limiting issue to
growth (that is, assuming the bacteria do not eventually run out of food and space implying exponential growth),
how many hours will it take the total bacteria population to reach 20,000?
VI. Partial Fractions
 Decompose an integrand using partial fractions and find the antiderivatives. (Be prepared to use distinct linear factors,
repeated linear factors, or quadratic factors.)
 Review Problems:
1.
2.
3.
4.
2x  3
dx
2
5
11x  15
 4 x 2  3x dx
x
 4 x 2  4 x  1 dx
2 x 3  5x 2  6 x  3
dx
 x 4  3x 2
x
VII. Logistic Growth
 The rate at which a population grows is directly proportional to the amount present times the difference between the carrying
capacity (M) and the population.
dP
 kP( M  P)
dt
M
1  Ae ( Mk ) t

Derive the General Logistic Equation using partial fractions. P 

Review Problems:
1. A certain wild animal preserve can support no more than 250 lowland gorillas. Twenty-eight gorillas were known to
be in the preserve in 1970. Assume that the rate of growth of the population is
2.
dP
 0.0004 P (250  P ) , where
dt
t is in years.
 Find a formula for the gorilla population in terms of t.
 How long will it take for the gorilla population to reach the carrying capacity of the preserve?
The table below shows the population of Virginia Beach, VA for selected years between 1950 and 2003.
Years after 1950
Population
0
5390
20
172,106
30
262,199
40
393,069
50
425,257
53
439,467
a.) Use logistic regression (on your calculator) to find a curve to model the data.
b.) Based on the regression equation, what number will the Virginia Beach population approach in the long run?
c.) Based on the regression equation, when will the Virginia Beach population first exceed 450,000 people?
d.) Write a logistic differential equation in the form of
Beach data in the table.
dP
 kP( M  P) that models the growth of the Virginia
dt