College Mathematics: Learning Worksheets Chapter 12
Name ________________________________ Date ______________ Class ____________
Goal: To solve differential equations that involve growth and decay.
Section 12-3 Differential Equations;
Growth and Decay
Theorem 1: Exponential Growth Law
If
dQ rQ
dt and 0
(0) ,QQthen 0,
rt
QQe
where
0
Qamount of Q at 0t
Table 1: Exponential Growth
Description Model Solution
Unlimited growth dy
dt ky
,0kt
(0)
y
c
kt
y
ce
Exponential decay dy
(0)yc
kt
y
Limited growth ()
dy
(0) 0y
kt
Logistic Growth ()
dy
,0kt
1
(0)
M
c
y
M
College Mathematics: Learning Worksheets Chapter 12
In Problems 1–4, find the general or particular solution, as indicated, for each
differential equation.
1. 8
14
dy
x
dx
The differential equation can be found by using the integration properties from
Section 6.1.
8
y
14
y
y
xdx
2.
18 ;
dx (0)7y
The differential equation can be found by using the integration properties from
Section 6.2. Let 6
3,uxtherefore, 5
18 .du x dxRewrite the original integral in terms of the
variable u and solve.
6
53
18 x
y
xe dx
Using the condition given, find the specific value of C.
6
6
3
3(0)
7
x
y
eC
eC
36.
x
3. 8
dy
y
dx
4. 7;
dx
x
dt (0)6x
The differential equation is in the form of unlimited growth; therefore, using the
unlimited growth model in Table 1 yield’s the following solution:
x
Using the condition given, find the specific value of C.
7
t
xce
x
5. Find the amount A in an account after t years if
0.02
dA
A
dt and (0) 8000A
The exponential growth law (unlimited growth) applies to the situation. Since we are
given the initial amount, the amount in an account after t years would be:
6. A single injection of a drug is administered to a patient. The amount Q in the body
decreases at a rate proportional to the amount present. For a particular drug, the rate
is 8% per hour. Thus, 0.08
dQ Q
dt and 0
(0) ,QQwhere t is time in hours.
a) If the initial injection is 10 milliliters [
(0) 10Q
], find ( )QQtsatisfying
both conditions.
b) How many milliliters (to two decimal places) are in the body after 6 hours?
c) How many hours (to two decimal places) will it take for half the drug to be left in
the body?
b) After 6 hours, 6,ttherefore
0.08(6)
(6) 10
(6) 10
Qe
Qe
c) Half the initial injection is 5 milliliters. We need to find t such that ( ) 5.Qt
0.08
() 10
t
Qt e
Therefore, it will take approximately 8.66 hours for the amount of the drug to be half
363
7. A company is trying to expose a new product to as many people as possible through
radio ads. Suppose that the rate of exposure to new people is proportional to the
number of those who have not heard of the product out of L possible listeners. No one
is aware of the product at the start of the campaign, and after 15 days, 50% of L are
aware of the product. Mathematically,
(),
dN kL N
dt
(0) 0,N
and (15) 0.5 .NL
a) Solve the differential equation.
b) What percent of L will have been exposed after 7 days of the campaign?
c) How many days will it take to expose 75% of L?
a) By the exponential growth law (limited growth) for the given conditions, we have
the equation
15
15
() (1 )
0.5 (1 )
kt
k
k
Nt L e
LL e
t
b) After 7 days, 7,ttherefore,
0.046(7)
(7) (1 )
NLe
College Mathematics: Learning Worksheets Chapter 12
364
c) Find t such that ( ) 0.75 .Nt L
0.046
0.046
0.046
() (1 )
0.75 (1 )
t
t
t
Nt L e
LL e
College Mathematics: Learning Worksheets Chapter 12
Name ________________________________ Date ______________ Class ____________
Goal: To calculate the values of definite integrals using the properties.
Section 12-4 The Definite Inte
g
ral
Theorem: Limits of Left and Right Sums
If () 0fxand is either increasing or decreasing on [a, b], then its left and right sums
approach the same real number as .n
Theorem: Limit of Riemann Sums
If f is a continuous function on [a, b], then the Riemann sums for f on [a, b] approach
a real number limit I as .n
Definition: Definite Integral
Let f be a continuous function on [a, b]. The limit I of Riemann sums for f on [a, b] is
called the definite integral of f from a to b and is denoted as () .
b
a
f
xdx
Properties: Definite Integrals
1. () 0
a
afxdx
f
ba
3. () () ,
bb
aa
kf x dx k f x dx
k a constant
f
bbb
f
College Mathematics: Learning Worksheets Chapter 12
In Problems 1 and 2, calculate the indicated Riemann sum n
Sfor the function 2
() 7 .
f
xx
1. Partition [–1, 9] into five subintervals of equal length, and for each subinterval
1
[,],
kk
x
x
let 1
()/2.
kk k
cx x
9(1) 10 2
55
x
11 0
13 2
35 4
(0) 7
f
(2) 11
f
(4) 23
f
57 6
79 8
(6) 43
f
(8) 71
f
512345
() () () () ()
S fc x fc x fc x fc x fc x
2. Partition [–4, 8] into four subintervals of equal length, and for each subinterval
1
[,],
kk
x
x
let 1
(2 ) / 3.
kkk
cxx
8 ( 4) 12 3
44
x
3
2(2) 5 3
3
c
4
2(5) 8 6
3
c
2
(3) 7 (3)
(3) 16
f
f
2
(6) 7 (6)
(6) 43
f
f
In Problems 3–7, calculate the definite integral, given that
6
018xdx
62
072xdx
92
6171xdx
3. 6
03
x
dx
x
4. 62
0(2 )
x
xdx
x
22
368
5. 92
02
x
dx
x
99
22
92
0
2 486
xdx
6. 0
67
x
dx
x
6
7. 72
7(215)
x
xdx
College Mathematics: Learning Worksheets Chapter 12
369
Name ________________________________ Date ______________ Class ____________
Goal: To use the fundamental theorem of calculus to solve problems.
In Problems 1–7, evaluate the integrals.
1. 63
2(8 3)
x
dx
6
4
63
2
2
83
(8 3) 41
xx
xdx
Section 12-5 The Fundamental Theorem
of Calculus
Theorem 1: Fundamental Theorem of Calculus
If f is a continuous function on [a, b], and F is any antiderivative of f, then
()()()
b
a
f
xdx Fb Fa
Definition: Average Value of a Continuous Function f over [a, b]
f
370
2. 5
0(2 )
x
edx
5
0
xx
3. 732
x
xx dx
4. 22
13
5
3dx
x
12
22 22
13 13
55( 3)
3
dx x dx
x
371
5. 134 6
02(3 2)
x
xdx
We will solve the problem using the substitution method of integration. Let
4
32,uxtherefore 33
12 6(2 ).du x dx du x dx Rewrite the original integral in
terms of the variable u and solve.
11
34 6 6
00
1
2(3 2) 6
x
x
xx dx udu
6.
2
2
132
325
53
xx dx
xx x
We will solve the problem using the substitution method of integration. Let
32
53,ux x xtherefore, 2
(3 2 5) .du x x dx Rewrite the original integral in
terms of the variable u and solve.
2
22
11
32
325 1
53
x
x
xx dx du
u
xx x
7. 22
136 1
x
xdx
We will solve the problem using the substitution method of integration. Let
2
61,uxtherefore, 12 4(3 ) .du xdx du x dx Rewrite the original integral in
terms of the variable u and solve.
12
22
2
11
1
36 1 4
x
x
xx dx udu
In Problems 8 and 9, find the average value of the function over the given interval.
8. 3
() 8 16 4;fx x x [1, 6]
63
11
( ) (8 16 4)
b
373
9. 0.3
() 2
x
gx e
[0,10]
We will solve the problem using the substitution method of integration. Let
0.3 ,ux therefore 0.3 .du dx Rewrite the original integral in terms of the
variable u and solve
10 0.3
0
10
0
11
() (2 )
10 0
2()
(10)( 0.3)
bx
a
xu
x
f
xdx e dx
ba
edu
10. The total cost (in dollars) of manufacturing x units of a product is
( ) 50,000 400 .Cx x
a) Find the average cost per unit if 400 units are produced.
[Hint: Recall that ()Cxis the average cost per unit.]
College Mathematics: Learning Worksheets Chapter 12
374
b) Find the average value of the cost over the interval [0, 400].
400
0
11
( ) (50,000 400 )
400 0
b
afxdx xdx
ba
11. A company manufactures a product and the research department produced the
marginal cost function
‘( ) 300 4
x
Cx 0 800,x
where ‘()Cxis in dollars and x is the number of units produced per month. Compute
the increase in cost going from a production level of 400 units per month to 800 units
per month. Set up a definite integral and evaluate it.
800
400
‘( ) (300 )
4
b
a
x
C x dx dx