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MATH 1432
Section 13470
M-F 12:00 - 2:00pm SEC 103
James West
[email protected]
620 PGH
Office Hours:
2:00 – 4:00pm MW in 620 PGH or by appointment
Section 7.6: Differential Equations and Exponential Growth/Decay
These type of differential equations are used in exponential growth and decay
models.
Exponential Growth and Decay models are used in:
Population Growth/Decay
Radioactive Decay
Investments
Mixing problems
Newton’s Law of Cooling
Formula:
A ( t ) = A0 e kt
A0= A(0) = initial amount
k = growth/decay rate
Note that we have a quantity that changes at a rate proportional to itself!
A 100-liter tank initially full of water develops a leak at the bottom. Given that
10% of the water leaks out in the first 5 minutes, find the amount of water left in
the tank 15 minutes after the leak develops if the water drains off at a rate that is
proportional to the amount of water present.
A certain species of virulent bacteria is being grown in a culture. It is observed
that the rate of growth of the bacterial population is proportional to the number
present. If there were 5000 bacteria in the initial population and the number
doubled after the first 60 minutes, how many bacteria will be present after 4
hours?
In a bacteria growing experiment, a biologist observes that the number of bacteria
in a certain culture triples every 4 hours. After 12 hours, it is estimated that there
are 1 million bacteria in the culture.
a. How many bacteria were present initially?
b. What is the doubling time for the bacteria population?
At what rate r of continuous compounding does a sum of money double in 10
years?
After 3 days a sample of radon-222 decayed to 58% of its original amount. What
is the half-life of radon-222? How long would it take the original sample to decay
to 10% of its original amount?
A deposit of $1000 is made into a fund with an annual interest rate of 10 percent.
Find the time (in years) necessary for the investment to double if the interest is
compounded continuously.
Section 7.7
Improper Integrals
The definition of the definite integral
∫ f ( x ) dx
b
a
requires that [a, b] be finite and
that f (x) be bounded on [a, b].
Also, the Fundamental Theorem of Calculus requires that f be continuous on [a,
b].
If one or both of the limits of integration are infinite or if f has a finite number of
infinite discontinuities on [a, b], then the integral is called an improper integral.
Types of improper integrals:
A. (one or both bounds are infinite)
∞
dx
,
x
∞
dx
∫1
∫−∞ x 2 + 1 are improper because one
or both bounds are infinite.
B. (infinite discontinuity at a boundary)
∫
5
1
3dx
∫−∞ x 4 + 5 and
1
dx
is improper because f ( x ) =
x −1
1
has an infinite
x −1
discontinuity at x = 1.
C. (infinite discontinuity in the interior)
2
dx
∫ ( x + 1)
−2
2
is improper because f ( x ) =
1
( x + 1)
2
has an infinite
discontinuity at x = –1, and –1 is between –2 and 2.
For the first type of improper integrals:
1) If f is continuous on [ a, ∞ ) , then
∞
∫ f ( x ) dx
a
= lim
b →∞
∫ f ( x ) dx .
b
a
2) If f is continuous on ( −∞,b ], then
∫
b
−∞
f ( x ) dx = lim
∫ f ( x ) dx .
b
a → −∞
a
3) If f is continuous on ( −∞, ∞ ) , then
∞
c
−∞
−∞
f ( x ) dx ∫
∫=
f ( x ) dx +
∞
∫ f ( x ) dx .
c
If the limit exists, then the improper integral is said to converge.
Otherwise, it diverges.
Examples for the first type of improper integral.
1.
∫
∞
1
dx
x
2.
∫
∞
2
e − x dx
3.
∫
∞
0
1
dx
x2 + 1
4.
ex
∫−∞ 1 + e2 x dx
∞
The second and third type of improper integral:
1.
If f is continuous on [ a, b ) but has an infinite discontinuity at b, then
∫
b
a
2.
f ( x ) dx = lim−
c→ b
∫ f ( x ) dx .
c
a
If f is continuous on ( a, b ] but has an infinite discontinuity at a, then
∫ f ( x ) dx
b
a
= lim+
c→ a
∫ f ( x ) dx .
b
c
3. If f is continuous on [a, b] except for some c in (a, b) at which f has an
infinite discontinuity, then
f ( x ) dx ∫ f ( x ) dx
∫=
b
c
a
a
+
∫ f ( x ) dx ,
b
c
provided both integrals on the right converge. If either integral on the right
diverges, we say that the integral on the left diverges.
Examples for the second type of improper integral.
1.
∫
1
0
dx
3
x
2.
∫
2
0
dx
x3
3.
∫
27
0
3
dx
27 − x
4.
∫
4
1
dx
x−2
Important examples:
∫
∞
1
dx
p =1
p
x
∫
∞
1
dx
p >1
xp
∫
∞
1
dx
0 < p <1
xp
∫
∞
1
dx
xp
Diverges for p ≤ 1
Converges for p > 1