Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #7
1. Show that exp( ) =
exp()
for all , .
exp()
Solution: Write exp() = exp( + ) so that
exp() = exp( ) exp().
Solving for exp( ) in (1) yields the result.
(1)
2. Let and denote real numbers and put ()
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #10
1. Let = () denote the number of radioactive isotopes of an element in a
sample at time . Assume that the number of atoms that decay in a unit of
time is a fraction, , of the number of isotopes pres
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #9
1. Assume that a certain strain of E Coli bacteria in a culture has a doubling time
of about 30 minutes.
(a) Assuming a Malthusian growth model for the bacteria, give an expression,
(), for the numb
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #8
1. Use the properties of ln and exp to compute the exact value of ln( ). Compare
your result with the approximation given by a calculator.
Solution: Compute
1
1
ln( ) = ln[1/2 ] = ln = .
2
2
The appr
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #6
2.5
1
< 1 by comparing the area under the graph of = 1/
1
from = 1 to = 2.5 with the sum of the areas of circumscribed rectangles of
width 0.25.
1. Show that
Use this result to conclude that 2.5 < .
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #1
1. Let denote a dierentiable on some, nonempty, open interval, . Assume
that () = 0 for all in the interval . Use the Fundamental Theorem of
Calculus to show that must be constant on .
Suggestion: Le
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #2
1. Let () denote the size of a bacterial population in culture at time . ()
can be measured by weight (e.g., grams), or by concentration via optical density
measurements. Assume that = () is twice di
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
Solutions to Assignment #4
1. Solve the initial value problem
= sin(2 );
(0) = 0,
for .
Solution: Compute
sin( 2 ) ,
() =
0
by making the change of variable = 2 ; so that, = 2 and
1
() =
2
2
sin()
0
=
2
1
[ cos()]
0
2
=
]
1[
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
Solutions to Assignment #5
1. Show that ln
()
= ln ln , for , > 0.
Solution: Write
= 1 so that
()
= ln(1 )
ln
= ln() + ln(1 )
= ln() + (1) ln()
= ln ln ,
which was to be shown.
2. Let () = ln
1 + 2 for all .
(a) Compute () and (
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Exam 2
1. In this problem you will solve the linear, rstorder dierential equation
= + .
(1)
(a) Use integration by parts to evaluate the integral
.
Solution: Set
=
then, =
so that
=
= ,
and
and
=
,
from wh
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Additional Review Problems
1. An initial population of 50, 000 inhabits a microcosm with carrying capacity
of 100, 000. Suppose that, after ve years, the population increases to 60, 000.
Determine the intrinsic gr
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions Review Problems for Exam #2
1. Suppose that the growth of a population of size = () follows the dierential
equation model
= ,
(1)
where and are positive parameters.
(a) Give an interpretation for the model in (1).
So
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions Review Problems for Exam #1
1. Water leaks out a barrel at a rate proportional to the square root of the depth
of the water at that time. If the water level starts at 36 inches and drops to 35
inches in 1 minute, how
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Exam 1
1. When people smoke, carbon monoxide is released into the air. Suppose that in
a room of volume 60 m3 , air containing 5% carbon monoxide is introduced at
a rate of 0.002 m3 /min. (This means that 5% of th
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Exam 2
Friday, December 2, 2011
Name:
Show all signicant work and justify all your answers. This is a closed book exam. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the followi
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Exam 1
Wednesday, October 12, 2011
Name:
Show all signicant work and justify all your answers. This is a closed book exam. Use
your own paper and/or the paper provided by the instructor. You have 50 minutes
to work on the foll
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #19
1. Consider the rst order dierential equation
d
= 5 6 + 2 .
d
(a) Find all equilibrium solutions of the equation, and determine the nature
of their stability.
Solution: Set () = 5 6 + 2 and factor t
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #17
1. Let () =
1
for > 1. Give the linear approximation to around
1+
= 0.
Solution: Compute
(; 0) = (0) + (0),
where
() =
1
,
2(1 + )3/2
for > 1.
Thus,
1
(; 0) = 1 ,
2
for .
2. Let () = for all . G
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #18
1. For the following rstorder dierential equations, nd all the equilibrium solutions and use the principle of linearized stability, when applicable, to determine
whether the equilibrium solutions ar
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #14
1. For any population, ignoring migration, harvesting, or predation, one can model
the relative growth rate by the following conservation principle
1
= birth rate (per capita) death rate (per capit
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #16
1. Logistic Growth1 . Suppose that the growth of a certain animal population is
governed by the dierential equation
1000
= 100 ,
(1)
where () denote the number of individuals in the population at
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #11
1. Use the method of separation of variables to nd all solutions to the dierential
equation
= .
Solution: Separate variables to obtain
= .
(1)
Evaluating the integrals in (1) yields
2
= + 1 ,
2
(2)
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #12
1. Solve the initial value problem
= + ,
(0) = 0.
(1)
Solution: Rewrite the equation as
+ =
and multiply by to obtain
+ = ,
which can be written as
[ ] = ,
(2)
by virtue of the product rule. Integ
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
1
Solutions to Assignment #13
1. Use the method of integrating factor discussed in Section 4.8.5 in the class
lecture notes to nd the general solution to the linear, rst order dierential
equation with constant coecients
= + ,
(1