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 th
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
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
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.
Solut
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
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
Cal
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 opt
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
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 sho
Calculus II with Applications to the Life Sciences
MATH 31S

Fall 2011
Math 31S. Rumbos
Fall 2011
Solutions to Assignment #3
1. Solve the initial value problem
= 2
(0) = 2.
Solution: Compute
2 = 2 +
() = 2 +
0
3
,
3
for all .
2. Solve the initial value problem
=
(1
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:
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 popu
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 paramete
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 star
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 introduce
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
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
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 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 >
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 applic
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 princ
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 (
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)
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 writte
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 ord