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Region Bounded - Calculus Two for Biological Sciences - Exam, Exams of Calculus

This is the Exam of Calculus Two for Biological Sciences which includes Numerical Sum, Approximation, Midpoint Rule, Terms, Modeled, Type Equilibrium Points, Starting Population etc. Key important points are: Region, Area, Region Is Rotated, Volume, Axis, Rest Length, Stretched Additional, Constant, Temperature Varies, Average Temperature

Typology: Exams

2012/2013

Uploaded on 02/18/2013

abhaya
abhaya 🇮🇳

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Final Exam, Math 30, May 16, 2008
Solve for y explicitly. Indefinite integrals must be in terms of x. Check out useful
information in the Appendix.
1) Find the volume of the solid obtained by rotating the given region about the y-
axis. (10 pts)
Region bounded by: 31),ln(
=xxy
Evaluate the following indefinite integrals (10 pts each) solutions must be in
terms of x.
2) dx
x
x)sin(ln
3) +dx
xx 5
1
22
4) Calculate the constant c for the probability density function p(x) (10 pts)
500)(
50)5()( 2
><=
=
xandxxp
xxxcxp
5) Find the solution to the differential equation (15 pts)
2)0(,
65
2=
++
=y
xx
y
dx
dy
6) Air freshener’s (AF) evaporation rate is given by daym
dt
dm /02., ==
λλ
and
m is the mass. If after 10 days 10.0 mg of AF is left, what was the starting mass of
AF? (10 pts)
7) A climate model for carbon dioxide concentration in atmosphere (C) in ppm is
represented by the following equation.
)350)(400()450( 2CCCk
dt
dC =
a) Find and identify by type all equilibrium points (10 pts)
pf3

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Final Exam, Math 30, May 16, 2008

Solve for y explicitly. Indefinite integrals must be in terms of x. Check out useful

information in the Appendix.

1) Find the volume of the solid obtained by rotating the given region about the y-

axis. (10 pts)

Region bounded by: y = ln( x ), 1 ≤ x ≤ 3

Evaluate the following indefinite integrals (10 pts each) solutions must be in

terms of x.

2) ∫ dx

x

sin(ln x )

dx x x 5

2 2

4) Calculate the constant c for the probability density function p(x) (10 pts)

2

p x x and x

p x c x x x

5) Find the solution to the differential equation (15 pts)

= y x x

y

dx

dy

6) Air freshener’s (AF) evaporation rate is given by m day

dt

dm

=−λ , λ=. 02 / and

m is the mass. If after 10 days 10.0 mg of AF is left, what was the starting mass of

AF? (10 pts)

7) A climate model for carbon dioxide concentration in atmosphere (C) in ppm is

represented by the following equation.

2 kC C C dt

dC = − − −

a) Find and identify by type all equilibrium points (10 pts)

b) Plot the solution if the starting concentration is 410 ppm. (4 pts)

c) Is there anything troubling about 450 ppm concentration? (3 pts)

d) By what minimum amount would you reduce the starting C to have a stable

concentration over time? (3 pts)

8) Consider a population P=P(t) where m and λ are constants representing

immigration rate and growth rate. m is always positive and λ can be either positive

or negative.

a) Solve the differential equation (10 pts)

P m P P o dt

dP

Find the conditions on λ in terms of m and P o that will lead population to:

b) Grow (2 pts)

c) Decline (2 pts)

d) Stay constant (2 pts)

9) Write down but don’t solve the differential equations (DEs) for the following

problems. Pay close attention to the wording. Proportionality constant k is

always positive. Make sure that you have correct signs. Analyze what is

changing and what is driving the change.

a) During a disease outbreak in a town the number healthy people decreases

proportionally to the product of healthy people and sick people. Write the DE for

the rate at which the number of healthy people decreases in terms of H (number of

healthy people), T(Total number of people) and k. (7 pts)

b) Cooling rate of a hot cup of coffee is proportional to difference between coffee

temperature (T) and ambient air temperature (Ta). Write the DE for the rate at

which coffee is cooling in terms of T, k, and T a. (7 pts)

10) For the following predator pray system determine which of the variables, x or

y, represent the prey population and which represent the predator population

(Explain) (5 pts). Find equilibrium solutions for predator and prey. (10 pts)

y xy dt

dy

x xy dt

dx