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Math 30 Final Exam: Integration, Diff. Equations, and Logistics Eq., Exams of Calculus

Guru Angad Dev Veterinary and Animal Sciences University Calculus

The final exam for math 30, held on december 15, 2007. The exam covers various topics including integration, differential equations, and the logistics equation. Students are required to find volumes, average values, evaluate indefinite integrals, solve differential equations, and determine equilibrium points. The document also includes useful identities and information about the logistics equation.

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Final Exam, Math 30, December 15, 2007

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 x-axis. (10 pts)

Region bounded by:

,2 xyxy ==

2) Find the average value of the function on the interval

)sin(10)( 2

xxxu =

[

]

,0 (10 pts)

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

terms of x.

∫dxxx )(cos)(sin 32

4) ∫−dx

xx 9

5) ∫++

+dx

6) Solve the differential equation that satisfies the given initial condition.

(10 pts)

dy sin

=, 2)

Partial preview of the text

Download Math 30 Final Exam: Integration, Diff. Equations, and Logistics Eq. and more Exams Calculus in PDF only on Docsity!

Final Exam, Math 30, December 15, 2007

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 x-axis. (10 pts)

Region bounded by:

y = 2 x , y = x

Find the average value of the function (^ )^10 sin( )on the interval

2 u x = x x

[ 0 , π] (10 pts)

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

terms of x.

sin ( x )cos ( x ) dx

2 3

−

x x 9

2 2

x x

5 6

6) Solve the differential equation that satisfies the given initial condition.

(10 pts)

y x dt

dy sin

2

, )^2 2

( =

π y

7) Solve the differential equation: x y

= + where y(0)=1.0 (10 pts)

Sketch a graph of the solutions that satisfy the given initial conditions on the

slope field map. (5 pts)

i) y(0) = 1.

ii) y(0) = -1.

Slope field for y’=x+y

10) The population of aphids on a rose plant increases at a rate proportional

to the number present.

a) Write a differential equation for population of aphids at time t in days.

(10 pts)

b) Find the solution to the differential equation where at t=0 there were 1000

aphids and population doubles every 10 days. (10 pts)

11) Newton’s Law of cooling states that the rate of cooling is proportional to

the temperature difference between the object and its surrounding. (10 pts)

The differential equation for temperature, T, of a coffee cup as a function of

time (t) where T A is the ambient temperature of air, T o > TA is the initial

temperature of the coffee and k is the proportionality constant is: (5 pts)

A) T’ = -kT(1- T/TA)

B) T’ = -k(T - TA)

C) T’ = k(T - T A)

D) T’ = kT/TA

E) T’ = -kT/TA

The solution to the differential equation is: (5 pts)

A) T(t)= T A/(1+Aexp(-kt)), where A = (T A – To)/To

B) T(t) = TA + Toexp(-kt)

C) T(t) = T Aexp(-kt) – TA + To

D) T(t) = TA + (To – TA)exp(-kt)

E) T(t) = (To/TA)exp(-kt)

Extra Credit (10 pts)

12) 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). Find equilibrium solutions for preditor and prey.

y xy dt

x xy dt

Appendix:

Identities:

( 1 cos( 2 )) 2

sin ( )

cos ( ) sin ( ) 1

sec ( ) tan ( ) 1

2 2

x x

tan( )sec( )

sec( )

sec ( )

tan( ) 2

Logistics Equation:

1 exp( )

P

K P

A

A kt

K

kPK P dt

Math 30 Final Exam: Integration, Diff. Equations, and Logistics Eq., Exams of Calculus

Related documents

Partial preview of the text

Download Math 30 Final Exam: Integration, Diff. Equations, and Logistics Eq. and more Exams Calculus in PDF only on Docsity!

Final Exam, Math 30, December 15, 2007

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 x-axis. (10 pts)

Region bounded by:

y = 2 x , y = x

[ 0 , π] (10 pts)

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

terms of x.

6) Solve the differential equation that satisfies the given initial condition.

(10 pts)

2

7) Solve the differential equation: x y

= + where y(0)=1.0 (10 pts)

Sketch a graph of the solutions that satisfy the given initial conditions on the

slope field map. (5 pts)

i) y(0) = 1.

ii) y(0) = -1.

10) The population of aphids on a rose plant increases at a rate proportional

to the number present.

a) Write a differential equation for population of aphids at time t in days.

(10 pts)

b) Find the solution to the differential equation where at t=0 there were 1000

aphids and population doubles every 10 days. (10 pts)

11) Newton’s Law of cooling states that the rate of cooling is proportional to

the temperature difference between the object and its surrounding. (10 pts)

The differential equation for temperature, T, of a coffee cup as a function of

time (t) where T A is the ambient temperature of air, T o > TA is the initial

temperature of the coffee and k is the proportionality constant is: (5 pts)

A) T’ = -kT(1- T/TA)

B) T’ = -k(T - TA)

C) T’ = k(T - T A)

D) T’ = kT/TA

E) T’ = -kT/TA

The solution to the differential equation is: (5 pts)

A) T(t)= T A/(1+Aexp(-kt)), where A = (T A – To)/To

B) T(t) = TA + Toexp(-kt)

C) T(t) = T Aexp(-kt) – TA + To

D) T(t) = TA + (To – TA)exp(-kt)

E) T(t) = (To/TA)exp(-kt)

Extra Credit (10 pts)

12) 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). Find equilibrium solutions for preditor and prey.

Appendix:

Identities:

Logistics Equation:

P

K P

A

K

ln(AB)=ln(A) + ln(B)

ln(2)=0.

ln(3)= 1.

ln(4)=1.

ln(5)=1.