Mu Individual_____________________MAO National Convention 2012

1

For each question, the letter choice “E, NOTA” means that none of the above answers are correct. Try your

best, and have fun!

1. The base of a solid is the region bounded by the parabolas

2

y x = and

2

2 y x = ÷ . Cross-sections

perpendicular to the x axis ÷ are squares with one side lying along the base. Find the volume of the solid.

A)

16

3

B)

64

3

C)

16

15

D)

64

15

E) NOTA

2. Using the substitution 4 v x y = + , solve the differential equation:

2

(4 )

dy

x y

dx

= + .

A) 2tan(2 ) 4 y x C x = + ÷ B)

1

2tan (2 ) y x C

÷

= + C)

1

2tan (2 ) y x C

÷

= +

D) 4tan(2 ) y x C = + E)NOTA

3. Frank-the-tank is thinking of two real numbers, x and y which satisfy

1

2 2 x

y

÷ = , 1 x > ÷ . If the expression

4 2 x y + is minimized, what is the value of

y

x

÷

?

A)1 B)

1

2

÷

C) 0 D)

3

2

÷

E) NOTA

4. Diego is the derivatives master. Diego stumbles across

the following expression:

.

.

2 .

2

2

x

x e

x e

y e

+

+

+

=

. Diego wants to find the value of the derivative of this

function at ( , 4) e . What value does

Diego come up with for the derivative at this point?

A)

1

3

e ÷

B)

3

1 e ÷

C)

3

e

D)

3

e

E) NOTA

5. Diego is also the limits master. When Diego evaluates:

lim

0 x ÷

sin

2

cos sin

sec

x x

e x e x

x

÷

, what does he come up

with?

A) 1 B) 0 C) -1 D) Does not Exist E) NOTA

6. Find

2

2

d y

dx

for the equation

2 2

6 x xy y ÷ + = at the point (6,4)

A) -21 B) -18 C) 0 D) 15 E) NOTA

7. Evaluate:

5

8tan xdx

}

Mu Individual_____________________MAO National Convention 2012

2

A)

6

4tan

3

x

C +

B)

4 2

tan tan

ln sec

4 2

x x

x C ÷ + +

C)

4 2

2tan 4tan 8ln sec x x x C ÷ + +

D)

10 2

4sec tan

ln sec

3 2

x x

x C ÷ + +

E) NOTA

8. Find

4

0

sin xdx

t

}

A) 0 B)

4

t

C)

3

4

t

D)

2

t

E)NOTA

9. When Wayne uses Newton’s iterations to approximate

2

x for

3 2

4 6 5 1 x x x ÷ + ÷ , (using

0

1 x = ), his answer

will be in the form

A

B

, where A and B are relatively prime. Find the number of factors in ( ) A B + .

A) 36 B) 20 C) 10 D) 2 E) NOTA

10.

2

( )

'( )

x

f x e = and (0) 10 f = . Using the Mean Value Theorem for Derivatives, we can conclude that

(1) X f Y < < for some numbers X and Y . What is the value of X Y ÷ ?

A) 1 e ÷ B) 1 e ÷ C) 1 e ÷ ÷ D) 1 e + E) NOTA

11. Evaluate:

3 2

4 2

2

3 2

x x x

dx

x x

+ + +

+ +

}

A)

3 2

4 2

2

ln

3 2

x x x

C

x x

+ + +

+

+ +

B)

1 2

tan ( ) ln( 2)

2

x x

C

÷

+ +

+

C)

1 2

2tan ( ) ln 2 x x C

÷

+ + +

D)

2

1

ln( 2)

tan ( )

2

x

x C

÷

+

+ +

E) NOTA

12. For the function

2012

( ) 2cos 3sin

x

f x x x x e

÷

= + ÷ + , find the 2011

th

derivative at 0 x = .

A) 2 B) 0 C) 2012! 2 + D) 2012! E) NOTA

Mu Individual_____________________MAO National Convention 2012

3

13. Peter the doughnut man is making his famous jelly-filled doughnuts. The outside of the doughnut is

modeled after the shape of the curve

2 4

1 x y + = , rotated about the vertical axis. This curve is symmetric about

the x-axis and the y-axis. How much cubic feet of jelly will it take to fill one of Peter’s famous jelly-filled

doughnuts?

A)

5

t

B)

2

5

t

C)

4

5

t

D)

8

5

t

E) NOTA

14. Chris, the 170 foot giant, is falling at night while horizontal when the moon is shining directly over his head.

He is falling at a rate of 4 feet per second when he is 150 feet from the ground. At that moment, how rapidly is

Chris’s shadow cast by the moon lengthening? (in feet per second)

A)

15

8

B)

15

6

C)

15

4

D)

15

2

E) NOTA

15. Andrew Chico is also the limit master. When Diego asks Chico the answer to the following question:

“What is the value of

lim

0 x

+

÷

3

ln x x ?”

Chico will answer what?

A) 0 B)

1

3

÷

C)

1

3

D) Does Not Exist E) NOTA

16. Dr. Smith is at his research center and gets hungry while trying to find a cure for cancer. He stops to get an

ice-cream cone. This ice-cream cone (vertex down) is 6 inches in diameter and 9 inches deep. Peter the ice-

cream man fills the ice-cream cone with ice-cream at a rate of 3 cubic inches per minute. Find the rate of change

of the depth of the ice-cream at the instant it is 6 inches deep.

A)

9

4t

B)

3

t

C)

3

4t

D)

9

t

E) NOTA

17. A particle’s velocity for 0 t > is given by the following function:

2

1

( )

3

v t

t

=

+

. Find the total distance the

particle has traveled for 0 1 t s s .

A)

6

t

B)

18

t

C)

3

18

t

D)

3

6

t

E) NOTA

18. Aaron is the master of computation. When asked to find the value of the 2

nd

derivative of

1

( ) ln(tan(sin ( ))) f x x

÷

=

at 2 x = , he will come up with what?

A)

1

4

÷

B)

1

2

÷

C)

1

2

D) Does Not Exist E) NOTA

Mu Individual_____________________MAO National Convention 2012

4

19. If

3 2

( ) 19 f x x x x = + ÷ +

has a relative minimum at

x A =

and the relative maximum at

x B =

, find

B

A

.

A) -1 B) 3 C)

1

3

D) 1 E) NOTA

20. Wilson is a master at finding tangent lines to functions. He wants to determine the equation of the tangent

line to ( ) 3cos(2 ) f x x = at

5

6

x

t

= . When he is done, he gets the tangent line of the equation in the following

form:

3 B

y Ax

C

t ÷

= +

. What is the value of 5 ? AB C ÷

A) 25 B) 35 C) 45 D) 55 E) NOTA

21. Jason is selling cookie-brownies at the 2012 National Mu Alpha Theta Bake-Sale stand. He sells cookie-

brownies at a fixed price of $200 0.05x ÷ , where x is the number of cookie-brownies he produces a day.

The cost of materials to make each cookie-brownie is 140 dollars, and authorization from the 2012 National Mu

Alpha Theta Convention is 9,500 dollars per day. How many cookie-brownies should Jason produce and sell

each day to maximize profit?

A) 600 B) 60 C) 68 D) 680 E) NOTA

22. Evaluate:

2

2 2

0

6 sin( ) sin(4 ) x x x dx

t

}

.

A)

1

5

÷

B)

3

10

÷

C)

3

5

÷

D)

4

5

÷

E) NOTA

23. Linda and Steve are fighting over differential approximation. The math question calls for the linear

approximation of

1

17

tan ( )

20

÷

. Linda wants to use the point

(1, )

4

t

to approximate

1

17

tan ( )

20

÷

. However,

Steve wants to use the point

(0, 0)

! If Linda’s answer can be expressed as

A B

C

t +

, and Steve’s answer

can be expressed as

D E

F

t +

, what is the value of

C

A B DE

F

÷ + +

?

A) 12 B)15 C)17 D)20 E) NOTA

Mu Individual_____________________MAO National Convention 2012

5

24.

5

( ) 3 2 f x x x = + ÷ Let

1

( ) ( ) j x f x

÷

= . What is the value of

'(2) j

?

A)

1

83

B)

1

8

C)1 D)83 E) NOTA

25. Ian is taking Dr. Fraser’s Calculus BC Sequences and Series Test. Ian has finished all but one question:

“Use the first three terms of the Maclaurin series of

ln( ) x e +

to approximate

1

2

0

ln( ) x e dx +

}

.”

Ian’s answer is found in the simplest form

2

2

Ae Be C

De

+ +

. What is the value of the expression

2

?

A

B C

D

+ +

A) 2 B) 3 C) 12 D) 15 E) NOTA

26. What is the radius of convergence of the Maclaurin series

( ) ln( ) f x x e = +

?

A) 1 B)

1

e

C) 0 D)

e

E) NOTA

27. Mario and Nic are farmers and introduce a flock of 100 Win-A-Saurus-Rexes into their farm. They predict

after

m

months that the rate of growth,

W

, will be modeled by the differential equation:

(600 )

5000

dW W W

dm

÷

=

. If the solution of their differential equation can be expressed as

Dm

A

W

B Ce

=

+

,

what is

2

? AB C D ÷

A) 597 B) 600 C) 603 D) 606 E) NOTA

28.

3 3 2

3

2 2

0

( )

(4 9)(4 9)

x

dx

x x + +

}

A)

3

4

B)

3

16

C)

3

32

D)

3

64

E) NOTA

Mu Individual_____________________MAO National Convention 2012

6

29.

ln2

3

0

ln( )

1

x

x

e A B

dx

e D C

= +

+

} , find BC AD ÷ .

A) 1 B) 4 C) 6 D) 8 E) NOTA

30. 2

0

3

x

x

e

dx

e

·

+

}

A)

3

9

t

B)

3

18

t

C)

3

6

t

D)

3

108

t

E) NOTA