Mu Individual_____________________MAO National Convention 2012
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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
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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
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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
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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
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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
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29.
ln2
3
0
ln( )
1
x
x
e A B
dx
e D C
= +
+
} , find BC AD ÷ .