mathematics

Published on November 2016 | Categories: Documents | Downloads: 52 | Comments: 0 | Views: 429
of 6
Download PDF   Embed   Report

review problems

Comments

Content


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




Sponsor Documents

Or use your account on DocShare.tips

Hide

Forgot your password?

Or register your new account on DocShare.tips

Hide

Lost your password? Please enter your email address. You will receive a link to create a new password.

Back to log-in

Close