x - Mu Alpha Theta

BC Calculus
FAMAT State Convention 2004
For all questions, E. NOTA means none of the above answers is correct.
1) Which of the following series converge?
∞
∞ 3
2
n
i) ∑ ii)
n ∑ iii)
2
n 1
n=0 3
n= 1 +
∑ ∞ 1
3
= 3+
n 1
n
∞
∑
iv) ( −1)
n=
1
n+
1 1
n
A) i, ii B) i, iv C) ii, iv D) iii, iv E) NOTA
2) Which of the following represents the surface area of revolution of the graph f ( x)
= x
2 + 2x
about the y-axis on the
interval[ 0,2]?
2
[ x + 2x
1]
2
2
A) π ( x + x)
4 3 2
2 ∫ 2 + dx B) 2π ∫ [ x x + 4x
+ 4x
+ 1]dx
0
0
2
2
2
C) 2π ∫ [ x 4x
+ 8x
+ 5] dx
D) 2π ∫ 2x + 3dx
E) NOTA
0
3) What is the area of one of the petals of the polar graph = 2sin( 4θ )
2
0
r ?
A) 8
π
B) 4
π
C) 3
π
D) 2
π
E) NOTA
4) What is the Maclaurin series for the function f ( x)
= arctan( x)
?
3 5 7 9
2 4 6 8
x x x x
x x x x
A) x − + − + ...
B) 1 − + − + ...
3! 5! 7! 9!
2! 4! 6! 8!
2 3 4 5
3 5 7 9
x x x x
x x x x
C) 1 + x + + + + ... D) x − + − + ...
E) NOTA
2! 3! 4! 5!
3 5 7 9
5) Evaluate
2
5x
+ 22x
+ 16
∫ dx
2
x +
( x 2)
A)
4
x
1 4
+ + C
x + 2 ( x + 2)
1
+ ln
x + 2
+
2
4 4 1
B) + + + C
2
x x + 2 ( x + 2)
4
ln + 2 4ln
E) NOTA
x + 2
C) 4 ln( x 2) − + x + C D) ( x ) − + x + C
6) Find
2
d y
2
dx
for the curve given
2
x = 3t and y = 2 t + 2 .
A)
1
− B)
2
3t
1
6t
C)
1
18t
2
D)
1
− E) NOTA
3
18t
7) Find the volume of revolution formed by revolving the ellipse x 2 + 16y
2 + 4x
− 96y
+ 132 = 0 about the line x = 6 .
π
2
2
2
2
A) 64 B) 48π C) 32π D) 16π
E) NOTA
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1 x
8) Evaluate∫ − cos2
dx
2
2
A) − cos x + C B)
sin
( 2x)
+ 2x
+ C
4
BC Calculus
FAMAT State Convention 2004
C)
sin
( 2x)
− 2x
+ C
4
2
D) − sin x + C E) NOTA
9) Find dy where f ( x)
= ( 4sin x) ( arcsin(
4x)
) .
dx
⎛
16sin x
⎞
A) ( 4cos x ) ( arcsin( 4x)
) + ⎜
⎟ B) ( 4cos x ) ( arcsin( 4x)
) + ⎜
2
⎝ 1−16x
⎠
⎟ 2
⎝ 1−16x
⎠
⎛
16sin x
⎞
⎛
4cos x
⎛ 4sin x ⎞
x E) NOTA
C) ( 4sin x ) ( arccos( 4x)
) + ⎜
⎟ D) ( 4cos ) ( arcsin( 4x)
) + ⎜
2
⎝ 1−16x
⎠
⎟ 2
⎝ 1−
4x
⎠
= on the interval ⎛ 3 3 ⎞
⎜ − , ⎟
⎝ 2 2 ?
⎠
A) 20.99 B) 28.20 C) 6.68 D) 5.31 E) NOTA
10) What is the arc length, to the nearest hundredth, of the graph of y −ln( cos x)
⎞
π
dx
11) Evaluate
∫ 2
x − 8x
+ 20
− π
⎛
⎜ln⎜
⎟ + ln⎜
⎟
2 ⎝ ⎝ 2 ⎠ ⎝ 2 ⎠
B)
1
( arctan π + 2 + arctan π − 2) )
4
D)
( ) ( )
A) 1 ⎛ π + 4 ⎞ ⎛ π − 4 ⎞⎞
⎟⎠
C) ( ) (
( π + 4) − ln( π − 4) arctan( π + 4) − arctan( π 4)
ln +
+
2
2
1 ⎛ ⎛ π + 4 ⎞ ⎛ π − 4 ⎞⎞
⎜arctan⎜
⎟ + arctan⎜
⎟⎟
E) NOTA
2 ⎝ ⎝ 2 ⎠ ⎝ 2 ⎠⎠
12) What is the arc length of the polar function r = sinθ
on the interval from
π
θ = to
4
2π
θ = .
3
A)
3 −
2
2
5π
B) 12
C)
2 −1
2
7π
D) 12
E) NOTA
13) Which of the following integrals represents the surface area of revolution of the graph of y = 2x
3 − 6x
about the x-axis on the
closed interval from x = 0 to x = 3 ?
3
3
3
2 B) π ( 2 − 6x)
4 2
A) π ∫ [ x 36x
− 72x
+ 37]dx
0
[ ]dx
∫
0
4 2
[ x 36x
− 72x
+ 37]dx
3
3
2 2
2 2
C) 2π ∫ 12x( x − 3)( x −1)
D)
∫ [ 12x( x − 3)( x −1)]dx
0
π E) NOTA
0
14) Which of the following improper integrals converges to a defined value?
∞
∞
∞
−
1
1
A)
∫ xe x dx
B)
∫ dx
C) dx
3
x
x
0
0
∫
D)
∫ cos πx dx
E) NOTA
1
∞
0
( )
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BC Calculus
FAMAT State Convention 2004
15) A tangent line can be drawn on the polar graph r = 3 − 3cosθ
such that the line is tangent to exactly two distinct points on the
graph. What are the polar coordinates of these two points?
A) ⎛ 9 2 ⎞
,
π ⎛ ⎞
⎜ ⎟ & ⎜
9 4
,
π ⎟ B)
⎛ 3 3 ⎞ ⎛
⎜ ⎟
⎝ 2 3 ⎠ ⎝ 2 3 ⎠ 1 , 3 − &
3 3 ⎞
⎜ + ⎟
⎝ 2 1, 3
⎠ ⎝ 2 ⎠
C) ( 6 , π ) & ( 0 , 0)
D)
⎛ ⎞
⎜
3 ,
π ⎟ & ⎛ 5 ⎞
⎜
3 ,
π ⎟ E) NOTA
⎝ 2 3 ⎠ ⎝ 2 3 ⎠
2
f '( )
( x) 2arcsin( 2x) + 3x
1−
2x
16) Find x of the function f
= .
A)
C)
1
3x
2
2
− 2x
− +
B)
2
2
4
1−
2x
2
1−
2x
+ 3 1−
2x
2
4
1−
4x
2
3x
−
1−
2x
2
D)
4
1−
4x
2
2
1−
4x
2
+ 3
+
1−
2x
1−
2x
2
2
−
−
6x
2
1−
2x
2
3x
1−
2x
2
2
E) NOTA
4 3
17) Evaluate
∫ ( sin φ cos φ)
dφ
5
4
sin φ cos φ
A) + + C
5 4
5
7
sin φ sin φ
B) − + C
5 7
5
7
cos φ cos φ
C) − + C
5 7
5
6
cos φ cos φ
D) + + C
5 6
E) NOTA
18) Evaluate
∫ ( arcsin x + arctan x)
dx
1
2
2
2
A) ( arcsin x + arctan x) + C
2
ln x + 1
2
2 ( )
C) x arctan x + xarcsin
x + 1−
x − + C
x
19) Evaluate lim
x→∞
2e
3
2x
− 2x
2
ln x + 1
2
( )
B) x x + x x + − + C
arctan
arcsin
1−
x
2 ( )
D) x + arcsin x + 1−
x − + C
2
2
ln x + 1
arctan E) NOTA
2
A) 0 B) 1 C) ∞ D) Does Not Exist E) NOTA
( )
n
2 4 6 8
2n
x x x x −1
x
20) 1 − + − + −...
+ + ... is the Maclaurin Series for which of the following?
2! 4! 6! 8! 2n
+ 1
A) y = sin x
B) y = ln x
C)
x
y = tan x
D) y = e
E) NOTA
∞
1
21) Make the following a true statement: The p-series ∑ , with n as a positive integer, ___________.
p
n
n=1
A) converges if p < 0 B) converges if p > 1 C) converges if 0 < p < 1 D) converges if p < 1 E) NOTA
2
22) What is the x-coordinate of the centroid of the region bounded by the functions f ( x)
= 6x
− x and g( x)
= 12 − 2x
?
A)
21
5
B) 3 C)
15
4
D) 4 E) NOTA
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BC Calculus
FAMAT State Convention 2004
2
23) What is the y-coordinate of the centroid of the region bounded by the functions f ( x)
= 6x
− x and g( x)
= 12 − 2x
?
A)
28
5
B) 4 C)
56
5
71
D) 10
E) NOTA
24) Find the slope and the concavity for the curve given by x = t 3 +1 and
y = t at the point ( 2 ,1)
.
A) 2
3 ; concave up B) 3
2 ; concave down C) 6
1 ; concave up D) 2
3 ; concave down E) NOTA
2
3
25) What is the volume of the figure formed by revolving the region bounded by y = x , y = 0 , and x = 3 about the line x = 3 ?
81 ( 9)
3
A) π
40
54 ( 3)
3
B) π
7
81 ( 9)
3
C) π
8 16 32
26) What is the sum of the infinite geometric series 4 + + + + ... ?
3 9 27
20
235 ( 3)
3
D) π
35
E) NOTA
A) 11 B) 12 C) 13 D)
49
4
E) NOTA
27) What is the volume of the three dimensional figure formed by taking equilateral triangle cross sections perpendicular to the x-axis
3
along the graph of x on the interval 0,4 ?
y = ( )
A) 16 3 B)
512
3
3
C)
4096
7
3
D)
16 3
3
E) NOTA
2
2x
− 2x
− 3
28) Evaluate
∫
dx
2
x −1
( ) ( )
( x + 1) ln( x −1)
ln x + 1 3ln x −1
ln
A) − − + 2x
+ C B) − + 2 + C
2 2
2 2
2
1 ⎛ 2x
− 2 ⎞
C) arctan
⎜ C
x
⎟ +
D) ln ( x + 1) − 3ln( x −1) + 2x
+ C E) NOTA
2 ⎝ −1
⎠
dy
29) f ( x)
= 2cosh( 2x)
. Find .
dx
A) − sinh ( 2x) B) 4 ( 2x)
C)
sinh − 2sinh( 2x)
D) 4sinh( 2x)
− E) NOTA
2
30) What is the moment about the y-axis of the region bounded by f ( x)
= −2x
+ 4 , g ( x)
= x , and the y-axis with uniform
density ρ ?
⎛10
5 −14
⎞ ⎛ 40 5 376 ⎞ ⎛ 80 5 752 ⎞ ⎛ 26 −10
5 ⎞
A) ρ ⎜ ⎟ B)
⎟
ρ ⎜ − C)
⎟
⎝ 3
ρ ⎜ − D)
⎠ ⎝ 3 15
ρ ⎜ ⎟ E) NOTA
⎠ ⎝ 3 15
⎠ ⎝ 3 ⎠
Page 4