Download Calculus Questions

Third Annual NCMATYC Math Competition
November 16, 2013
Calculus Test
Please do NOT open this booklet until given the signal to begin.
You have 90 minutes to complete this 40-question multiple choice calculus test. Approximately two-thirds
of the questions pertain to topics covered in a typical Calculus I course; the rest pertain to topics covered
in a typical Calculus II course. Answer the questions on the electronic grading form by giving the best
answer to each question.
The scoring will be done by giving one point for each question answered correctly and zero points for
each question answered incorrectly or left blank. Note that the scoring does not include a “guessing
penalty” and so it is to your advantage to answer all the questions, even if you have to guess. If there is a
tie, question number 19 will be used as a tie-breaker.
This test was designed to be a CHALLENGE. There will surely be questions that you will not be able to
answer. We hope you will enjoy working on and take pride in those questions which you are able to
answer.
You may write in the test booklet. You may keep your test booklet and any of your scrap papers. Only the
electronic grading form will be collected and graded.
Good luck!
NCMATYC Calculus Tournament Fall 2013
1. Suppose f and g are differentiable functions for all real numbers and h  x   f  g  x   . Furthermore, it
is known that g  1  4; g   1  5; g   4  2; f  1  4; f (1)  9; f (4)  1. Determine h  1 .
A. 5
B. 16
2. What is lim
t 0
A.
3
4
C. 45
D. 16
E. 18
C. 
D. 0
E. does not exist
27  3t 2  27
?
4t

B.
3
4
3. What is the 20th derivative of the function f  x   cos  x  sin  x  ?
A. f  20  x   220  cos  x    219
2
B. f  20  x   220 cos  x  sin  x 
D. f  20  x   219  220  cos  x  
E. f 
2
20 
C. f  20  x   220 cos  x  sin  x 
 x   220  cos2  x   sin 2  x  
4. Let f and g be differentiable functions for all real numbers, what is lim
f  g  x   h  f  g  x 
h 0
A. f   g  x  
B. f  g   x  
C.
 f  g  x 
D. f   g   x  
h
?
E. none of these
5. Which of the following types of functions must have a derivative equal to 0 at some value in its
domain?
A. Cubic
B. Quartic
C. Exponential
1
D. Tangent
E. none of these
NCMATYC Calculus Tournament Fall 2013
6. Let f and g be differentiable functions such that g  x   x 3 f  x 2  1 , f  5  3 , and f   5  2 . What
is the value of g   2 ?
A. −12
B. −18
7. What is lim
x 0
A. 
B. undefined
8. Determine
A. 
E. 0
C.
1
2
D. 0
E. none of these
d2y
2
 2
D.
2
dx
9x
E.
d2y
for y 3  x 2 .
dx 2
d 2 y y  4x

A.
dx 2
x2

D. 12
x
?
x x
1
2
9. Compute
C. 20
d 2 y y  4x
B.

dx 2
3y2
d 2 y y  2x
C.

dx 2
9 y2
d2y
2
 2
2
dx
9y
1  ln  x 
dx
x2
1
 ln  x   1  C
x
D. 
B.
1
ln  x   C
x
1
 ln  x   1  C
x3
C. 
E.
1
ln  x   C
x2
10. Which of the following is an inflection point on the graph of y 
A. There is no inflection point
 10 
B.  8, 
 49 
2 

C.   4, 
25 

2
1
 3  2 ln  x    C
x3
x2
?
( x  1) 2
2 

D.   8, 
27 

E. none of these
NCMATYC Calculus Tournament Fall 2013
11. What is the equation of the tangent line to the graph of y 
A. y  7 x  2
B. y   x  2
 1
12. Evaluate: lim 1  
x 
 x
( x  2)3 ( x  1)2
at x  0 ?
( x  2)2
C. y  2 x  2
D. y  7 x  2
C. e2
D. 0
E. none of these
2x
A. e
B. 2e
E. 
13. For how many integer values of c does lim x   x  equal a real number?
x c
A. 1
B. 0
C. 2
D. infinitely many
E. impossible to determine
14. For how many values of the domain does the function f  x   xe x cos  x  have a horizontal tangent
line?
A. 0
B. 1
C. 2
D. 4
E. infinitely many
15. Which of the following is the derivative of the function F  x   f  x  g  x  h  x  ?
i.
F   x    f  x  g   x   f   x  g  x   h  x   f  x  g  x  h  x 
ii.
F   x    g  x  h  x   g   x  h  x   f  x   f   x  g  x  h  x 
iii.
F   x    f  x  h  x   f   x  h  x   g  x   f  x  g   x  h  x 
A. i
B. ii
C. iii
D. none of these
3
E. all of these
NCMATYC Calculus Tournament Fall 2013
16. Evaluate

2
0
x 2  1dx
A. 2
B.
2
3
C.
17. Evaluate the first derivative of y  x

dy  sin  x 
A.

 cos  x  ln  x  
dx  x

D.
18. Define h  x   
D.
3
8
3
E. none of these
.
1
B.

dy  sin  x 

 cos  x  ln  x   xsin x 
dx  x

dy
 xsin  x 1 sin  x 
dx
dy sin  x 

 cos  x  ln  x 
dx
x
2
3 x2
sin  x 
4
3
E.
dy
 x cos x  sin  x 
dx
1  r dr . Evaluate h  x  .
2
A. h  x    6 x  3 1  9 x 4 
B. h  x    6 x  3 1  9 x 4 
E. h  x    2 x  3 1  9 x 4 
D. h  x    3 1  9 x 4 
19. Compute the local extrema of y 
A. No local extrema for x  3
C. h  x   3 1  9 x4 
5ln  x 
ln  ln  x  
2
for x  3 .

10 
B.  e 2 ,
 ln  2  



C. e 5 ,  ln  5
1


D. e e ,5e

E. none of
these
20. Suppose the radius of a cylinder is measured at 2 cm with a possible error of ±0.01 cm and the height is
measured at 5 cm with a possible error of ± 0.1 cm. Estimate the error in the calculated volume using
differentials to the nearest hundredth of a cubic centimeter.
A. 2.2π cm3
B. 2.4π cm3
C. 0.6π cm3
4
D. 0.8π cm3
E. 1.0π cm3
NCMATYC Calculus Tournament Fall 2013
21. Let the quadratic function f  x   ax2  bx  c be non-negative on the interval  1,1 . Which of the
following is
A.
 f  x dx ?
1
1
2
 f  1  2 f  0   f 1 
3
D.
B.
1
 f  1  4 f  0   f 1 
2
1
 f  1  4 f  0   f 1 
3
C.
1
 f  1  2 f  0   f 1 
4
E. None of these
22. Which of the following is represents the volume of the solid formed by revolving the region bounded by
the graphs of y  ln  x  , x-axis, and the line x  3 about the line x  3 .


A.   9   ln  x   dx
3
1
2
B.  
ln  3
0
9  e dy
D.    ln  x   dx
3
C.  
2y
ln  3
0
3  e  dy
2
y
2
E. None of these
1
23. The intersection of y  x 2  14 and y  x creates two tangent lines, one to each curve. Which of the
following expressions gives the angle created by the intersection of these two tangent lines?
1
A. tan 1  2   tan 1  
2
1
B. tan 1  8   tan 1  
4
1
D. tan 1  8   tan 1  
8
1
C. tan 1  4   tan 1  
4
1
E. tan 1  4   tan 1  
2
24. L be the tangent line to x 2  3x 2 y  y 3  5 at the point  1,1 . What is the sum of the intercepts of L?
A. 
5
6
B.
49
12
C.
7
12
5
D.
5
6
E. 
11
30
NCMATYC Calculus Tournament Fall 2013
25. What is the arc length of y  e0.5 x  e0.5 x from x  0 to x  2ln  3 ?
A.
2
3
B.
4
3
C.
8
3
D.
16
3
E.
32
3
26. Determine the minimal distance from the point  5,0 to the graph y  x 2  1 .
A.
15
B. 4 5
C.
D.
34
E. 2 5
26
27. A circle centered at the origin passes through the point  4, 2  . Determine the area in Quadrant IV
between the tangent line to the circle at  4, 2  and the circle.
A. 16  5
C. 32 10
B. 32  5
D. 50 10
E. 25  5
28. Let R be the region in Quadrant I bounded by y  1  x3 , the line y  0 , and the line x  0 . Calculate the
volume of the solid of revolution generated by revolving R about the line x  1 .
A.
9
10
29. What is the value of
A. 1
B.
9
14
C.
4
x2
0
x2   4  x 

B. 2
2
5
14
D.
3
5
E.
5
3
dx ?
C. 4
D. 6
6
E. 8
NCMATYC Calculus Tournament Fall 2013
30. Determine
A.
d2y
if 𝑦 4 − 2𝑦 = 𝑥
dx 2
12 y 2
 4 y3  2
2
B.
12 y 2
 4 y3  2
C. 
3
12 y 2
 4 y3  2
2
D. 
12 y 2
 4 y3  2
3
E. None of these
31. A particular radio station plays eight minutes of music followed by two minutes of commercials. If a
listener tunes in randomly, what is the average number of minutes he will listen before he hears a
commercial?
A. 3.2 min
B. 4 min
C. 2 min
D. 2.8 min
E. 3.6 min
32. Determine the area of the largest rectangle that can be inscribed in a semicircle of radius 4 cm.
A. 8 cm2
B. 16 cm2
C. 12 cm2
D. 10 cm2
E. 32 cm2
33. Determine the equation of the normal line to the curve y   x 4  1 ln  x  1 at x  0 .
3
B. 2 y  x
A. y   x
34. Evaluate
A.
16
e


0
D. y  x
E. y  2 x
ecost  sin  2t  dt .
B. 4e
35. What is the length of the curve y 
A. 2 2
C. y  2 x
B. 1
C.
2
e 1
D.
4
e 1
D.
2
E.
4
e
 x  x   sin  x  ?
2
1
C. 2
7
E. none of these
NCMATYC Calculus Tournament Fall 2013
36. Let
2
dv
 2tv  3t 2 et and v  0  8 . What is v  t  ?
dt
A. v  t   t 2et  8et
2
2
B. v  t   tet  8et
2
2
C. v  t   t 3et  8et
2
D. v  t   8et
2
2
E. none of these
1  2n
37. Evaluate  n .
n 1 9

A. It is divergent.
B.
1
5
C.
15
42
D.
3
8
E.
38. Which of the following is the general solution for the differential equation x
  3e x
B. y  x 4   
  x
 3e x 
A. y   
dx  C
 x 
D. y  x 4


dx  C 


  3xe dx  C 
23
56
dy
 4 y  3x 4e x ?
dx
C. y    3 x 4 e x dx  C
E. y    3xe x dx  C
x
39. A rope is 40 feet long and weighs 0.5 lb/ft. If it hangs over the edge of a building 200 feet tall, how much
work is done by pulling half of the rope to the top of the building?
A. 240 ft-lb
B. 300 ft-lb
C. 260 ft-lb
D. 380 ft-lb
E. none of these
40. Find the centroid of the region bounded by the curve 4x2 – 24x + y2 + 10y + 45 = 0
A. (0, 0)
B. (1, -3)
C. (2, -4)
8
D. (3, -5)
E. (4, -6)