Third Annual NCMATYC Calculus Tournament
November 22, 2014
Morning Component
Good morning!
Please do NOT open this booklet until given the signal to begin.
There are 40 multiple choice questions and you will be given 90 minutes to complete the test. Record your
answers 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. Thus, it is to your advantage to answer as many questions as
possible, even if you have to guess. If there is a tie, question number 24 will be used as a tie-breaker.
This test was designed to be a CHALLENGE. Do not waste time on questions you are unable to answer; focus
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!
Do Not Open Until Signaled.
NCMATYC Calculus Tournament Fall 2014
1. A company that manufactures digital cameras estimates that the profit, P, in millions of dollars, can be
modeled by P  x   0.02 x3  0.01x 2  1.2 x 1.1 for 0  x  8 , where x is the amount spent on
advertising in hundreds of thousands of dollars. Margaret computed the value of the instantaneous rate of
change at x  6 and found that P  6  0.84 . Which of the following is the correct interpretation of this
value in context of the problem?
A. If the company spends $600,000 on advertising then the company is losing $840,000.
B. For the first $600,000 spent on advertising, the company loses $840,000.
C. When the company spends $600,000 on advertising, the instantaneous rate of change is decreasing at a
rate of $840,000 per additional hundred thousand dollars.
D. When the company spends $600,000 on advertising, the profit is decreasing at a rate of $840,000 per
hundred thousand dollars.
E. None of the above.
2. Suppose f and g are differentiable functions such that f ( x)  x ln  g  x   , f (1) 
Compute the value of g (1) .
A. 2
B. 1
C. 0
2
 1 , and g (1)  e .
e
D. e
E. e 2
3. Determine the equation of the tangent line to ( x  y) 3  ( y  x) 4  0 at 1, 2  .
A. x  y  3
B. x  y  1
C. x  2 y  3
D. 2 x  y  0
E. x  2 y  5
4. If g  x   f  h  x   , h 1  2 , g 1  8 , h 1  1 , and g  1  12 , compute f   2 .
A. 12
B. 8
5. What is the lim
x 5
A. 
1
2
x 5
10  2 x
C. 2
D. 1
E. none of these
D. 
E. undefined
?
B. 0
C.
1
2
1
NCMATYC Calculus Tournament Fall 2014
 f 1x  
d
6. Let f  x   ln  x  and g  x   f  x
 . Compute
 g  x  .


dx


B. e x
A. 0
C. e
D. 1
E. ee
7. At which of the following x-values will y  8  x 2 have a tangent line whose slope is 1 ?
A. 2 2
B. 0
C.
D. 2
2
E. none of these
8. Let f  x   ax3  bx2  cx  d be a cubic function that has horizontal tangent lines at  3, 66 and  3, 42 .
What is the value of a  b  c  d ?
A. 14
B. −7
C. 0
D. 7
E. −14
 2 x  sin  x  
9. Compute: lim 

x  3 x  8cos  x  


A.
7
11
B.
2
3
C.
23
12
D.
5
8
E. 
10. Which of the following curves has an inflection point but no minima or maxima? Assume a, b  0 .
A. f  x   a  x
B. f  x   loga  x 
C. f  x   ax3  b
D. f  x   ax4  bx
E. none of these
11. How many of the lines listed below are tangent to the graph of f  x   x2  2 x  4 ?
L1 : y  9 x
A. 0
B. 1
L2 : y  6 x
L3 : y  2 x
C. 2
L4 : y  7 x
D. 3
2
E. 4
NCMATYC Calculus Tournament Fall 2014
12. Determine
A.
d2y
for x 2  9 y 2  72 .
dx 2
d 2 y x2  9 y 2

dx 2
81y 3
B.
d 2 y  x2  9 y 2

dx 2
81y3
C.
d 2 y x2  9 y 2
d 2 y  x2  9 y 2
d 2 y  x2  9
D.
E.



dx 2
81y 3
dx 2
81y3
dx 2
81y 3
13. Suppose f  x  is continuous on 0,1 and f  0  6 and f 1  2 . Which of the following is implied by
the Intermediate Value Theorem for continuous functions?
1
A. f    2
2
B. There is a unique c in 0,1 , where f  c   5 .
C. There is a value c in  0,1 , where f  c  equals the circumference of a circle with radius
D. There is a value c in between 0 and 1, where f  c  equals 6 or 2 .
1
.
2
E. None of the above
14. What is the minimum degree of an odd polynomial with at least two inflection points?
A. 4
B. 5
C. 7
 
d f f  f  x 
15. If f  x   cos  x  , compute
dx




A.  sin  x  sin  cos  x   sin cos  cos  x  
B.  cos  x  cos  cos  x   sin cos  cos  x  

C. sin  x  sin  cos  x   sin cos  cos  x  

D. 9
 .

D. cos  x  cos  cos  x   sin cos  cos  x  

E. none of the above
3
E. not enough information
NCMATYC Calculus Tournament Fall 2014
16. For what x-values is f  x  
x2  x
non-differentiable?
x cos  x 
A. 0 and k where k is any integer
B.
k
where k is any odd integer
2
C.
k
where k is any integer
2
D. 0 and
k
where k is any odd integer
2
E. The function is differentiable everywhere.
17. If f 1  10 and f   x   2 for 1  x  4 , how small can f  4  possibly be?
A. 16
B. 10
C. 8
D. 14
E. None of these
18. What is the maximum value for f  t   t 3e at , a  0 ?
A. 27a 3e3
B.
19. What is the lim
x 
A. undefined
ln  x 
ln  ln  x  
27a 3
e3
C.
27e3
a3
D.
1
27a 3e3
E.
27
a 3e 3
?
B. 0
C. 
D. 1
4
E. 
NCMATYC Calculus Tournament Fall 2014
2 x 0  x  3
 2x 0  x  2


20. If f  x    x
and g  x    2 x
. Let h  x   f  g  x   . Evaluate h  4  .
3

x

6

2

x

6
 3
 4
A.
1
2
B. 
2
3
C.
2
3
D. 2
E. 
1
6
21. What time could it be if the tip of the hour hand on a clock is in a position such that the ratio of the rate of
change in its vertical position to the rate of change in its horizontal position is - 3 :1 ?
I. 2:00
A. I only
B. II only
II. 5:00
C. I and III
III. 8:00
D. I, II, and III
E. none of these
 3x  2 
22. What is the lim 
 ?
x 
 x 1 
x
1
A.  3
e
B.
3
e
C.  3 e
D.
1
e
3
E. 
23. A light in a lighthouse 1 km offshore from a straight shoreline is rotating at 2 revolutions per minute. How
fast is the beam moving along the shoreline (in km/min) when it passes a point ½ km from the point on the
shoreline closest to the lighthouse?
A. 5
B. 4
D. 2
C. 3
E. 
24. A wire of length 100 inches is cut into two pieces; one is bent to form a square, the other an equilateral
triangle. If the cut is to be made such that the sum of the two areas is to be a minimum, find the perimeter
of the square.
A.
100 3
3
B.
400 3
3
C.
100 3
94 3
5
D.
400 3
94 3
E. none of these
NCMATYC Calculus Tournament Fall 2014
25. Let f(x) and g(x) be such that  2 f  x  g  x  dx  [ f  x ]2  C . Which of the following must be true?
A. f  x   g  x 
B. f   x   g  x 
C. f  x   g   x 
D. f  x   2 g  x 
E. g  x   1
26. What is the volume of the solid obtained when the triangle with vertices (1,0), (5,0), and (3,4) is rotated
around the x-axis?
A.
8
3
B.
32
3
C.
48
3
D.
64
3
E. none of these
 n  
3   3 
27. Evaluate lim     4  2  i   1     .
n  
n    n   
 i 1   
A. 31
B. 35
C. 49
D. 51
E. 39
28. Calculate the area of an ellipse with major axis 10 units and minor axis 6 units.
A. 3.75
B. 7.5
C. 8
D. 12
E. 15
B. 1
C. 1
D. undefined
E. none of these
C. 
D. 
E. 2
 k 2

29.    cos( x)dx  

k 1  0


10
A. 0
 x2 1 
  x4  1  dx =
1 
1
30.
A. 0
B.

2
6
NCMATYC Calculus Tournament Fall 2014
0  x 1
 x
31. Define f  x  to be a periodic function on all reals with period two units such that f  x   
2  x 1  x  2
on the interval 0, 2 . Evaluate
10
 f  x dx .
0
A. 0
B. 2
C. 5
D. 10
E. undefined
32. A rectangle with vertices of the form P  a, b  is inscribed in the ellipse x 2  9 y 2  72 . Maximize the
area of the inscribed rectangle.
A. 24 2
B. 48 3
C. 48
D. 96
E. 48 6
33. Let R be the region bounded by y  x 4 , x  1 , and the x-axis. Let V1 be the volume of the solid of revolution
formed by revolving R about the x-axis. Let V2 be the volume of the solid of revolution formed by revolving
R about the y-axis. What is the ratio of V2 to V1 ?
A. 1:3
B. 3:1
C. 3:2
D. 2:3
E. 1: 3
34. A cylinder of radius 5 feet and depth 10 feet is filled with oil. If the cost in dollars per cubic foot to pump
out the oil is given by C  h   12 1  0.02h 2  where h is the depth of the oil in the cylinder, what is the
total cost to empty the cylinder?
A. $200
B. $200
C. $5000
D. $5000
E. $3300
35. A 50-foot chain that weighs 5 lbs/ft is hanging from a ledge. How much work is done in winding it up until
its lowermost link is 25 feet from the ledge?
A. 4687.5 ft-lbs
B. 3125 ft-lbs
C. 3750 ft-lbs
7
D. 1562.5 ft-lbs
E. 7500 ft-lbs
NCMATYC Calculus Tournament Fall 2014
 f  x   f  2 
36. If f  x  is differentiable at x  2 , evaluate lim 
.
x 2
x

2


A. f  2 
B. 2 f   2 
C.
2 f   2
2
D. 2 2 f   2 
E. does not exist
37. Evaluate  x5 1  x3 dx .
A.
5 1  x3 
6
5
2
1  x 

3
3
C
2
D. 
5 1  x3 
6
2 1  x3 
2
5
2
B.
1  x 

3
2
3
15
5
2

2 1  x3 
9
3
2
C
C. 
2 1  x3 
15
5
2

2 1  x3 
3
9
2
C
2
C
E. None of these
38. What is the derivative of tan  x  with respect to sin  x  ?
A. sec  x  tan  x 
B. sec2  x 
C. sec2  x  sin  x 
D. sec2  x  cos  x  E. sec3  x 
39. A particle moves along the curve y  x 2  2 x . Find the Cartesian coordinates of the point on the curve
where the rate of change for the x and y coordinates of the particle is the same.
1 3
A.  , 
2 4
 1 3
B.   , 
 2 4
1 3
C.  ,  
2 4
40. What is the solution of the differential equation 2 xy
A. 3 y 2  cx  3  y 3 
B. 1  y 2  cx
E. None of these
dy
 1 y2 ?
dx
C. 1  y 2  cx 2
8
 1 3
D.   ,  
 2 4
D. 3x 2  cx  3  y 3 
E. 3 y 2  cx  3  x 3 