Calculus Applications
___ FAMAT State Convention 2015
Remember—NOTA means “None of the Above”
1. What is the maximum of f ( x, y )  x y, given that x  y  1?
2
3
3
a)
b)
2
2
2
 3
9
3
9
c) 
2
3
d)
e) NOTA
2. Let f ( x)  5  6 x  12 x  2 x  x and let g ( x)  f ( x) for    x  . At what value of x
2
3
4
/
is g ( x) increasing most rapidly?
a) -2
b) - ½
c) ½
d) 1
e) NOTA
3. Let z  x  y , where x and y are increasing at the constant rates of 2 units/second and 3
2
2
units/second, respectively. What is the set of points for which the rate of change of
a)
( x, y) :
3 y  2 x
b) ( x, y) : 3 y  2 x
d) ( x, y) : 2y  3x
x
4. If f ( x ) 


3
c ( x, y) : 2y  3x
z is zero?
e) NOTA
sin tdt , then at what value of x in the interval  0,2  is f ( x) a maximum?
/2
a) 0
b)

2
c)

d)
3
2
e) NOTA
5. A rectangle is to be inscribed in a semicircle of radius 10, with one side lying on the diameter of
the semicircle. What is the maximum possible area of this rectangle?
a)
5 2
b) 50
c) 100
d) 145
e) NOTA
Calculus Applications—page 2
___ FAMAT State Convention 2015
6. A solid rectangular brick is to be made from 1 cu. ft. of clay. The brick must be 3 times as long
as it is wide. For what value of the ratio of height to width (in ft) will the brick have a
minimum surface area?
a)
1
27
b)
1
9
c)
2
9
d)
2
3
e) NOTA
7. Let x be the length of one of the equal sides of an isosceles triangle, and let
between them. If x is increasing at the rate of

180
radians/hour, then at what rate, in
x  12 m and  
a)
36 2

4
 be the angle
1
m / hr , and  is increasing at the rate of
12
m2 / hr, is the area of the triangle increasing when
?
 73 
 2
 2
b) 
c)
3 

2
5
d)
1  
2  
2 5
e) NOTA
8. The radius of a right circular cylinder increases at the rate of 0.1 cm/min, and the height
decreases at the rate of 0.2 cm/min. What is the rate of change of the volume of the cylinder,
3
in cm /min, when the radius is 2 cm and the height is 3 cm?
a) 2
b)
8
5
c)
3
5
d)
2
5
e) NOTA
9. Find the volume of the solid obtained by revolving about the x-axis the region R bounded by
the graph of y 
a)

2
1
and below by the x-axis for 1  x  .
x
3
b) 
c)
2
d) 2
e) NOTA
Calculus Applications—page 3
____ FAMAT State Convention 2015
10. Find the surface area of the band of the sphere generated by revolving the arc of the circle
x 2  y 2  r 2 lying above the interval  a, a  , a  r, about the x-axis.
a)
 ar
b) 2 ar
c) 6 ar
d) 8 ar
e) NOTA
11. Which of the following statements below are TRUE?
I.
An absolute maximum or minimum must occur at a critical point or at an endpoint.
II.
To find the linear approximation to a function at x  a, you need to know the first
derivative of that function.
III.
/
Newton’s Method fails if f ( a ) is equal to zero and a is used as an initial value in the
iteration process.
a) I and II only
b) I and III only
c) II and III only
d) All true
e) NOTA
12. A 20 ft cable weighs 80 lbs and hangs from the ceiling of a building without touching the
floor. Determine the work (in ft-lbs) that must be done to lift the bottom end of the chain all
the way up until it touches the ceiling.
a) 200
b) 400
c) 600
d) 800
e) NOTA
13. Determine the volume of the solid obtained by rotating the region bounded by
y  ( x  1)( x  3)2 and the x-axis (above it) and above the interval [1, 3] about the y-axis.
a)
12
5
b)
24
5
c)
12
25
d)
24
25
14. Determine the area of the region enclosed by y  sin x, y  cos x, x=
a) 2 2
b) 2 2 - 1
c) 2 2 - 2
e) NOTA

2
d) 2 2 - 4
and the y  axis.
e) NOTA
Calculus Applications—page 4
__ FAMAT State Convention 2015
15. Determine the average value of the following function on the given interval:
R( z )  sin(2 z )e(1cos(2 z )) on   ,  .
a) 1
b)
1

c)
1
2
d)
1
4
e) NOTA
16. Determine the volume of the solid obtained by rotating the region bounded
by x  ( y  2) and y  x about the line y  1.
2
a)
63
8
b)
63
2
c)
128
5
d)
64
5
e) NOTA
17. We have a cable that weighs 2 lbs/ft attached to a bucket filled with coal that weighs 800
lbs. The bucket is initially at the bottom of a 500 ft mine shaft. Determine the amount of work
(in ft-lbs) required to lift the bucket all the way up the shaft.
a) 250,000
b) 450,000
c) 650,000
d) 750,000
e) NOTA
18. Let F(x) be a strictly decreasing continuously differentiable function on [ a, b ]. Then
b
 F ( x) dx must equal
/
a
a) F (b)  F (a)
b) F (a)  F (b)
c) F (b)  F (a)
d) F (a)  F (b)
e) NOTA
c) -2 sin 2x
d) -2 sin 2x + c
e) NOTA
2x
19. Find Dx

cos 2tdt.
2015
a) -4 sin 2x
b) -4 sin 2x + c
Calculus Applications—page 5
___ FAMAT State Convention 2015
20. The region S is bounded by y  x  2 x  3 , y  0, x  0and x  9. Which of the following is
2
the approximation to the area of S obtained by computing the sum of the areas of the 3
inscribed rectangles with bases [0, 3], [3, 6], and [6, 9] (lower Riemann sum)?
a)105
b) 108
c) 117
d) 189
e) NOTA

(sin x) n
21. Which is the following is the interval of convergence for the series 
?
n
n 1
   
, 
2
2

a) (0, )
b) 
   
   
c) 
, 
2
2

d) 
, 
2
2

e) NOTA

22. Which of the following are sufficient conditions for the convergence of
a ?
n 1
I.
lim an  0
II.
The sequence of partial sums is bounded.
n
n

a
III.
n 1
n
converges
a) None are sufficient
b) I only
c) II only

d) III only


23. What is the set of limit points of the sequence sin  n
a)
0
b) 1

c) 1,1
2
e) NOTA


:
n

0,1,...
?

2

d) 0,1
e) NOTA
24. Determine the following limits for each number below if lim f ( x)  A and lim f ( x)  B.
x0
x0
Then sum their answers.
1)
4)
lim f ( x2  x)
2)
x0
lim ( f ( x3 )  f ( x))
x0
a) A + 2B
5)
lim ( f ( x2 )  f ( x))
x0
3)
lim f ( x3  x)
x0
lim f ( x2  x)
x1
b) 2A + 3B
c) 2A + B
d) 3A + 3B
e) NOTA
Calculus Applications—page 6
___ FAMAT State Convention 2015
1  sin 2 x 2  cos x 2
25. Evaluate: lim
x0
x3 tan x
3
a) ½
b) 1
c) 3/2
d) 2
e) NOTA
26. Find the maximum possible surface area of a cylinder inscribed in a hemisphere of radius 1.
a)
 2

b) 1 

2 


2 
c) 2 
d)
2 2
e) NOTA
27. If f is a continuous function, then find the sum of the answers to parts (i) and (ii) below.
x2
i)
Find f (4) if
 f (t )dt  x sin  x for all x.
0
f ( x)
ii)
Find f (4) if

t 2 dt  x sin  x for all x.
0
a) 0
b) 1
c)

2
d)
28. Select the TRUE statement associated with the function f ( x) 
a)
b)
c)
d)
e)

e) NOTA
sin x
.
x2
The graph of the function is symmetric about the x-axis.
The function does NOT have a horizontal asymptote.
The graph of the function passes thru the origin.
The function has a vertical asymptote at x  0.
None of the above statements are always true.
29. Which of the following is TRUE about the graph of y  ln x 2  1 in the interval (-1, 1)?
a) It is increasing
b) It has a relative min at (0, 0)
d) It has an asymptote at x = 0.
c) It is concave down
E) NOTA

30. Evaluate the integral
a

b
a) log 
b)
Arctan ax  Arctan bx
dx where a and b are positive numbers.
x
0

1
a
log  
2
b
c)
a
 
 log  
b
d)

a
log  
2
b
e) NOTA