Gateway Calculus Bowl
2011
Practice
Problems
Colorado Youth
Education Connection,
2011
Problem 1
d
2  ?
dx
(a) 0
(b) 
(c) 2
(d)  ln 
2
(e)  2  2 ln 
Problem 2
d 2
x
dx
 ?
x 
(a) 0
(b) 
(c) 2
(d)  2 ln 
(e)  2  2 ln 
Problem 3
If
y  12 cos3t
what is the constant A for which y
satisfies the equation
Ay’’+y = 0 ?
(a) 0
(b) 3
(c) 1/3
(d) -1/9
(e) 1/9
Problem 4
d
cos(ln 3 x)  ?
dx
(a) -sin(ln 3 x)
(b) -3sin(ln 3 x)ln 2 x
-3sin(ln 3 x)ln 2 x
(c)
x2
-3sin(ln 3 x)ln 2 x
(d)
x3
-3sin(ln 3 x)ln 2 x
(e)
x
Problem 5
sin 7t
lim
t 0 sin 9t
(a) 0
(b) 1
 ?
(c) 7/9
(d) 9/7
(e) Does not exist
Problem 6
A function f has a point of inflection when
(best answer):
(a) f’ = 0
(b) f’ < 0 and f’’ > 0
(c) f’ = 0 and f’’ < 0
(d) f’’ = 0
(e) the curvature changes sign at this point
Problem 7
If
f ( x)  sin 2 (5-x) then f '(0) = ?
(a) -2 cos(5)
(b) 2 sin(5) cos(5)
(c) -2 sin(5) cos(5)
(d) 2 sin 2 (5)
(e) 10 sin(5)cos(5)
Problem 8
The line segment drawn indicates an inflection point of
the curve when
a) It is blue
b) It is green
c) It is red
d) It is blue and horizontal
e) Never, because this curve has no inflection points
Problem 9
sin 7t
lim
t 1 sin 9t
(a) 0
(b) 1
 ?
(c) 7/9
(d) 9/7
(e) sin 7 / sin 9
Problem 10
What is the average value of the function f(x) = x 3 -2x
on the closed interval [0, 2]?
(a) 0
(b) 1
(c) -1
(d) 2
(e) -3/2
Problem 11
x
Define F ( x )   -1 f (t ) dt, where f (t ) is the
piecewise function shown in the figure. What
is the value of F ( 2) closest to? y
(a)  2
2
(b)  1
1
(c) 1
(d) 2
(e) 4
x
-1
1
-1
2
Problem 12
If r (t ) represents the rate at which a country's debt is
growing in dollars per year in the year t , then the proper
1990
units of the quantity

r (t ) dt is
1980
(a) dollars
(b) per year
(c) per year per year
(d) dollars per year
(e) dollars per year per year
Problem 13
If 0  g ( x )  | sin x | for all x in the interval
 4 , 2 ,
(a)
(b)
(c)
(d)
then lim g ( x )
x 0
does not exist
is 0
is 1
is  1
1
(e) is
2
Problem 14
“calculus” is:
(a) A Latin word meaning a small stone used for counting.
(b) A branch of mathematics including the study of limits,
derivatives, integrals, sequences and series.
(c) Any method or system of calculation based on the
symbolic manipulation of expressions.
(d) A form of hardened dental plaque.
(e) (b) and (c)
(f) All of the above
Problem 15
It wouldbe most reasonablet o det ermine
x
x
e
e
 1 dx  ?

by t hefollowingmet hodor met hods(best answer) :
(a) Use int egrat ion - by - part swit h u  e x  1 and dv  e x dx
(b) Use subst it ut ion u  e x .
(c) Use subst it ut ion u  e x or u  e x  1.
(d) Use t he t rigonomet ricsubst it ut ion u  sin x.
(e) Use t hemet hodof part ialfract ions.
Problem 16
Det ermine
x
e

(a)
(b)
(c )
e x  1 dx  ?
1 x x
e (e  1) 3 / 2  C
3
2 x
(e  1) 3 / 2  C
3
(e x  1) 3 / 2  C
e(3 / 2) x
(d )
 ex  C
3/ 2
(e)
ex 1  C
Problem 17
Let R (t ) be the area of the region bounded by the y - axis,
a positive continuous function y  f ( x ), a negative continuous
function y  g ( x ), and the line x  t , where t  0. Which
of the following must be equal to dR
?
dt
(a)

(b)

t
( f ( x )  g ( x )) dx
0
t
( f ( x )  g ( x )) dx
0
(c) f (t )  g (t )
(d) f (t )  g (t )
(e) f (t )  g (t )
y
y=f(x)
R(t)
y=g(x)
x=t
Problem 18
Which of the following integrals gives the area between
the graphs of y  x and y = x between x  0 and x  1?
(a)
(c)
(e)



1
( x  x ) dx
0
(b)

1
( x 1) 2 dx
0
1
( x  x) 2 dx
0
(d)
1
( x  x ) 2 dx
0

1
( x  x) dx
0
Problem 19
Given a function f , which is a true statement?
(a) If f ''(a)  0, then the graph of y  f ( x) is concave upward at x  a.
(b) If f '(a)  0 and f '(b)  0 then the graph of y  f ( x) is decreasing
between x  a and x  b.
(c) If f is a continuous function between x  a and x  b,
f ( a)  0, f (b)  0, then somewhere between x  a and x  b,
f ( x)  0.
(d) If f is a continuous function between x  a and x  b, then
f is differentiable between x  a and x  b.
(e) If f (a)  0, f (b)  0, then somewhere between x  a and x  b,
f ( x)  0.
Problem 20
Which one of the following functions could have the
graph shown in red below? ( x and y scales may be
unequal)
y
(a)
(b)
(c)
(d)
(e)
y  2.72 x
y  0.01  0.001x
y  27.9  0.1x
y  5.7  200x
y  x / 3.14
x
Problem 21
If the graph of y  f ( x) has x-intercepts x  1 and
x
x  2, then the graph of y  f   has which of the
2
following x-intercepts?
(a)  1, 2
(b)  12 ,
5
2
(c)  32 ,
3
2
(d)  12 , 1
(e)  2, 4
?
Problem 22
Evaluate lim
x 
9x 1
x2  2
(a) 0
(b) 3
(c) 3
(d) 9
(e) the limit does not exist
Problem 23
If a function k ( x ) is such that 0  k ( x )  1 for 0  x  1,
then which of the following statements must be true?
(a) k ( x ) has an inverse
(b) k ( x ) is continuous for 0  x  1
(c) k ( x ) is increasing for 0  x  1
(d) sin( k ( x ) )  0 for 0  x  1
(e) ln( k ( x ))  0 for 0  x  1
Problem 24
The statement of the Mean Value Theorem (MVT) is
a) The statement that all functions have a mean value.
b) The statement that mean value can be computed via
definite integrals.
c) The statement that continuous functions over an
interval [a,b] take on their mean value somewhere in
that interval.
d) The statement that for functions with a continuous
derivative over an interval [a,b] , at at least one point
in that interval the derivative takes on the value of
the average slope.
e) A false conjecture first published by Parameshvara
(1370-1460).
Problem 25
If
x 2  xy  y 2  0, which of the following is true?
(a) 2 x  dy
 2y  0
dx
dy
(b) 2 x  dy

2
y
0
dx
dx
(c) 2 x  y  x dy
 2y  0
dx
dy
(d) 2 x  x dy

2
y
0
dx
dx
dy
(e) 2 x  y  x dy

2
y
0
dx
dx
Problem 26
A particle moves along the x-axis with
t
x
(
t
)

e
cost
position at time t given by
for 0  t  2 . Find the time at which
the particle is furthest to the right.
(a) 0
(b) π
(c) 2π
(d) 3π/2
(e) 3π/4
Problem 27
The average value of sin x on the interval
   ,   is
 2 2 
(a)
2

(b) 0
(c) 1
(d) 2
(e) 2
Problem 28
The function f ( x )  x 2  4 is not differentiable at
(a) x  2 only
(b) x  2 only
(c) x  0 only
(d) x  2 , x  2
(e) x  2 , x  2 , x  0
Problem 29
The arc length of the curve f ( x )  4  x 2 from
x  0 to x  2 is
(a) 8
(b) 4
(c) 2
(d) 
(e) 2
Problem 30
 Which of the following is a true story about a famous woman mathematician?
Hypatia was the first recognized women mathematician. She was born in Alexandria, Egypt,
around 350AD and was a recognized scholar. Besides being a mathematician, she was an
astute astronomer and a philosopher. An angry Christian mob killed her.
* She wrote a commentary on the 13th volume of the famous Greek mathematics text
book, 'Arithmetica'.
* She edited Ptolemy's famous version of the 'Almagest'.
* She edited her father's commentary on 'Euclid's Elements'.
Maria Gaetana Agnesi was a child prodigy, an Italian linguist and a math wizard. Born in a
wealthy family in 1718, she was the 21st child of her parents. She was fluent in 6
languages. Some of her contributions are as follows:
* She wrote the first book introducing integral and differential calculus.
* She determined the equation of a peculiar curve, which came to be known as the 'Witch
of Agnesi'.
Marie Sklodowska Curie was born in 1868 in Poland, eventually becoming a famous
biologist and mathematician in Germany. She was the first woman professor at the
University of Stutgart and developed the mathematical theory behind radiocarbon dating.
A.Hypatia
B.Hypatia and Maria Agnesi
C.Maria Agnesi and Marie Curie
D. Hypatia and Marie Curie
E. All of the above
Problem 31
y
 2 x  1 is the tangent line to the graph of a function f at x  3.
2
Consequently, f '(3) 
(a)  1
(b) 0
(c) 1
(d) 2
(e)
4
Problem 32
In mathematics, a sequence is
(a) An ordered list.
(b) An infinite series.
(c) Any quantity of numbers added together.
(d) Logically connected assertions.
(e) A periodic table of numbers.
Problem 33
Which of the following represents the area
of the shaded region in the figure ?
y
(a)
(b)


d
c
b
a
f ( y )dy
d
y=f(x)
f ( x ) dx
(c) f (b)  f ( a )
(d) (b  a ) [ f (b)  f ( a )]
(e) ( d  c ) [ f (b)  f ( a )]
c
O
x
a
b
Problem 34
Which of the following
functions could have the
graph shown on the right?
(a) y  x ( x  4)
(b) y  x ( 4  x )
(c) y  x 2 ( x  4) 2
(d) y  x 2 ( x  4)
(e) y  x 2 ( 4  x ) 2
y
Problem 35
If v (t ) models the velocity of a moving object,
then

b
v (t ) dt with a  b signifies
a
(a) the total speed between time a and time b
(b) the difference in accelerati on at time a and time b
(c) displaceme nt from time a to time b
(d) arc length of v (t ) from time a to time b
(e) the distance travelled between time a and time b
Problem 36
Assume that f   x   g   x  for any x, where
a  x  b. Then which one of the following
for true?
a xb ?
statements is true
always
(a) f  x   g  x  .
(b) f  x   g  x   c where a  c  b.
(c) f  x   g  x   c for some constant c.
(d) f  x   g  x 
(e) f  c   g  c  for some c where a  c  b.
Problem 37
If f (2)  2, f (2)  2, g (2)  4, and g (2)  3,
then (f g ) (2) is
(a) 12
(b) 2
(c)  4
(d)  6
(e) Not enough information
Problem 38
The region bounded by the graphs of y  x and
y  x 3 has area given by
1
(b)  1 ( x 3  x ) dx
1
(d)  0 ( x 3  x ) dx
(a)  1 ( x  x 3 ) dx
(c)  0 ( x  x 3 ) dx
(e)
1
0
( x  x 3 ) dx
1
1