Transcript Document

Calculus Bowl 2015
Sample
Problems
Sponsored by Colorado Youth Education
Connection, Northrop Grumman, and Aurora Public
Schools
For more information, see http://coyec.org/calcbowl
Problem 1
d
30
cos
(
9

)
=


dx
(a) 270 (cos (9 )) 29  sin (9 )
(b) 30 (cos (9 )) 29
(c) 30 (cos (9 )) 29  sin (9 )
(d) 0
(e) The function is not differentiable
Problem 2
If f (t ) is measured in meters/second and t is measured
in seconds, what are the units of f (t ) ?
(a) no units
(b) meters
(c) meters /sec
(d) meters / sec 2
(e) seconds
Problem 3
The slope of the line y  2 is
(a) 0
(b) 2
(c)  2
(d) 1
(e) undefined
Problem 4
What is the period of y = 4 cos ( x - 2) + 1 ?
(a) 1
(b) 2
(c) 
(d) 2
(e) 4
Problem 5
Assume that the polynomial f ( x) has exactly
two local maxima and one local minimum,
and that these are the only critical points
of f ( x). Then the largest number of
x-intercepts the graph of f ( x) could have is
(a) 8
(b) 6
(d) 2
(e) 0
(c) 4
Problem 6
The graph of which of the following equations is
a straight line passing through the origin :
(a) x  2 y  1  0
(b) 3x  2 y  3
(c) 4 x  3  0
(d) 5 x  4 y
(e) 7 y  9  0
Problem 7
If
s t   10 t 2 is the position of a particle
moving on a line, then its accelerati on is
(a) 5
(b) 9.8
(c) 10
(d) 20
(e) 20 t
Problem 8
dy
If e  x, t hen

dx
y
(a)
ex
(b)
1
x
(c)
1
y
(d) ln x
(e) lny
Problem 9
If f ( a )  0 at a certain point a, which statement
concerning a is correct?
(a)
(b)
(c)
(d)
f (a )
f (a )
f (a )
f (a )
is a local maximum.
is a local minimum.
is a point of inflection .
is a point at which the tangent line is
parallel to the y - axis.
(e) f ( a ) is a point at which the tangent line is
parallel to the x - axis.
Problem 10
The graph of y  p( x) is shown below. Define
q ( x) by q ( x)  p (2 x). What is q( 1) ?
P(x)
(a) 2
(b) 1
(c) 0.5
(d) 0
(e)  1
(-1, -1)
(2,2)
Problem 11
After investing $1000 at the annual interest rate of 7 % compounded
continuously, your balance B is measured in dollars and defined by
B  f (t ). The most appropriate interpretation of the expression
f
1
(3000) is
(a) your balance in dollars after 3000 months.
(b) your balance in dollars after 3000 years.
(c) the number of years required to gain $3000 in interest.
(d) the number of years required to gain $2000 in interest.
(e) the interest earned on a $3000 deposit in t years.
Problem 12
The graphs of a function y  f  x  and its inverse
y  f 1  x  are symmetric with respect to the line
(a) y  0
(b) x  0
(c) y  x
(d) y   x
(e) y 
1
x
Problem 13
According to the graph of p( x) below, p 1 (1.4) is approximately
(b)
(c)
1
1.4
1.3
2
(d)
3
(a)
(e)
1.4
1
y
y=p(x)
Problem 14
The graph shown below is that of y  g ( x ). What is
( g  g )( 2) ?
g(x)
(a)
(b)
(c)
(d)
(e)
2
1
0.5
0.25
0
(2,1)
Problem 15
The graph of y  mx  b has how many
points of inflection?
(a)
none
(b) one
(c)
two
(d) m
(e) infinitely many
Problem 16
If the amplitude of y   k1
then the period must be
(a) 
(b) 2
(c) 4
(d) 8
(e) 16
 cos (k 2 ) is 2,
Problem 17
Determine the domain of the real - valued function
f ( x)  5  x .
(a) x  0
(b) x  25
(c) x  25
(d) 0  x  5
(e) 0  x  25
Problem 18
lim
n 
2n 2

2
n  1000n
1
(a)
500
(b) 1
(c) 2
(d) 
(e)  
Problem 19
8
If
 f ( x) dx  6
5
 f ( x) dx =
-2
(a)  4
(b)  2
(c) 2
(d) 4
(e) 8
and
 f ( x) dx =2
-2
5
then
8
Problem 20
a
If
 f ( x) dx  0
for some a  0, which of the following
-a
statements about f is true ?
(a) f is an even function
(b)
(c)
(d)
(e)
f is an odd function
f is neither even nor odd
f is both even and odd
We do not know enough about f to say
whether it's even or odd.
Problem 21
If the function f has a continuous derivative on
[0, c ] , then
c
0
f ( x ) dx 
(a) f ( c )  f (0)
(b) f (0)  f ( c )
(c) f ( c )
(d) f ( x )  c
(e) f ( c )  f (0)
Problem 22
The graph below may be a part of which of the following
functions?
(a) y  3x 2
y
(b) y  x 4
(c) y   cos ( x )  1
(d) y  e x  e  x
(e) may be any of the above
x
Problem 23
10
If
 f ( x) dx  4
1
 f ( x) dx 
1
(a)
(b)
(c)
(d)
(e)
and
 f ( x )  7
3
3
then
10
3
0
3
10
11
Problem 24
The average value of the piecewise linear function
shown over the interval [ 4, 2] is
y
(a) 1
(b)
3
2
(c) 2
(d)
5
5
(e) 9
Problem 25
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 26
b
e
cos x dx 
sin x
0
(a)
esin b cos b  1
(b) e
sin b
1
(c) esin b cos b
(d) esin b
(e) e
sin b
1
Problem 27
The most reasonable area of the shaded region
shown below is
y
(a) 3
(b)
6
5
(c) 1
(d)
(e)
5
6
1
2
Problem 28
Suppose F ( x)  g ( x) and that
g (1)  0,
F (1)  3,
g (2)  5,
g (3)  10,
F (2)  2,
F (3)  6.
(b) 5
(c) 2
3
 g ( x) dx 
Then
1
(a) 10
(d)
3
(e)  9
Problem 29
If r (t ) represents the rate at which a country's debt is
growing, then the increase in its debt between 1980
and 1990 is given by
(a)
r (1990)  r (1980)
1990  1980
(b) r (1990) - r (1980)
1990
(c)
1
10

1990
r (t ) dt
1980
1990
(e)
1
10

1980
(d)

1980
r (t ) dt
r (t ) dt
Problem 30
Consider approximating the area of the shaded region in
the figure using numerical methods. In case the number
of subdivisions is the same, order the approximate areas
from smallest to largest using the following methods.
I. Left-hand sum
y
II. Right-hand sum
III. Exact integration
(a) I, II, III
(c) II, III, I
(e) III, II, I
(b) I, III, II
(d) III, I, II
Problem 31
Let f ( x) be a continuous function on the closed
interval [1, 4]. If 5  f ( x)  9 on this interval,
4
then the value of
 f ( x) dx
1
(a) 12
(c) 18
(e) 27
(b) 15
(d) 21
cannot be