Calculus Bowl 2011

Sample Problems

Sponsored by Colorado Youth Education Connection, Northrop Grumman, and Aurora Public SchoolsFor more information, see http://coyec.org/calcbowl

Problem 1

30

29

29

29

cos ( 9 ) =

(a) 270 (cos (9 )) sin (9 )

(b) 30 (cos (9 ))

(c) 30 (cos (9 )) sin (9 )

(d) 0

(e) The function is not differentiable

d

dx

Problem 2

2

If ( ) is measured in meters/second and is measured

in seconds, what are the units of ( ) ?

(a) no units

(b) meters

(c) meters /sec

(d) meters / sec

(e) seconds

f t t

f t

Problem 3

The slope of the line is

(a) 0

(b) 2

(c)

d) 1

(e) undefined

y

2

2

(

Problem 4

4 (e)

2 (d)

(c)

2 (b)

1 (a)

? 1+2)-( cos 4= of period theisWhat

xy

Problem 5

Assume that the polynomial ( ) has exactlytwo local maxima and one local minimum, and that these are the only critical points of ( ). Then the largest number of -intercepts

f x

f xx the graph of ( ) could have is

(a) 8 (b) 6 (c) 4

(d) 2 (e) 0

f x

Problem 6

097 (e)

45 (d)

034 (c)

323 (b)

012 (a)

:origin the throughpassing linestraight a

is equations following theof whichof graph The

y

yx

x

yx

yx

Problem 7

t

tts

20 (e)

20 (d)

10 (c)

9.8 (b)

5 (a)

is onaccelerati its thenline,a on moving

particlea of position theis 10 If 2

Problem 8

lny (e)

ln (d)

(c)

(b)

(a)

then, If

y1

x1

x

e

dx

dyxe

x

y

Problem 9

axis.- the toparallel

is line tangent theat whichpoint a is )( (e)

axis.- the toparallel

is line tangent theat whichpoint a is )( (d)

.inflection ofpoint a is )( (c)

minimum. locala is )( (b)

maximum. locala is )( (a)

correct? is concerning

statement which,point certaina at 0)( If

x

af

y

af

af

af

af

a

aaf

Problem 10

The graph of ( ) is shown below. Define

( ) by ( ) (2 ). What is ( 1)?

(a) 2

(b) 1

(c) 0.5

(d) 0

(e) 1

y p x

q x q x p x q

P(x)

(-1, -1)

(2,2)

Problem 11

After investing $1000 at the annual interest rate of 7 % compounded

continuously, your balance is measured in dollars and defined by

( ). The most appropriate interpretation of the expres

B

B f t 1

sion

(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 r

f

equired to gain $2000 in interest.

(e) the interest earned on a $3000 deposit in years. t

1

1

The graphs of a function and its inverse

are symmetric with respect to the line

(a) 0

(b) 0

(c)

(d)

(e) x

y f x

y f x

y

x

y x

y x

y

Problem 12

Problem 13

1According to the graph of ( ) below, (1.4) is approximately

1 (a)

1.4(b) 1.3

(c) 2

(d) 3

1.4 (e)

1

p x p

y=p(x)

y

Problem 14

0 (e)

0.25 (d)

0.5 (c)

1 (b)

2 (a)

?)2)((

isWhat ).( of that isbelow shown graph The

gg

xgy

g(x) (2,1)

Problem 15The graph of has how manypoints of inflection?

(a) none

(b) one

(c) two

(d)

e) infinitely many

y mx b

m

(

Problem 16

16 (e)

8 (d)

4 (c)

2 (b)

(a)

bemust period thethen

2, is )( cos of amplitude theIf 21 ky k

Problem 17

250 (e)

50 (d)

25 (c)

25 (b)

0 a)(

. 5)(

function valued-real theof domain theDetermine

x

x

x

x

x

xxf

Problem 182

2

2

1000

1(a)

500(b) 1

(c) 2

(d)

(e)

limn

n

n n

8 8

5 -2

5

-2

If ( ) 6 and ( ) =2

then ( ) =

(a) 4

(b) 2

(c) 2

(d) 4

(e) 8

f x dx f x dx

f x dx

Problem 19

a

-a

If ( ) 0 for some 0, which of the following

statements about is true ?

(a) is an even function

(b) is an odd function

(c) is neither even nor odd

(d) is both even

f x dx a

f

f

f

f

f

and odd

(e) We do not know enough about to say

whether it's even or odd.

f

Problem 20

Problem 21

)0()( (e)

)( (d)

)( (c)

)()0( (b)

)0()( a)(

)( then, ] ,0[

on derivative continuousa has function theIf

0

fcf

cxf

cf

cff

fcf

dxxfc

f

c

Problem 22

above theofany bemay (e)

(d)

1)( cos (c)

(b)

3 (a)

functions?

following theof whichofpart a bemay below graph The

4

2

xx eey

xy

xy

xy

y

x

Problem 23

10 10

1 3

3

1

If ( ) 4 and ( ) 7

then ( )

(a) 3

(b) 0

(c) 3

(d) 10

(e) 11

f x dx f x

f x dx

Problem 24

32

55

The average value of the piecewise linear function

shown over the interval [ 4, 2] is

(a) 1

(b)

(c) 2

(d)

(e) 9

y

Problem 25

1990

1980

If ( ) represents the rate at which a country's debt is

growing in dollars per year in the year , then the proper

units of the quantity ( ) is

(a) dollars

(b) per year

(c) per y

r t

t

r t dt

ear per year

(d) dollars per year

(e) dollars per year per year

Problem 26sin

0

sin

sin

sin

sin

sin

cos

(a) cos 1

(b) 1

(c) cos

(d)

(e) 1

bx

b

b

b

b

b

e x dx

e b

e

e b

e

e

Problem 27

21

65

56

(e)

(d)

1 (c)

(b)

3 (a)

isbelow shown

region shaded theofarea reasonablemost The

y

Problem 28

3

1

Suppose ( ) ( ) and that

(1) 0, (2) 5, (3) 10,

(1) 3, (2) 2, (3) 6.

Then ( )

(a) 10 (b) 5

F x g x

g g g

F F F

g x dx

(c) 2

(d) 3 (e) 9

Problem 29If ( ) represents the rate at which a country's debt is

growing, then the increase in its debt between 1980

and 1990 is given by

(1990) (1980)(a) (b) (1990) - (1980

1990 1980

r t

r rr r

1990 19901

101980 1980

19901

101980

)

(c) ( ) (d) ( )

(e) ( )

r t dt r t dt

r t dt

Problem 30Consider 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

II. Right-hand sum

III. Exact integration

(a) I, II, III (b) I, III, II

(c) II, III, I (d) III, I, II

(e) III, II, I

y

Problem 31

4

1

Let ( ) be a continuous function on the closed

interval [1, 4]. If 5 ( ) 9 on this interval,

then the value of ( ) cannot be

(a) 12 (b) 15

(c) 18 (d)

f x

f x

f x dx

21

(e) 27