2001 AP CALCULUS AB FREE RESPONSE ANSWERS

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AP Calculus AB 2001 Scoring Guidelines

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AP CALCULUS AB 2001 SCORING GUIDELINES

Inquiry 1

Let R and S be the regions in the first quadrant shown in the

number over. The region R is bounded by the x-axis and also the

graphs of 32y x and also tany x . The region S is bounded by

the y-axis and the graphs of 32y x and tany x .

(a) Find the area of R.

(b) Find the location of S.

(c) Find the volume of the solid generated as soon as S is radvanced

about the x-axis.

Point of intersection 32 tanx x at ( , ) (0.902155,1.265751)A B

(a) Area R = 3 2 3

0tan 2

A

Ax dx x dx = 0.729

or

Area R = 1/ 3 10

(2 ) tanB

y y dy = 0.729 or

Area R = 3 2 3 3

0 02 2 tan

Ax dx x x dx = 0.729

3 :

1 : limits

1 : integrand

1 : answer

(b) Area S = 30

2 tanA

x x dx = 1.160 or 1.161 or

Area S = 21 1/ 3

0tan (2 )

B

By dy y dy = 1.160 or 1.161

or

Area S

= 2 1/ 3 1/ 3 1

0 0(2 ) (2 ) tan

By dy y y dy

= 1.160 or 1.161

3 :

1 : limits

1 : integrand

1 : answer

(c) Volume = 23 20

2 tanA

x x dx = 2.652 or 8.331 or 8.332

3 :

1 : boundaries and constant

1 : integrand

1 : answer

AP CALCULUS AB 2001 SCORING GUIDELINES

Inquiry 2

The temperature, in degrees Celsius (C), of the water in a pond is a

differentiable attribute W of time t. The table over mirrors the water

temperature as tape-recorded eexceptionally 3 days over a 15-day duration.

(a) Use data from the table to uncover an approximation for (12)W . Show the

computations that result in your answer. Indicate systems of meacertain.

(b) Approximate the average temperature, in levels Celsius, of the water

over the moment interval 0 15t days by making use of a trapezoidal

approximation via subintervals of length 3t days.

(c) A student proposes the function P, provided by ( / 3)( ) 20 10 tP t te , as a version for the

temperature of the water in the pond at time t, wbelow t is measured in days and also ( )P t is

measured in degrees Celsius. Find (12)P .

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Using proper devices, explain the interpretation of

your answer in regards to water temperature.

(d) Use the feature P defined in part (c) to discover the average value, in levels Celsius, of ( )P t

over the moment interval 0 15t days.

(a) Difference quotient; e.g.

(15) (12) 1(12)15 12 3

W WW

C/day or

(12) (9) 2(12)12 9 3

W WW

C/day or

(15) (9) 1(12)15 9 2

W WW

C/day

2 : 1 : distinction quotient

1 : answer (through units)

(b) 3 20 2(31) 2(28) 2(24) 2(22) 21 376.52

Median temperature 1 (376.5) 25.115

C

2 : 1 : trapezoidal method

1 : answer

(c) / 3 / 312

10(12) 103

t t

tP e te

430 0.549e C/day

This implies that the temperature is decreasing at the

price of 0.549 C/day as soon as t = 12 days.

2 : 1 : (12) (through or without units)

1 : interpretation

P

(d) 15 / 3

0

1 20 10 25.75715

tte dt C

3 :

1 : integrand

1 : limits and

average value constant

1 : answer

t (days)

( )W t(C)

0 3 6 9 12 15

20 31 28 24 22 21

AP CALCULUS AB 2001 SCORING GUIDELINES

Concern 3

A auto is traveling on a straight road via velocity

55 ft/sec at time t = 0. For 0 18t secs, the

cars acceleration ( )a t , in ft/sec2, is the piecewise

linear function characterized by the graph over.

(a) Is the velocity of the auto raising at t = 2

seconds? Why or why not?

(b) At what time in the interval 0 18t , other than t = 0, is the velocity of the vehicle

55 ft/sec? Why?

(c) On the moment interval 0 18t , what is the cars absolute maximum velocity, in ft/sec,

and also at what time does it occur? Justify your answer.

(d) At what times in the interval 0 18t , if any type of, is the cars velocity equal to zero? Justify

your answer.

(a) Since (2) (2)v a and also (2) 15 0a , the velocity is

enhancing at t = 2.

1 : answer and also factor

(b) At time t = 12 because 12

0(12) (0) ( ) 0v v a t dt .

2 : 1 : 12

1 : reason

t

(c) The absolute maximum velocity is 115 ft/sec at

t = 6.

The absolute maximum have to occur at t = 6 or

at an endallude.

6

0(6) 55 ( )

155 2(15) (4)(15) 115 (0)2

v a t dt

v

18

6( ) 0a t dt so (18) (6)v v

4 :

1 : 6

1 : absolute maximum velocity

1 : identifies 6 and

18 as candidays or

shows that boosts,

decreases, then increases

1 : eliminates 18

t

t

t

v

t

(d) The cars velocity is never before equal to 0. The absolute

minimum occurs at t = 16 where 16

6(16) 115 ( ) 115 105 10 0v a t dt .

2 : 1 : answer

1 : reason

AP CALCULUS AB 2001 SCORING GUIDELINES

Advanc

Concern 4 Let h be a function characterized for all 0x such that (4) 3h and the derivative of h is offered

by 2 2

( )x

h xx

for all 0x .

(a) Find all worths of x for which the graph of h has a horizontal tangent, and also determine

whether h has actually a local maximum, a regional minimum, or neither at each of these values.

Justify your answers.

(b) On what intervals, if any, is the graph of h concave up? Justify your answer.

(c) Write an equation for the line tangent to the graph of h at x = 4.

(d) Does the line tangent to the graph of h at x = 4 lie over or listed below the graph of h for

4x ? Why?

(a) ( ) 0h x at 2x

Local mi

4 :

1 : 2

1 : analysis

2 : conclusions 1 > not dealing with

x

(b) ( ) 1h x

the grap

(c) 1(4)h

3y

(d) The tang

the graph

+

x

( )h x +

nima at x

discontinuity at 0

22 0x

fo

h of h is conc

6 2 74 2

7 ( 4)2

x

ent line is be

of h is conc

2 2

0

0

2001 bygram and also A

2 an

r all x

ave up f

low the

ave up fo

0

und

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5

d at 2x

0 . Because of this,

or all 0x . 3 :

1 : ( )

1 : ( ) 0

1 : answer

h x

h x

1 : tangent line equation

graph because

r 4x .

1 : answer with factor

AP CALCULUS AB 2001 SCORING GUIDELINES