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AP Calculus BC2004 Free-Response Questions

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2004 AP CALCULUS BC FREE-RESPONSE QUESTIONS

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2

CALCULUS BC SECTION II, Part A

Time45 minutes Number of problems3

A graphing calculator is required for some problems or parts of problems.

1. Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The

traffic flow at a particular intersection is modeled by the function F defined by

( ) ( )82 4sin 2tF t = + for 0 30,t

where ( )F t is measured in cars per minute and t is measured in minutes.

(a) To the nearest whole number, how many cars pass through the intersection over the 30-minute period?

(b) Is the traffic flow increasing or decreasing at 7 ?t = Give a reason for your answer.

(c) What is the average value of the traffic flow over the time interval 10 15 ?t Indicate units of measure.

(d) What is the average rate of change of the traffic flow over the time interval 10 15 ?t Indicate units of measure.

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3

2. Let f and g be the functions given by ( ) ( )2 1f x x x= - and ( ) ( )3 1g x x x= - for 0 1.x The graphs of f and g are shown in the figure above.

(a) Find the area of the shaded region enclosed by the graphs of f and g.

(b) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved about the horizontal line 2.y =

(c) Let h be the function given by ( ) ( )1h x kx x= - for 0 1.x For each 0,k > the region (not shown) enclosed by the graphs of h and g is the base of a solid with square cross sections perpendicular to the x-axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k.

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4

3. An object moving along a curve in the xy-plane has position ( ) ( )( ),x t y t at time t 0 with ( )23 cos .dx tdt = +

The derivative dydt

is not explicitly given. At time t 2,= the object is at position ( )1, 8 .

(a) Find the x-coordinate of the position of the object at time 4.t =

(b) At time 2,t = the value of dydt

is 7.- Write an equation for the line tangent to the curve at the point

( ) ( )( )2 , 2 .x y

(c) Find the speed of the object at time 2.t =

(d) For 3,t the line tangent to the curve at ( ) ( )( ),x t y t has a slope of 2 1.t + Find the acceleration vector of the object at time 4.t =

END OF PART A OF SECTION II

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5

CALCULUS BC SECTION II, Part B


No calculator is allowed for these problems.

4. Consider the curve given by 2 24 7 3 .x y xy+ = +

(a) Show that 3 2

.8 3

dy y xdx y x

-

=

-

(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P.

(c) Find the value of 2

2d y

dx at the point P found in part (b). Does the curve have a local maximum, a local

minimum, or neither at the point P ? Justify your answer.

5. A population is modeled by a function P that satisfies the logistic differential equation

( )1 .5 12dP P Pdt

= -

(a) If ( )0 3,P = what is ( )lim ?t

P t

If ( )0 20,P = what is ( )lim ?t

P t

(b) If ( )0 3,P = for what value of P is the population growing the fastest?

(c) A different population is modeled by a function Y that satisfies the separable differential equation

( )1 .5 12dY Y tdt

= -

Find ( )Y t if ( )0 3.Y =

(d) For the function Y found in part (c), what is ( )lim ?t

Y t

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6

6. Let f be the function given by ( ) ( )sin 5 ,4f x xp

= + and let ( )P x be the third-degree Taylor polynomial

for f about 0.x =

(a) Find ( ).P x

(b) Find the coefficient of 22x in the Taylor series for f about 0.x =

(c) Use the Lagrange error bound to show that ( ) ( )1 1 1 .10 10 100f P- <

(d) Let G be the function given by ( ) ( )0

.x

G x f t dt= Write the third-degree Taylor polynomial for G about 0.x =

END OF EXAMINATION

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AP Calculus BC2004 Scoring Guidelines





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AP CALCULUS BC 2004 SCORING GUIDELINES


2

Question 1

Traffic flow is defined as the rate at which cars pass through an intersection, measured in cars per minute. The traffic flow at a particular intersection is modeled by the function F defined by

( ) ( )82 4sin 2tF t = + for 0 30,t where ( )F t is measured in cars per minute and t is measured in minutes.

(a) To the nearest whole number, how many cars pass through the intersection over the 30-minute period?

(b) Is the traffic flow increasing or decreasing at 7 ?t = Give a reason for your answer.

(c) What is the average value of the traffic flow over the time interval 10 15 ?t Indicate units of measure.

(d) What is the average rate of change of the traffic flow over the time interval 10 15 ?t Indicate units of measure.

(a) ( )30

02474F t dt = cars 3 :

1 : limits1 : integrand1 : answer

(b) ( )7 1.872 or 1.873F = Since ( )7 0,F < the traffic flow is decreasing

at 7.t =

1 : answer with reason

(c) ( )15

101 81.899 cars min5 F t dt = 3 :

1 : limits1 : integrand1 : answer

(d) ( ) ( )15 10 1.51715 10F F

=

or 21.518 cars min

1 : answer

Units of cars min in (c) and 2cars min in (d)

1 : units in (c) and (d)

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3

Question 2

Let f and g be the functions given by ( ) ( )2 1f x x x= and

( ) ( )3 1g x x x= for 0 1.x The graphs of f and g are shown in the figure above.

(a) Find the area of the shaded region enclosed by the graphs of f and g.

(b) Find the volume of the solid generated when the shaded region enclosed by the graphs of f and g is revolved about the horizontal line 2.y =

(c) Let h be the function given by ( ) ( )1h x k x x= for 0 1.x For each 0,k > the region (not shown) enclosed by the graphs of h and g is the

base of a solid with square cross sections perpendicular to the x-axis. There is a value of k for which the volume of this solid is equal to 15. Write, but do not solve, an equation involving an integral expression that could be used to find the value of k.

(a) Area ( ) ( )( )

( ) ( )( )

1

01

02 1 3 1 1.133

f x g x dx

x x x x dx

=

= =

2 : { 1 : integral1 : answer

(b) Volume ( )( ) ( )( )( )1 2 20 2 2g x f x dx= ( )( ) ( )( )( )1 2 20 2 3 1 2 2 1

16.179

x x x x dx=

=

4 :

( ) ( )( )2 2

1 : limits and constant 2 : integrand 1 each error Note: 0 2 if integral not of form

1 : answer

b

ac R x r x dx

(c) Volume ( ) ( )( )1 20h x g x dx=

( ) ( )( )1 20

1 3 1 15k x x x x dx =

3 : { 2 : integrand1 : answer

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4

Question 3

An object moving along a curve in the xy-plane has position ( ) ( )( ),x t y t at time 0t with

( )23 cos .dx tdt = + The derivative dydt is not explicitly given. At time 2,t = the object is at position

( )1, 8 . (a) Find the x-coordinate of the position of the object at time 4.t =

(b) At time 2,t = the value of dydt is 7. Write an equation for the line tangent to the curve at the point

( ) ( )( )2 , 2 .x y (c) Find the speed of the object at time 2.t =

(d) For 3,t the line tangent to the curve at ( ) ( )( ),x t y t has a slope of 2 1.t + Find the acceleration vector of the object at time 4.t =

(a) ( ) ( ) ( )( )( )( )

4 22

4 22

4 2 3 cos

1 3 cos 7.132 or 7.133

x x t dt

t dt

= + +

= + + =

3 : ( )( )4 22 1 : 3 cos

1 : handles initial condition 1 : answer

t dt +

(b)

22

7 2.9833 cos 4tt

dydy dtdx dx

dt=

=

= = = +

( )8 2.983 1y x =

2 : 2 1 : finds

1 : equationt

dydx =

(c) The speed of the object at time 2t = is

( )( ) ( )( )2 22 2 7.382 or 7.383.x y + =

1 : answer

(d) ( )4 2.303x =

( ) ( ) ( )( )22 1 3 cosdy dy dxy t t tdt dx dt = = = + + ( )4 24.813 or 24.814y =

The acceleration vector at 4t = is 2.303, 24.813 or 2.303, 24.814 .

3 :

( )1 : 4

1 :

1 : answer

xdydt

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5

Question 4

Consider the curve given by 2 24 7 3 .x y x y+ = +

(a) Show that 3 2 .8 3dy y xdx y x

=

(b) Show that there is a point P with x-coordinate 3 at which the line tangent to the curve at P is horizontal. Find the y-coordinate of P.

(c) Find the value of 2

2d ydx

at the point P found in part (b). Does the curve have a local maximum, a

local minimum, or neither at the point P ? Justify your answer.

(a) ( )

2 8 3 38 3 3 2

3 28 3

x y y y x yy x y y x

y xy y x

+ = + =

=

2 : 1 : implicit differentiation

1 : solves for y

(b) 3 2 0; 3 2 08 3y x y xy x

= =

When 3,x = 3 6

2yy

==

2 23 4 2 25+ = and 7 3 3 2 25+ =

Therefore, ( )3, 2P = is on the curve and the slope is 0 at this point.

3 : ( )( )

1 : 0

1 : shows slope is 0 at 3, 21 : shows 3, 2 lies on curve

dydx

=

(c) ( )( ) ( )( )( )

2

2 28 3 3 2 3 2 8 3

8 3y x y y x yd y

dx y x =

At ( )3, 2 ,P = ( )( )( )

2

2 216 9 2 2 .716 9

d ydx

= =

Since 0y = and 0y < at P, the curve has a local maximum at P.

4 : ( )

2

2

2

2

2 :

1 : value of at 3, 2

1 : conclusion with justification

d ydx

d ydx

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6

Question 5

A population is modeled by a function P that satisfies the logistic differential equation

( )1 .5 12dP P Pdt = (a) If ( )0 3,P = what is ( )lim ?

tP t

If ( )0 20,P = what is ( )lim ?t

P t

(b) If ( )0 3,P = for what value of P is the population growing the fastest? (c) A different population is modeled by a function Y that satisfies the separable differential equation

( )1 .5 12dY Y tdt = Find ( )Y t if ( )0 3.Y = (d) For the function Y found in part (c), what is ( )lim ?

tY t

(a) For this logistic differential equation, the carrying capacity is 12. If ( )0 3,P = ( )lim 12.

tP t

=

If ( )0 20,P = ( )lim 12.t

P t

=

2 : 1 : answer1 : answer

(b) The population is growing the fastest when P is half the carrying capacity. Therefore, P is growing the fastest when 6.P =

1 : answer

(c) ( ) ( )1 1 115 12 5 60t tdY dt dtY = = 2

ln 5 120t tY C= +

( )2

5 120t t

Y t Ke

= 3K =

( )2

5 1203t t

Y t e

=

5 :

1 : separates variables 1 : antiderivatives 1 : constant of integration 1 : uses initial condition 1 : solves for 0 1 if is not exponential

YY

Note: max 2 5 [1-1-0-0-0] if no constant of integration Note: 0 5 if no separation of variables

(d) ( )lim 0t

Y t

=

1 : answer 0 1 if Y is not exponential

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7

Question 6

Let f be the function given by ( ) ( )sin 5 ,4f x x = + and let ( )P x be the third-degree Taylor polynomial for f about 0.x =

(a) Find ( ).P x

(b) Find the coefficient of 22x in the Taylor series for f about 0.x =

(c) Use the Lagrange error bound to show that ( ) ( )1 1 1 .10 10 100f P < (d) Let G be the function given by ( ) ( )

0.

xG x f t dt= Write the third-degree Taylor polynomial

for G about 0.x =

(a) ( ) ( ) 20 sin 4 2f = = ( ) ( ) 5 20 5cos 4 2f = = ( ) ( ) 25 20 25sin 4 2f = = ( ) ( ) 125 20 125cos 4 2f = =

( ) ( ) ( )2 32 5 2 25 2 125 2

2 2 2 2! 2 3!P x x x x= +

4 : ( )P x

1 each error or missing term

deduct only once for ( )4sin evaluation error

deduct only once for ( )4cos evaluation error

1 max for all extra terms, ,+ misuse of equality

(b) ( )225 2

2 22!

2 : 1 : magnitude

1 : sign

(c) ( ) ( ) ( ) ( ) ( )( )( )1

10

44

0

4

1 1 1 1max10 10 4! 10

625 1 1 14! 10 384 100

cf P f c

= 1, as shown above.

(a) Find x dxn0

1

z in terms of n.

(b) Let T be the triangular region bounded by , the x-axis, and the line x = 1. Show that the area of T is 12n

.

(c) Let S be the region bounded by the graph of y xn= , the line , and the x-axis. Express the area of S in terms of n and determine the value of n that maximizes the area of S.

END OF EXAMINATION

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AP Calculus BC2004 Scoring Guidelines

Form B





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AP CALCULUS BC 2004 SCORING GUIDELINES (Form B)


2

Question 1

A particle moving along a curve in the plane has position ( ) ( )( ),x t y t at time t, where

4 9dx tdt = + and 2 5t tdy e edt

= +

for all real values of t. At time 0,t = the particle is at the point (4, 1). (a) Find the speed of the particle and its acceleration vector at time 0.t = (b) Find an equation of the line tangent to the path of the particle at time 0.t = (c) Find the total distance traveled by the particle over the time interval 0 3.t (d) Find the x-coordinate of the position of the particle at time 3.t = (a) At time 0:t = Speed 2 2 2 2(0) (0) 3 7 58x y = + = + = Acceleration vector ( ) ( )0 , 0 0, 3x y = =

2 : 1 : speed1 : acceleration vector

(b) ( )( )0 7

30ydy

dx x

= =

Tangent line is ( )7 4 13y x= +

2 : 1 : slope1 : tangent line

(c) Distance ( ) ( )3 2 240

9 2 5

45.226 or 45.227

t tt e e dt= + + +

=

3 :

2 : distance integral 1 each integrand error

1 error in limits 1 : answer

(d) ( )3 40

3 4 9

17.930 or 17.931

x t dt= + +

= 2 : 1 : integral1 : answer

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3

Question 2

Let f be a function having derivatives of all orders for all real numbers. The third-degree Taylor polynomial for f about 2x = is given by ( ) ( ) ( )2 37 9 2 3 2 .T x x x= (a) Find ( )2f and ( )2 .f (b) Is there enough information given to determine whether f has a critical point at 2 ?x = If not, explain why not. If so, determine whether ( )2f is a relative maximum, a relative minimum,

or neither, and justify your answer. (c) Use ( )T x to find an approximation for ( )0 .f Is there enough information given to determine

whether f has a critical point at 0 ?x = If not, explain why not. If so, determine whether ( )0f is a relative maximum, a relative minimum, or neither, and justify your answer.

(d) The fourth derivative of f satisfies the inequality ( ) ( )4 6f x for all x in the closed interval [ ]0, 2 . Use the Lagrange error bound on the approximation to ( )0f found in part (c) to explain why

( )0f is negative. (a) ( ) ( )2 2 7f T= =

( )2 92!f

= so ( )2 18f =

2 : ( )( )

1 : 2 71 : 2 18ff

= =

(b) Yes, since ( ) ( )2 2 0,f T = = f does have a critical point at 2.x =

Since ( )2 18 0,f = < ( )2f is a relative maximum value.

2 : ( )

( )( )

1 : states 2 01 : declares 2 as a relative maximum because 2 0

ff

f

=



4

Question 3

A test plane flies in a straight line with positive velocity ( ) ,v t in miles per minute at time t minutes, where v is a differentiable function of t. Selected values of ( )v t for 0 40t are shown in the table above. (a) Use a midpoint Riemann sum with four subintervals of equal length and values from the table to

approximate ( )40

0.v t dt Show the computations that lead to your answer. Using correct units,

explain the meaning of ( )40

0v t dt in terms of the planes flight.

(b) Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal zero on the open interval 0 40?t< < Justify your answer.

(c) The function f, defined by ( ) ( ) ( )76 cos 3sin ,10 40t tf t = + + is used to model the velocity of the plane, in miles per minute, for 0 40.t According to this model, what is the acceleration of the plane at 23 ?t = Indicates units of measure.

(d) According to the model f, given in part (c), what is the average velocity of the plane, in miles per minute, over the time interval 0 40?t

(a) Midpoint Riemann sum is

( ) ( ) ( ) ( )[ ][ ]

10 5 15 25 3510 9.2 7.0 2.4 4.3 229

v v v v + + += + + + =

The integral gives the total distance in miles that the plane flies during the 40 minutes.

3 : ( ) ( ) ( ) ( )1 : 5 15 25 35

1 : answer 1 : meaning with units

v v v v+ + +

(b) By the Mean Value Theorem, ( ) 0v t = somewhere in the interval ( )0, 15 and somewhere in the interval ( )25, 30 . Therefore the acceleration will equal 0 for at least two values of t.

2 : 1 : two instances1 : justification

(c) ( )23 0.407 or 0.408f = miles per minute2

1 : answer with units

(d) Average velocity ( )40

01405.916 miles per minute

f t dt=

=

3 : 1 : limits1 : integrand1 : answer

t (min) 0 5 10 15 20 25 30 35 40 ( )v t (mpm) 7.0 9.2 9.5 7.0 4.5 2.4 2.4 4.3 7.3

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5

Question 4

The figure above shows the graph of ,f the derivative of the function f, on the closed interval 1 5.x The graph of f has horizontal tangent lines at 1x = and 3.x = The function f is twice differentiable with

( )2 6.f = (a) Find the x-coordinate of each of the points of inflection of the graph

of f. Give a reason for your answer. (b) At what value of x does f attain its absolute minimum value on the

closed interval 1 5 ?x At what value of x does f attain its absolute maximum value on the closed interval 1 5 ?x Show the analysis that leads to your answers.

(c) Let g be the function defined by ( ) ( ).g x x f x= Find an equation for the line tangent to the graph of g at 2.x =

(a) 1x = and 3x = because the graph of f changes from

increasing to decreasing at 1,x = and changes from decreasing to increasing at 3.x =

2 : 1 : 1, 3

1 : reasonx x= =

(b) The function f decreases from 1x = to 4,x = then increases from 4x = to 5.x = Therefore, the absolute minimum value for f is at 4.x =

The absolute maximum value must occur at 1x = or at 5.x =

( ) ( ) ( )5

15 1 0f f f t dt

=



6

Question 5

Let g be the function given by ( ) 1 .g xx

=

(a) Find the average value of g on the closed interval [ ]1, 4 .

(b) Let S be the solid generated when the region bounded by the graph of ( ) ,y g x= the vertical lines 1x = and 4,x = and the x-axis is revolved about the x-axis. Find the volume of S.

(c) For the solid S, given in part (b), find the average value of the areas of the cross sections perpendicular to the x-axis.

(d) The average value of a function f on the unbounded interval [ , )a is defined to be

( )lim .b

ab

f x dx

b a

Show that the improper integral ( )

4g x dx

is divergent, but the average value of g on the interval [4, ) is finite.

(a) 44

1 1

1 1 1 4 2 223 3 3 3 3dx xx= = =

2 : 1 : integral1 : antidifferentiation

and evaluation

(b) Volume 4 4

11

1 ln ln 4dx xx = = =

2 : 1 : integral1 : antidifferentiation

and evaluation

(c) The cross section at x has area ( )21 xx = Average value

4

1

1 1 ln 43 3dxx = =

1 : answer

(d) ( ) ( )4 4

1lim lim 2 4b

b bg x dx dx b

x

= = =

This limit is not finite, so the integral is divergent.

( )

4

4

1 1 2 44 4 4

bbg x dx bdxb b bx

= =

2 4lim 04bbb

=

4 :

( )

( )

4

4

1 : 2 4

1 : indicates integral diverges1 2 4 1 : 4 4

1 : finite limit as

b

b

g x dx b

bg x dxb bb

= =

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7

Question 6

Let be the line tangent to the graph of ny x= at the point (1, 1), where 1,n > as shown above.

(a) Find 1

0nx dx in terms of n.

(b) Let T be the triangular region bounded by , the x-axis, and the

line 1.x = Show that the area of T is 1 .2n

(c) Let S be the region bounded by the graph of ,ny x= the line , and the x-axis. Express the area of S in terms of n and determine the value of n that maximizes the area of S.

(a) 111

00

11 1

nn xx dx n n

+= =+ +

2 : 1 : antiderivative of 1 : answer

nx

(b) Let b be the length of the base of triangle T.

1b is the slope of line , which is n

( ) ( )1 1Area 12 2T b n= =

3 :

1 : slope of line is 1 1 : base of is

1 1 : shows area is 2

n

T n

n

(c)

( ) ( )1

0Area Area

1 11 2

nS x dx T

n n

=

= +

( ) 2 21 1Area 0

( 1) 2d Sdn n n

= + =+

( )222 1n n= + ( )2 1n n= +

1 1 22 1

n = = +

4 :

1 : area of in terms of 1 : derivative1 : sets derivative equal to 0

1 : solves for

S n

n

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AP Calculus BC 2005 Free-Response Questions

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GO ON TO THE NEXT PAGE. 2




1. Let f and g be the functions given by ( ) ( )1 sin4

f x xp= + and ( ) 4 .xg x -= Let R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first

quadrant enclosed by the graphs of f and g, as shown in the figure above.

(a) Find the area of R.

(b) Find the area of S.

(c) Find the volume of the solid generated when S is revolved about the horizontal line y 1.= -

WRITE ALL WORK IN THE TEST BOOKLET.

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2. The curve above is drawn in the xy-plane and is described by the equation in polar coordinates ( )sin 2r q q= + for 0 ,q p where r is measured in meters and q is measured in radians. The derivative of r with respect

to q is given by ( )1 2cos 2 .drd qq = +

(a) Find the area bounded by the curve and the x-axis.

(b) Find the angle q that corresponds to the point on the curve with x-coordinate 2.-

(c) For 2 ,3 3p p

q< < drdq is negative. What does this fact say about r ? What does this fact say about the

curve?

(d) Find the value of q in the interval 02p

q that corresponds to the point on the curve in the first

quadrant with greatest distance from the origin. Justify your answer.


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4

Distance x (cm)

0 1 5 6 8

Temperature ( )T x ( )C 100 93 70 62 55

3. A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the

temperature ( ),T x in degrees Celsius ( )C , of the wire x cm from the heated end. The function T is decreasing and twice differentiable.

(a) Estimate ( )7 .T Show the work that leads to your answer. Indicate units of measure.

(b) Write an integral expression in terms of ( )T x for the average temperature of the wire. Estimate the average temperature of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.

(c) Find ( )8

0,T x dx and indicate units of measure. Explain the meaning of ( )

8

0T x dx in terms of the

temperature of the wire.

(d) Are the data in the table consistent with the assertion that ( ) 0T x > for every x in the interval 0 8 ?x< < Explain your answer.



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4. Consider the differential equation 2 .dy x ydx = -

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point ( )0, 1 .

(Note: Use the axes provided in the pink test booklet.)

(b) The solution curve that passes through the point ( )0, 1 has a local minimum at ( )3ln .2x = What is the y-coordinate of this local minimum?

(c) Let ( )y f x= be the particular solution to the given differential equation with the initial condition ( )0 1.f = Use Eulers method, starting at 0x = with two steps of equal size, to approximate ( )0.4 .f -

Show the work that leads to your answer.

(d) Find 2

2d ydx

in terms of x and y. Determine whether the approximation found in part (c) is less than or

greater than ( )0.4 .f - Explain your reasoning.


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6

5. A car is traveling on a straight road. For 0 24t seconds, the cars velocity ( ),v t in meters per second, is modeled by the piecewise-linear function defined by the graph above.

(a) Find ( )24

0.v t dt Using correct units, explain the meaning of ( )

24

0.v t dt

(b) For each of ( )4v and ( )20 ,v find the value or explain why it does not exist. Indicate units of measure.

(c) Let ( )a t be the cars acceleration at time t, in meters per second per second. For 0 24,t< < write a piecewise-defined function for ( ).a t

(d) Find the average rate of change of v over the interval 8 20.t Does the Mean Value Theorem guarantee a value of c, for 8 20,c< < such that ( )v c is equal to this average rate of change? Why or why not?

6. Let f be a function with derivatives of all orders and for which ( )2 7.f = When n is odd, the nth derivative

of f at 2x = is 0. When n is even and 2,n the nth derivative of f at 2x = is given by ( ) ( ) ( )1 !2 .3

nn

nf

-=

(a) Write the sixth-degree Taylor polynomial for f about 2.x =

(b) In the Taylor series for f about 2,x = what is the coefficient of ( )22 nx - for 1 ?n (c) Find the interval of convergence of the Taylor series for f about 2.x = Show the work that leads to your

answer.


END OF EXAM

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AP Calculus BC 2005 Scoring Guidelines




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Copyright 2005 by College Board. All rights reserved. Visit apcentral.collegeboard.com (for AP professionals) and www.collegeboard.com/apstudents (for AP students and parents).

2

Question 1

Let f and g be the functions given by ( ) ( )1 sin4f x x= + and ( ) 4 .xg x = Let

R be the shaded region in the first quadrant enclosed by the y-axis and the graphs of f and g, and let S be the shaded region in the first quadrant enclosed by the graphs of f and g, as shown in the figure above. (a) Find the area of R. (b) Find the area of S. (c) Find the volume of the solid generated when S is revolved about the horizontal

line 1.y =

( ) ( )f x g x= when ( )1 sin 44xx + = .

f and g intersect when 0.178218x = and when 1.x = Let 0.178218.a =

(a) ( ) ( )( )0

0.064a

g x f x dx = or 0.065


(b) ( ) ( )( )1

0.410a

f x g x dx =


(c) ( )( ) ( )( )( )1 2 21 1 4.558a f x g x dx + + = or 4.559

3 : { 2 : integrand1 : limits, constant, and answer

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3

Question 2

The curve above is drawn in the xy-plane and is described by the equation in polar coordinates ( )sin 2r = + for 0 , where r is measured in meters and is measured in radians. The derivative of r with respect to is

given by ( )1 2cos 2 .drd = +

(a) Find the area bounded by the curve and the x-axis. (b) Find the angle that corresponds to the point on the curve with

x-coordinate 2.

(c) For 2 ,3 3 < < drd is negative. What does this fact say about r ? What does this fact say about the curve?

(d) Find the value of in the interval 0 2 that corresponds to the point on the curve in the first quadrant

with greatest distance from the origin. Justify your answer.

(a) Area

( )( )

20

20

121 sin 2 4.3822

r d

d

=

= + =

3 : 1 : limits and constant

1 : integrand 1 : answer

(b) ( ) ( )( ) ( )2 cos sin 2 cosr = = + 2.786 =

2 : { 1 : equation1 : answer

(c) Since 0drd < for 2 ,3 3

< < r is decreasing on this

interval. This means the curve is getting closer to the origin.

2 : { 1 : information about 1 : information about the curver

(d) The only value in 0, 2

where 0drd = is .3

=

r 0 0

3 1.913

2 1.571

The greatest distance occurs when .3 =

2 : 1 : or 1.04731 : answer with justification

=

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4

Question 3

Distance x (cm) 0 1 5

6 8

Temperature ( )T x ( )C 100 93 70 62 55

A metal wire of length 8 centimeters (cm) is heated at one end. The table above gives selected values of the temperature ( ) ,T x in degrees Celsius ( )C , of the wire x cm from the heated end. The function T is decreasing and twice

differentiable. (a) Estimate ( )7 .T Show the work that leads to your answer. Indicate units of measure. (b) Write an integral expression in terms of ( )T x for the average temperature of the wire. Estimate the average temperature

of the wire using a trapezoidal sum with the four subintervals indicated by the data in the table. Indicate units of measure.

(c) Find ( )8

0,T x dx and indicate units of measure. Explain the meaning of ( )

8

0T x dx in terms of the temperature of the

wire. (d) Are the data in the table consistent with the assertion that ( ) 0T x > for every x in the interval 0 8 ?x< < Explain

your answer.

(a) ( ) ( )8 6 55 62 7 C cm8 6 2 2T T = =

1 : answer

(b) ( )8

018 T x dx

Trapezoidal approximation for ( )8

0:T x dx

100 93 93 70 70 62 62 551 4 1 22 2 2 2A+ + + += + + +

Average temperature 1 75.6875 C8 A =

3 : ( )

8

01 1 : 8

1 : trapezoidal sum 1 : answer

T x dx

(c) ( ) ( ) ( )8

08 0 55 100 45 CT x dx T T = = =

The temperature drops 45 C from the heated end of the wire to the other end of the wire.

2 : { 1 : value1 : meaning

(d) Average rate of change of temperature on [ ]1, 5 is 70 93 5.75.5 1 =

Average rate of change of temperature on [ ]5, 6 is 62 70 8.6 5 =

No. By the MVT, ( )1 5.75T c = for some 1c in the interval ( )1, 5 and ( )2 8T c = for some 2c in the interval ( )5, 6 . It follows that T must decrease somewhere in the interval ( )1 2, .c c Therefore T is not positive for every x in [ ]0, 8 .

2 : { 1 : two slopes of secant lines1 : answer with explanation

Units of C cm in (a), and C in (b) and (c) 1 : units in (a), (b), and (c)

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5

Question 4

Consider the differential equation 2 .dy x ydx =

(a) On the axes provided, sketch a slope field for the given differential equation at the twelve points indicated, and sketch the solution curve that passes through the point ( )0, 1 . (Note: Use the axes provided in the pink test booklet.)

(b) The solution curve that passes through the point ( )0, 1 has a local minimum at ( )3ln .2x = What is the y-coordinate of this local minimum?

(c) Let ( )y f x= be the particular solution to the given differential equation with the initial condition ( )0 1.f = Use Eulers method, starting at 0x = with two steps of equal size, to approximate ( )0.4 .f

Show the work that leads to your answer.

(d) Find 2

2d ydx

in terms of x and y. Determine whether the approximation found in part (c) is less than or

greater than ( )0.4 .f Explain your reasoning.

(a) 3 :

( )

1 : zero slopes 1 : nonzero slopes1 : curve through 0, 1

(b) 0dydx = when 2x y=

The y-coordinate is ( )32ln .2 2 : 1 : sets 0

1 : answer

dydx

=

(c) ( ) ( ) ( ) ( )( ) ( )

0.2 0 0 0.21 1 0.2 1.2

f f f + = + =

( ) ( ) ( )( )( )( )

0.4 0.2 0.2 0.21.2 1.6 0.2 1.52

f f f + + =

2 : ( )1 : Euler's method with two steps 1 : Euler approximation to 0.4f

(d) 2

2 2 2 2d y dy x ydxdx

= = +

2

2d ydx

is positive in quadrant II because 0x < and 0.y >

( )1.52 0.4f< since all solution curves in quadrant II are concave up.

2 :

2

2 1 :


d ydx

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6

Question 5

A car is traveling on a straight road. For 0 24t seconds, the cars velocity ( ) ,v t in meters per second, is modeled by the piecewise-linear function defined by the graph above.

(a) Find ( )24

0.v t dt Using correct units, explain the meaning of ( )

24

0.v t dt

(b) For each of ( )4v and ( )20 ,v find the value or explain why it does not exist. Indicate units of measure.

(c) Let ( )a t be the cars acceleration at time t, in meters per second per second. For 0 24,t< < write a piecewise-defined function for ( ).a t

(d) Find the average rate of change of v over the interval 8 20.t Does the Mean Value Theorem guarantee a value of c, for 8 20,c< < such that ( )v c is equal to this average rate of change? Why or why not?

(a) ( ) ( )( ) ( )( ) ( )( )24

01 14 20 12 20 8 20 3602 2v t dt = + + =

The car travels 360 meters in these 24 seconds.

2 : { 1 : value1 : meaning with units

(b) ( )4v does not exist because ( ) ( ) ( ) ( )

4 4

4 4lim 5 0 lim .4 4t tv t v v t v

t t + = =

( ) 220 0 520 m sec16 24 2v = =

3 : ( )( )

1 : 4 does not exist, with explanation 1 : 20 1 : units

vv

(c)

( )

5 if 0 4 0 if 4 16

5 if 16 242

tta tt

<



7

Question 6

Let f be a function with derivatives of all orders and for which ( )2 7.f = When n is odd, the nth derivative

of f at 2x = is 0. When n is even and 2,n the nth derivative of f at 2x = is given by ( ) ( ) ( )1 !2 .3

nn

nf =

(a) Write the sixth-degree Taylor polynomial for f about 2.x =

(b) In the Taylor series for f about 2,x = what is the coefficient of ( )22 nx for 1 ?n (c) Find the interval of convergence of the Taylor series for f about 2.x = Show the work that leads to your

answer.

(a) ( ) ( ) ( ) ( )2 4 66 2 4 61! 1 3! 1 5! 17 2 2 22! 4! 6!3 3 3

P x x x x= + + +

3 : ( )6

1 : polynomial about 2 2 : 1 each incorrect term 1 max for all extra terms, , misuse of equality

xP x

=

+

(b) ( ) ( ) ( )2 22 1 ! 1 1

2 !3 3 2n nn

n n

=

1 : coefficient

(c) The Taylor series for f about 2x = is

( ) ( )221

17 2 .2 3

nn

nf x x

n=

= +

( ) ( ) ( )( )

( )

( ) ( )( )

2 12 1

22

222

2 2

1 1 22 1 3lim 1 1 22 322 3lim 2 92 1 3 3

nn

n nn

n

nn

xnLxn

xn xn

++

+

=

= =

+

1L < when 2 3.x < Thus, the series converges when 1 5.x < <

When 5,x = the series is 2

21 1

3 1 17 7 ,22 3

n

nn n nn= =

+ = +

which diverges, because 1

1 ,n n=

the harmonic series, diverges.

When 1,x = the series is 2

21 1

( 3) 1 17 7 ,22 3

n

nn n nn= =

+ = +

which diverges, because 1

1 ,n n=

the harmonic series, diverges.

The interval of convergence is ( )1, 5 .

5 :

1 : sets up ratio1: computes limit of ratio

1: identifies interior of interval of convergence1 : considers both endpoints1 : analysis/conclusion for

both endpoints

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AP Calculus BC 2005 Free-Response Questions

Form B




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2005 AP CALCULUS BC FREE-RESPONSE QUESTIONS (Form B)






1. An object moving along a curve in the xy-plane has position ( ) ( )( ),x t y t at time t 0 with

212 3dx t tdt

= - and ( )( )4ln 1 4 .dy tdt = + -

At time t 0,= the object is at position ( )13, 5 .- At time 2,t = the object is at point P with x-coordinate 3.

(a) Find the acceleration vector at time 2t = and the speed at time t 2.=

(b) Find the y-coordinate of P.

(c) Write an equation for the line tangent to the curve at P.

(d) For what value of t, if any, is the object at rest? Explain your reasoning.


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2. A water tank at Camp Newton holds 1200 gallons of water at time 0.t = During the time interval 0 18t hours, water is pumped into the tank at the rate

( ) ( )295 sin 6tW t t= gallons per hour.

During the same time interval, water is removed from the tank at the rate

( ) ( )2275sin 3tR t = gallons per hour.

(a) Is the amount of water in the tank increasing at time 15 ?t = Why or why not?

(b) To the nearest whole number, how many gallons of water are in the tank at time 18 ?t =

(c) At what time t, for t0 18, is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.

(d) For 18,t > no water is pumped into the tank, but water continues to be removed at the rate ( )R t until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.


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4

3. The Taylor series about 0x = for a certain function f converges to ( )f x for all x in the interval of convergence. The nth derivative of f at 0x = is given by

( )( )( ) ( )

( )

1

2

1 10

5 1

nn

n

nf

n

+- + !

=

-

for 2.n

The graph of f has a horizontal tangent line at 0,x = and ( )0 6.f =

(a) Determine whether f has a relative maximum, a relative minimum, or neither at 0.x = Justify your answer.

(b) Write the third-degree Taylor polynomial for f about 0.x =

(c) Find the radius of convergence of the Taylor series for f about 0.x = Show the work that leads to your answer.



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4. The graph of the function f above consists of three line segments.

(a) Let g be the function given by ( ) ( )4

.x

g x f t dt-

= For each of ( )1 ,g - ( )1 ,g - and ( )1 ,g - find the value or state that it does not exist.

(b) For the function g defined in part (a), find the x-coordinate of each point of inflection of the graph of g on

the open interval 4 3.x- < < Explain your reasoning.

(c) Let h be the function given by ( ) ( )3

.x

h x f t dt= Find all values of x in the closed interval 4 3x- for which ( ) 0.h x =

(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning.


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6

5. Consider the curve given by 2 2 .y xy= +

(a) Show that .2

dy ydx y x

=

-

(b) Find all points ( ),x y on the curve where the line tangent to the curve has slope 1 .2

(c) Show that there are no points ( ),x y on the curve where the line tangent to the curve is horizontal.

(d) Let x and y be functions of time t that are related by the equation 2 2 .y xy= + At time 5,t = the value

of y is 3 and 6.dydt

= Find the value of dxdt

at time 5.t =

6. Consider the graph of the function f given by ( )1

2f x

x=

+ for 0,x as shown in the figure above. Let R be

the region bounded by the graph of f, the x- and y-axes, and the vertical line ,x k= where 0.k

(a) Find the area of R in terms of k.

(b) Find the volume of the solid generated when R is revolved about the x-axis in terms of k.

(c) Let S be the unbounded region in the first quadrant to the right of the vertical line x k= and below the graph of f, as shown in the figure above. Find all values of k such that the volume of the solid generated when S is revolved about the x-axis is equal to the volume of the solid found in part (b).


END OF EXAM

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Form B




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2

Question 1

An object moving along a curve in the xy-plane has position ( ) ( )( ),x t y t at time 0t with

212 3dx t tdt = and ( )( )4ln 1 4 .dy tdt = +

At time 0,t = the object is at position ( )13, 5 . At time 2,t = the object is at point P with x-coordinate 3.

(a) Find the acceleration vector at time 2t = and the speed at time 2.t =

(b) Find the y-coordinate of P.

(c) Write an equation for the line tangent to the curve at P.

(d) For what value of t, if any, is the object at rest? Explain your reasoning.

(a) ( ) ( ) 322 0, 2 1.88217x y = = =

( )2 0, 1.882a = Speed ( )( )2212 ln 17 12.329 or 12.330= + =

2 : 1 : acceleration vector

1 : speed

(b) ( ) ( ) ( )( )400 ln 1 4t

y t y u du= + +

( ) ( )( )2 402 5 ln 1 4 13.671y u du= + + = 3 :

( )( )2 40 1 : ln 1 41 : handles initial condition

1 : answer

u du +

(c) At 2,t = slope ( )ln 17 0.23612

dydtdxdt

= = =

( )13.671 0.236 3y x =

2 :

1 : slope1 : equation

(d) ( ) 0x t = if 0, 4t = ( ) 0y t = if 4t =

4t =

2 : 1 : reason1 : answer

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3

Question 2

A water tank at Camp Newton holds 1200 gallons of water at time 0.t = During the time interval 0 18t hours, water is pumped into the tank at the rate

( ) ( )295 sin 6tW t t= gallons per hour. During the same time interval, water is removed from the tank at the rate

( ) ( )2275sin 3tR t = gallons per hour. (a) Is the amount of water in the tank increasing at time 15 ?t = Why or why not?

(b) To the nearest whole number, how many gallons of water are in the tank at time 18 ?t =

(c) At what time t, for 0 18,t is the amount of water in the tank at an absolute minimum? Show the work that leads to your conclusion.

(d) For 18,t > no water is pumped into the tank, but water continues to be removed at the rate ( )R t until the tank becomes empty. Let k be the time at which the tank becomes empty. Write, but do not solve, an equation involving an integral expression that can be used to find the value of k.

(a) No; the amount of water is not increasing at 15t = since ( ) ( )15 15 121.09 0.W R = <


(b) ( ) ( )( )18

01200 1309.788W t R t dt+ = 1310 gallons


(c) ( ) ( ) 0W t R t = 0, 6.4948, 12.9748t =

t (hours) gallons of water 0 1200

6.495 525 12.975 1697

18 1310 The values at the endpoints and the critical points show that the absolute minimum occurs when

6.494 or 6.495. t =

3 :

1 : interior critical points 1 : amount of water is least at 6.494 or 6.4951 : analysis for absolute minimum

t

=

(d) ( )18

1310k

R t dt =

2 : 1 : limits1 : equation

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4

Question 3

The Taylor series about 0x = for a certain function f converges to ( )f x for all x in the interval of convergence. The nth derivative of f at 0x = is given by

( ) ( ) ( ) ( )( )

1

21 105 1

nn

nnf

n

+ + !=

for 2.n

The graph of f has a horizontal tangent line at 0,x = and ( )0 6.f =

(a) Determine whether f has a relative maximum, a relative minimum, or neither at 0.x = Justify your answer.

(b) Write the third-degree Taylor polynomial for f about 0.x =

(c) Find the radius of convergence of the Taylor series for f about 0.x = Show the work that leads to your answer.

(a) f has a relative maximum at 0x = because ( )0 0f = and ( )0 0.f < 2 :

1 : answer1 : reason

(b) ( ) ( )0 6, 0 0f f = =

( ) ( )2 2 3 23! 6 4!0 , 0255 1 5 2

f f = = =

( )2 3

2 32 3 2

3! 4! 3 16 6 25 1255 2! 5 2 3!x xP x x x= + = +

3 : ( )P x 1 each incorrect term

Note: 1 max for use of extra terms

(c) ( ) ( ) ( ) ( )

( )

1

20 1 1

! 5 1

nnn n

n nf nu x xn n

+ += =

( ) ( )

( ) ( )( )

( )( )

21

1 211

2

2

1 251 15 1

2 1 11 5

nn

nnnn n

n

n xu nu n x

n

n n xn n

++

+++

+

= +

+ =+

1 1 1lim 5n

nn

u xu+

= < if 5.x <

The radius of convergence is 5.

4 :

1 : general term 1 : sets up ratio 1 : computes limit1 : applies ratio test to get radius of convergence

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5

Question 4

The graph of the function f above consists of three line segments.

(a) Let g be the function given by ( ) ( )4

.x

g x f t dt

= For each of ( )1 ,g ( )1 ,g and ( )1 ,g find the value or state that it does not exist.

(b) For the function g defined in part (a), find the x-coordinate of each point of inflection of the graph of g on the open interval 4 3.x < < Explain your reasoning.

(c) Let h be the function given by ( ) ( )3

.x

h x f t dt= Find all values of x in the closed interval 4 3x for which ( ) 0.h x =

(d) For the function h defined in part (c), find all intervals on which h is decreasing. Explain your reasoning.

(a) ( ) ( ) ( )( )1

41 151 3 52 2g f t dt

= = = ( ) ( )1 1 2g f = = ( )1g does not exist because f is not differentiable

at 1.x =

3 : ( )( )( )

1 : 11 : 11 : 1

ggg

(b) 1x = g f = changes from increasing to decreasing at 1.x =

2 : 1 : 1 (only)

1 : reasonx =

(c) 1, 1, 3x = 2 : correct values 1 each missing or extra value

(d) h is decreasing on [ ]0, 2 0h f = < when 0f >

2 : 1 : interval1 : reason

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6

Question 5

Consider the curve given by 2 2 .y xy= +

(a) Show that .2dy ydx y x=

(b) Find all points ( ),x y on the curve where the line tangent to the curve has slope 1 .2

(c) Show that there are no points ( ),x y on the curve where the line tangent to the curve is horizontal.

(d) Let x and y be functions of time t that are related by the equation 2 2 .y xy= + At time 5,t = the

value of y is 3 and 6.dydt = Find the value of dxdt at time 5.t =

(a) 2y y y x y = + ( )2y x y y =

2yy y x

=

2 : 1 : implicit differentiation

1 : solves for y

(b) 12 2y

y x =

2 2y y x= 0x =

2y = ( ) ( )0, 2 , 0, 2

2 : 1 1 : 2 2

1 : answer

yy x

=

(c) 02y

y x =

0y = The curve has no horizontal tangent since

20 2 0x + for any x.

2 : 1 : 01 : explanation

y =

(d) When 3,y = 23 2 3x= + so 7 .3x =

2dy dy ydx dxdt dx dt y x dt= =

At 5,t = 3 96 7 116 3

dx dxdt dt= =

5

223t

dxdt =

=

3 : 1 : solves for 1 : chain rule

1 : answer

x

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7

Question 6

Consider the graph of the function f given by

( ) 1 2f x x= + for 0,x as shown in the figure above. Let R be the region bounded by the graph of f, the x- and y-axes, and the vertical line ,x k= where 0.k

(a) Find the area of R in terms of k.

(b) Find the volume of the solid generated when R is revolved about the x-axis in terms of k.

(c) Let S be the unbounded region in the first quadrant to the right of the vertical line x k= and below

the graph of f, as shown in the figure above. Find all values of k such that the volume of the solid generated when S is revolved about the x-axis is equal to the volume of the solid found in part (b).

(a) Area of R ( ) ( )0

1 ln 2 ln 22k

dx kx= = + +

2 : 1 : integral1 : antidifferentiation and

evaluation

(b) ( )20

0

12

2 2 2

k

R

k

V dxx

x k

=+

= = + +

3 :

1 : limits 1 : integrand1 : antidifferentiation and

evaluation

(c) ( )2

12

lim 2 2

Sk

n

n k

V dxx

x k

=+

= =+ +

S RV V=

2 2 2k k = + +

2 12 2k =+

2k =

4 :

1 : improper integral1 : antidifferentiation and

evaluation 1 : equation 1 : answer

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2

Question 1

Let R be the shaded region bounded by the graph of lny x= and the line

2,y x= as shown above. (a) Find the area of R. (b) Find the volume of the solid generated when R is rotated about the horizontal

line 3.y =

(c) Write, but do not evaluate, an integral expression that can be used to find the volume of the solid generated when R is rotated about the y-axis.

( )ln 2x x= when 0.15859x = and 3.14619.

Let 0.15859S = and 3.14619T =

(a) Area of ( ) ( )( )ln 2 1.949T

SR x x dx= =

3 : 1 : integrand

1 : limits1 : answer

(b) Volume ( )( ) ( )( )2 2ln 3 2 334.198 or 34.199

T

Sx x dx= + +

=

3 : { 2 : integrand1 : limits, constant, and answer

(c) Volume ( )( )2 222

( 2)T

y

Sy e dy

= +

3 : { 2 : integrand1 : limits and constant

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3

Question 2

At an intersection in Thomasville, Oregon, cars turn

left at the rate ( ) ( )260 sin 3tL t t= cars per hour over the time interval 0 18t hours. The graph of

( )y L t= is shown above. (a) To the nearest whole number, find the total

number of cars turning left at the intersection over the time interval 0 18t hours.

(b) Traffic engineers will consider turn restrictions when ( ) 150L t cars per hour. Find all values of t for which ( ) 150L t and compute the average value of L over this time interval. Indicate units of measure.

(c) Traffic engineers will install a signal if there is any two-hour time interval during which the product of the total number of cars turning left and the total number of oncoming cars traveling straight through the intersection is greater than 200,000. In every two-hour time interval, 500 oncoming cars travel straight through the intersection. Does this intersection require a traffic signal? Explain the reasoning that leads to your conclusion.

(a) ( )18

01658L t dt cars

2 : { 1 : setup 1 : answer (b) ( ) 150L t = when 12.42831,t = 16.12166

Let 12.42831R = and 16.12166S = ( ) 150L t for t in the interval [ ],R S

( )1 199.426S

RL t dtS R = cars per hour

3 : ( )1 : -interval when 150

1 : average value integral 1 : answer with units

t L t

(c) For the product to exceed 200,000, the number of cars turning left in a two-hour interval must be greater than 400.

( )15

13431.931 400L t dt = >

OR The number of cars turning left will be greater than 400

on a two-hour interval if ( ) 200L t on that interval. ( ) 200L t on any two-hour subinterval of

[ ]13.25304, 15.32386 . Yes, a traffic signal is required.

4 : [ ]

( )2

1 : considers 400 cars1 : valid interval , 2

1 : value of

1 : answer and explanation

h

h

h h

L t dt+

+

OR

4 : ( )

1 : considers 200 cars per hour 1 : solves 2001 : discusses 2 hour interval


L t

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4

Question 3

An object moving along a curve in the xy-plane is at position ( ) ( )( ),x t y t at time t, where

( )1sin 1 2 tdx edt = and 34

1dy tdt t

=+

for 0.t At time 2,t = the object is at the point ( )6, 3 . (Note: 1sin arcsinx x = )

(a) Find the acceleration vector and the speed of the object at time 2.t = (b) The curve has a vertical tangent line at one point. At what time t is the object at this point? (c) Let ( )m t denote the slope of the line tangent to the curve at the point ( ) ( )( ), .x t y t Write an expression for

( )m t in terms of t and use it to evaluate ( )lim .t

m t

(d) The graph of the curve has a horizontal asymptote .y c= Write, but do not evaluate, an expression involving an improper integral that represents this value c.

(a) ( )2 0.395 or 0.396, 0.741 or 0.740 a = Speed ( ) ( )2 22 2 1.207x y = + = or 1.208

2 : { 1 : acceleration 1 : speed

(b) ( )1sin 1 2 0te = 1 2 0te =

ln 2 0.693t = = and 0dydt when ln 2t =

2 : ( )1 : 01 : answer

x t =

(c) ( ) ( )3 14 1

1 sin 1 2 ttm tt e

= +

( ) ( )

( )

3 1

1

4 1lim lim1 sin 1 2

10 0sin 1

tt ttm tt e

= +

= =

2 : ( ) 1 : 1 : limit value

m t

(d) Since ( )lim ,t

x t

=

( ) 324lim 3

1ttc y t dtt

= = ++

3 :

1: integrand 1: limits

1: initial value consistent

with lower limit

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5

Question 4

t

(seconds) 0 10 20 30 40 50 60 70 80

( )v t (feet per second)

5 14 22 29 35 40 44 47 49

Rocket A has positive velocity ( )v t after being launched upward from an initial height of 0 feet at time 0t = seconds. The velocity of the rocket is recorded for selected values of t over the interval 0 80t seconds, as shown in the table above. (a) Find the average acceleration of rocket A over the time interval 0 80t seconds. Indicate units of

measure.

(b) Using correct units, explain the meaning of ( )70

10v t dt in terms of the rockets flight. Use a midpoint

Riemann sum with 3 subintervals of equal length to approximate ( )70

10.v t dt

(c) Rocket B is launched upward with an acceleration of ( ) 31

a tt

=+

feet per second per second. At time

0t = seconds, the initial height of the rocket is 0 feet, and the initial velocity is 2 feet per second. Which of the two rockets is traveling faster at time 80t = seconds? Explain your answer.

(a) Average acceleration of rocket A is

( ) ( ) 280 0 49 5 11 ft sec80 0 80 20v v = =

1 : answer

(b) Since the velocity is positive, ( )70

10v t dt represents the

distance, in feet, traveled by rocket A from 10t = seconds to 70t = seconds.

A midpoint Riemann sum is

( ) ( ) ( )[ ][ ]

20 20 40 6020 22 35 44 2020 ft

v v v+ += + + =

3 : ( ) ( ) ( ) 1 : explanation1 : uses 20 , 40 , 60

1 : valuev v v

(c) Let ( )Bv t be the velocity of rocket B at time t.

( ) 3 6 11B

v t dt t Ct

= = + ++

( )2 0 6Bv C= = + ( ) 6 1 4Bv t t= + ( ) ( )80 50 49 80Bv v= > = Rocket B is traveling faster at time 80t = seconds.

4 : ( ) ( )

1 : 6 1 1 : constant of integration 1 : uses initial condition1 : finds 80 , compares to 80 ,

and draws a conclusionB

t

v v

+

Units of 2ft sec in (a) and ft in (b) 1 : units in (a) and (b)

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6

Question 5

Consider the differential equation 2 65 2dy xdx y= for 2.y Let ( )y f x= be the particular solution to this

differential equation with the initial condition ( )1 4.f =

(a) Evaluate dydx and 2

2d ydx

at ( )1, 4 .

(b) Is it possible for the x-axis to be tangent to the graph of f at some point? Explain why or why not. (c) Find the second-degree Taylor polynomial for f about 1.x = (d) Use Eulers method, starting at 1x = with two steps of equal size, to approximate ( )0 .f Show the work

that leads to your answer.

(a) ( )1, 4

6dydx =

( )2

22 10 6 2

d y dyx y dxdx= +

( ) ( )

2

2 21, 4

110 6 6 96

d ydx

= + =

3 :

( )

( )

1, 42

2

2

21, 4

1 :

1 :

1 :

dydx

d ydxd ydx

(b) The x-axis will be tangent to the graph of f if ( ), 0

0.k

dydx =

The x-axis will never be tangent to the graph of f because

( )

2

, 05 3 0

k

dy kdx = + > for all k.

2 : 1 : 0 and 0


dy ydx = =

(c) ( ) ( ) ( )294 6 1 12P x x x= + + +

2 : { 1 : quadratic and centered at 1 1 : coefficients x =

(d) ( )1 4f =

( ) ( )1 14 6 12 2f + = ( ) ( )1 5 50 1 22 4 8f + + =

2 : ( )1 : Euler's method with 2 steps1 : Euler's approximation to 0f

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7

Question 6

The function f is defined by the power series

( ) ( )2 3 12 3

2 3 4 1L Ln nnxx x xf x n

= + + + ++

for all real numbers x for which the series converges. The function g is defined by the power series

( ) ( )( )2 3 11 2! 4! 6! 2 !L L

n nxx x xg x n= + + + +

for all real numbers x for which the series converges. (a) Find the interval of convergence of the power series for f. Justify your answer. (b) The graph of ( ) ( )y f x g x= passes through the point ( )0, 1 . Find ( )0y and ( )0 .y Determine whether y

has a relative minimum, a relative maximum, or neither at 0.x = Give a reason for your answer.

(a) ( ) ( )( )

( )( )( )

1 211 1 112 21

n n

n nn x nn xn n nnx

+ + + ++ = + +

( )( )( )

21lim 2nn x xn n

+ =

+

The series converges when 1 1.x < <

When 1,x = the series is 1 2 32 3 4 + +L

This series does not converge, because the limit of the individual terms is not zero.

When 1,x = the series is 1 2 32 3 4+ + +L

This series does not converge, because the limit of the individual terms is not zero. Thus, the interval of convergence is 1 1.x < <

5 :

1 : sets up ratio 1 : computes limit of ratio1 : identifies radius of convergence

1 : considers both endpoints 1 : analysis/conclusion for both endpoints

(b) ( ) 21 4 92 3 4f x x x = + +L a

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