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AP微积分AB 2005 真题解答

AP微积分AB 2005 真题解答
AP微积分AB 2005 真题解答

AP? Calculus AB

2005 Scoring Guidelines

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Question 1

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

sin 4

f x x π=

+ and ()4.x g 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 =?

The tide removes sand from Sandy Point Beach at a rate modeled by the function R, given by

()(425sin

.25t R t π=+

A pumping station adds sand to the beach at a rate modeled by the function S, given by

()15.13t S t t

=

+ Both ()R t and ()S t have units of cubic yards per hour and t is measured in hours for 0 6.t ≤≤ At time 0,t = the beach contains 2500 cubic yards of sand.

(a) How much sand will the tide remove from the beach during this 6-hour period? Indicate units of measure. (b) Write an expression for (),Y t the total number of cubic yards of sand on the beach at time t .

(c) Find the rate at which the total amount of sand on the beach is changing at time 4.t =

(d) For 06,t ≤≤ at what time t is the amount of sand on the beach a minimum? What is the minimum value?

Justify your answers.

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

,T x dx ′∫ and indicate units of measure. Explain the meaning of ()8

T 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 08?x << Explain

your answer.

Question 4

x 0 01x << 1 12x << 2 23x << 3 34x << ()f x –1 Negative 0 Positive 2 Positive 0 Negative ()f x ′ 4 Positive 0 Positive DNE Negative –3 Negative ()f x ′′

–2 Negative 0 Positive DNE Negative 0 Positive

Let f be a function that is continuous on the interval [)0,4. The function f is twice differentiable except at 2.x = The

function f and its derivatives have the properties indicated in the table above, where DNE indicates that the derivatives of f do not exist at 2.x = (a) For 04,x << find all values of x at which f has a relative extremum. Determine whether f has a relative maximum

or a relative minimum at each of these values. Justify your answer.

(b) On the axes provided, sketch the graph of a function that has all the characteristics of f .

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

(c) Let g be the function defined by ()()1

x

g x f t dt =

∫ on the open interval ()0,4. For

04,x << find all values of x at which g has a relative extremum. Determine whether g has a

relative maximum or a relative minimum at each of these values. Justify your answer.

(d) For the function g defined in part (c), find all values of x , for 04,x << at which the graph of g has a point of

inflection. Justify your answer.

(a) f

has a relative maximum at 2x = because f ′ changes from

positive to negative at 2.x = 2 :

{

1 : relative extremum at 2

1 : relative maximum with justification

x =(b)

2 : () 1 : points at 0,1,2,

3 and behavior at 2,21 : appropriate increasing/decreasing and concavity behavior x =???????

(c) ()()0g x f x ==′ at 1,3.x =

g ′ changes from negative to positive at 1x = so g has a relative

minimum at 1.x = g ′ changes from positive to negative at 3x = so g has a relative maximum at 3.x = 3 : ()() 1 : 1 : critical points 1 : answer with justification

g x f x ′=??

??? (d) The graph of g has a point of inflection at 2x = because g f ′′′=

changes sign at 2.x =

2 :

{

1 : 2

1 : answer with justification

x =

Question 5

A car is traveling on a straight road. For 024t ≤≤ seconds, the car’s velocity (),v t in meters per second, is modeled by the piecewise-linear function defined by the graph above.

(a) Find ()240

.v t dt ∫

Using correct units, explain the meaning of ()240

.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 car’s acceleration at time t , in meters per second per second. For 024,t << write a

piecewise-defined function for ().a t (d) Find the average rate of change of v over the interval 820.t ≤≤ Does the Mean Value Theorem guarantee

a value of c , for 820,c << such that ()v c ′ is equal to this average rate of change? Why or why not?

Question 6

Consider the differential equation

2.dy x dx y

=? (a) On the axes provided, sketch a slope field for the given differential equation at the

twelve points indicated.

(Note: Use the axes provided in the pink test booklet.) (b) Let ()y f x = be the particular solution to the differential equation with the initial

condition ()1 1.f =? Write an equation for the line tangent to the graph of f at ()1,1? and use it to approximate ()1.1.f (c) Find the particular solution ()y f x = to the given differential equation with the initial

condition ()1 1.f =?

The line tangent to f at ()1,1? is 1y +=()1.1 is approximately 0.8.? 2x ?

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