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 ?