APR3

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2004 SCORING GUIDELINES (Form B)
Question 2
APR3 - Solutions
For 0 I t I 3 1, the rate of change of the number of mosquitoes on Tropical Island at time t days is
(3
modeled by R ( t ) = 5ficos - mosquitoes per day. There are 1000 mosquitoes on Tropical Island at
time t = 0.
(a) Show that the number of mosquitoes is increasing at time t = 6.
(b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of
mosquitoes increasing at a decreasing rate? Give a reason for your answer.
(c) According to the model, how many mosquitoes will be on the island at time t = 3 l ? Round your
answer to the nearest whole number.
(d) To the nearest whole number, what is the maximum number of mosquitoes for 0 I t I 3 l ? Show
the analysis that leads to your conclusion.
(a) Since R ( 6 ) = 4.438 > 0, the number of mosquitoes is
increasing at t = 6.
(b)
R'(6) = -1.913
Since R'(6) < 0, the number of mosquitoes is
increasing at a decreasing rate at t = 6.
1 : shows that R ( 6 ) > 0
2:
{
1 : considers R'(6)
1 : answer with reason
2 : { 1 : integral
1 : answer
To the nearest whole number, there are 964
mosquitoes.
(d) R ( t ) = 0 when t = 0 , t = 2 . 5 ~or, t = 7 . 5 ~
R ( t ) > 0 on 0 < t < 2 . 5 ~
R ( t ) < 0 on 2 . 5 <
~t <7.5~
R ( t ) > 0 on 7 . 5 <
~ t < 31
The absolute maximum number of mosquitoes occurs
at t = 2 . 5 ~or at t = 3 1.
There are 964 mosquitoes at t = 3 1, so the maximum
number of mosquitoes is 1039, to the nearest whole
number.
2 : absolute maximum value
1 : integral
1 : answer
2 : analysis
1 : computes interior
critical points
. 1 : completes analysis
'
APR3S
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Final Draft for Scoring
AP Calculus AB-5
APR3 - Solutions
A cubic polynomial function f is defined b y
f(x) = 4x3 + u2+ bz
+k
where a, b, and k are constants. The function f has a local minimum at x = - 1 , and the graph
o f f has a point of inflection at z = - 2 .
(a) Find the values of a and b.
( b ) If
J1( z )& = 32, what is the value of k ?
U
- 1 : f I(x)
(a) ff(z)= 122' +2m + b
ftl(x) = 24s
+ 2a
1 :f"(x)
5 : . 1 : fl(-1) = 0
1 : ff'(-2) = 0
1:a, b
-
(b) Ju1(4r3 + 2 4 x 2 + 3 6 x + t)&
= Z'
+ 8 2 + 18x2 + klZ='
= 27 -tk
e=O
2 : antidifferentiation
< - 1 > each error
4 : .
1 : expression in I;
l : k
APR3S
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AP Calculus AB-5
/
BC-5
APR3 - Solutions
Consider the curve given by q2- x3y = 6 .
dy
3x2y - y2
(a) Show that - =
dx
2 q - x3 '
(b) Find all points on the curve whose scoordinate is 1, and write an equation for the tangent line at
each of these points.
(c) Find the scoordinate of each point on the curve where the tangent line is vertical.
1 : implicit differentiation
dY
1 : verifies expression for dx
(b) When x = 1, y2 - y = 6
4
y2 - y - 6 = 0
(Y - 3)(Y + 2) = 0
y = 3, y = - 2
1
1 : y2 - y = 6
1 : solves for y
2 : tangent lines
Note: 014 if not solving an equation of the
form y2 - y = k
Tangent line equation is y = 3
Tangent line equation is y
+ 2 = 2(x - 1)
(c) Tangent line is vertical when 2 q - x3 = 0
1
~ ( 2 ~ 2 -) =
x O givesx= Oor y = -x2
2
There is no point on the curve with
scoordinate 0.
1
1
1
When y = - x 2 , -x5 - - x
2
4
2
5
=6
3
I
dY equal to 0
1 : sets denominator of dx
1
1 : substitutes y = -x2 or x = f
a
2
into the equation for the curve
I
1 : solves for x-coordinate
1I
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