AP
®
CALCULUS AB
2010 SCORING GUIDELINES (Form B)
Question 2
© 2010 The College Board.
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The function
g
is defined for
0
x
>
with
( )
1
2,
g
=
(
)
(
)
1
sin
,
g
x
x
x
′
=
+
and
(
)
(
)
2
1
1
1
cos
.
g
x
x
x
x
⎛
⎞
′′
=
−
+
⎜
⎟
⎝
⎠
(a) Find all values of
x
in the interval 0.12
1
x
≤
≤
at which the graph of
g
has a horizontal tangent line.
(b) On what subintervals of
(
)
0.12, 1
, if any, is the graph of
g
concave down? Justify your answer.
(c) Write an equation for the line tangent to the graph of
g
at
0.3.
x
=
(d) Does the line tangent to the graph of
g
at
0.3
x
=
lie above or below the graph of
g
for 0.3
1 ?
x
<
<
Why?
(a)
The graph of
g
has a horizontal tangent line when
(
)
0.
g
x
′
=
This occurs at
0.163
x
=
and
0.359.
x
=
2 :
(
)
1 : sets
0
1 : answer
g
x
′
=
⎧
⎨
⎩
(b)
(
)
0
g
x
′′
=
at
0.129458
x
=
and
0.222734
x
=
The graph of
g
is concave down on
(
)
0.1295, 0.2227
because
(
)
0
g
x
′′
<
on this interval.
2 :
{
1 : answer
1 : justification
(c)
(
)
0.3
0.472161
g
′
= −
(
)
(
)
0.3
1
0.3
2
1.546007
g
g
x
dx
′
=
+
=
∫
An equation for the line tangent to the graph of
g
is
(
)
1.546
0.472
0.3 .
y
x
=
−
−
4 :
(
)
(
)
1 :
0.3
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 Fall '10
 JUAN
 Calculus, Derivative, The College Board, 0.472161 g, 1 0.3 g

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