is 1. Determine the interval of convergence. Show
1 .
g
(d) The Maclaurin series for
g
, evaluated at
1,
x
=
is a convergent alternating series with individual terms
that decrease in absolute value to 0. Show that your approximation in part (c) must differ from
( )
1
g
by
less than
1
.
5
(a)
(
)
6
9
12
3
1
3
1
2
3
4
n
n
x
x
x
x
x
n
+
−
+
−
+
+
−
⋅
+
"
"
2 :
{
1 : first four terms
1 : general term
(b)
The interval of convergence is centered at
0.
x
=
At
1,
x
= −
the series is
1
1
1
1
1
,
2
3
4
n
−
−
−
−
−
−
−
"
"
which
diverges because the harmonic series diverges.
At
1,
x
=
the series is
(
)
1
1
1
1
1
1
1
,
2
3
4
n
n
+
−
+
−
+
+
−
⋅
+
"
"
the
alternating harmonic series, which converges.
Therefore the interval of convergence is
1
1.
x
−
<
≤
2 : answer with analysis
(c)
The Maclaurin series for
(
)
,
f
x
′
(
)
2
,
f
t
′
and
(
)
g x
are
(
)
(
)
1
3
1
2
5
8
11
1
:
1
3
3
3
3
3
n
n
n
f
x
x
x
x
x
x
∞
+
−
=
′
−
⋅
=
−
+
−
+
∑
"
(
)
(
)
1
2
6
2
4
10
16
22
1
:
1
3
3
3
3
3
n
n
n
f
t
t
t
t
t
t
∞
+
−
=
′
−
⋅
=
−
+
−
+
∑
"
(
)
(
)
6
1
5
11
17
23
1
1
3
3
3
3
3
:
1
6
1
5
11
17
23
n
n
n
x
x
x
x
x
g x
n
∞
−
+
=
−
⋅
=
−
+
−
+
−
∑
"
Thus
( )
3
3
18
1
.
5
11
55
g
≈
−
=
4 :
(
)
(
)
(
)
2
2
1 : two terms for
1 : other terms for
1 : first two terms for
1 : approximation
f
t
f
t
g x
⎧
′
⎪
⎪
′
⎨
⎪
⎪
⎩
(d) The Maclaurin series for
g
evaluated at
1
x
=
is alternating, and the
terms decrease in absolute value to 0.
Thus
( )
17
18
3 1
3
1
1
.
55
17
17
5
g
⋅
−
<
=
<
1 : analysis

AP
®
CALCULUS BC
2012 SCORING GUIDELINES
Question 4
© 2012 The College Board.
Visit the College Board on the Web: .
x
1
1.1
1.2
1.3
1.4
( )
f
x
¢
8
10
12
13
14.5
The function
f
is twice differentiable for
0
x
>
with
( )
1
15
f
=
and
( )
1
20.
f
′′
=
Values of
,
f
′
the derivative of
f
, are given for selected values of
x
in the table above.
(a)
Write an equation for the line tangent to the graph of
f
at
1.
x
=
Use this line to approximate
(
)
1.4 .
f
(b)
Use a midpoint Riemann sum with two subintervals of equal length and values from the table to
approximate
(
)
1.4
1
.
f
x
dx
′
Use the approximation for
(
)
1.4
1
f
x
dx
′
to estimate the value of
(
)
1.4 .
f
Show
the computations that lead to your answer.
(c)
Use Euler’s method, starting at
1
x
=
with two steps of equal size, to approximate
(
)
1.4 .
f
Show the
computations that lead to your answer.
(d)
Write the second-degree Taylor polynomial for
f
about
1.
x
=
Use the Taylor polynomial to approximate
(
)
1.4 .
f
(a)
( )
1
15,
f
=
( )
1
8
f
′
=
An equation for the tangent line is
(
)
15
8
1 .
y
x
=
+
−
(
)
(
)
15
8 1.4
1.4
1
18.2
f
≈
+
−
=
2 :
{
1 : tangent line
1 : approximation
(b)
(
)
(
)(
)
(
)(
)
1.4
1
0.2
10
0.2
13
4.6
dx
f
x
≈
+
=
′
(
)
( )
(
)
1.4
1
1.4
1
f
f
f
dx
x
′
=
+
(
)
15
4.6
9
1
1
6
.4
.
f
≈
+
=
3 :
1 : midpoint Riemann sum
1 : Fundamental Theorem of Calculus
1 : answer
(c)
(
)
( )
(
)( )
1
0.2
1.2
8
16.6
f
f
≈
+
=
(
)
(
)(
)
1.4
16.6
0.2
12
19.0
f
≈
+
=
2 :
{
1 : Euler’s method with two steps
1 : answer
(d)
(
)
(
)
(
)
(
)
(
)
2
2
2
20
15
8
1
1
2!
0
1
1
1
8
5
1
T
x
x
x
x
x
=
+
−
+
−
+
−
−
=
+
(
)
(
)
(
)
2
15
8 1.4
1
10 1.4
1
19
.
.
1 4
8
f
≈
+
−
+
−
=
2 :
{
1 : Taylor polynomial
1 : approximation

AP
®
CALCULUS BC
2012 SCORING GUIDELINES
Question 6
© 2012 The College Board.