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AP Calculus BC
Unit 10: Infinite Sequences and Series
10.13 Finding Maclaurin Series for a Function
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In the Maclaurin series formula, \( f^{(n)}(0) \) represents the \( n \)th derivative of \( f \) evaluated at
zero
The Maclaurin series is a special case of the Taylor series centered at
zero
What is the Maclaurin series for \( \sin(x) \)?
sin
(
x
)
=
\sin(x) =
sin
(
x
)
=
∑
n
=
0
∞
(
−
1
)
n
x
2
n
+
1
(
2
n
+
1
)
!
\sum_{n = 0}^{\infty} ( - 1)^{n} \frac{x^{2n + 1}}{(2n + 1)!}
∑
n
=
0
∞
(
−
1
)
n
(
2
n
+
1
)!
x
2
n
+
1
What is the Maclaurin series centered at?
x
0
=
x_{0} =
x
0
=
0
0
0
What does
f
(
n
)
(
0
)
f^{(n)}(0)
f
(
n
)
(
0
)
represent in the Maclaurin series formula?
n
n
n
th derivative at x = 0</latex>
What is the Maclaurin series formula?
f
(
x
)
=
f(x) =
f
(
x
)
=
∑
n
=
0
∞
f
(
n
)
(
0
)
n
!
x
n
\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}
∑
n
=
0
∞
n
!
f
(
n
)
(
0
)
x
n
What is the Maclaurin series for \( e^x \)?
e
x
=
e^{x} =
e
x
=
∑
n
=
0
∞
x
n
n
!
\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}
∑
n
=
0
∞
n
!
x
n
The Maclaurin series for \( \sin(x) \) involves only odd powers of \( x \).
True
Steps to find the Maclaurin series for a function
1️⃣ Calculate the
n
n
n
th derivative of the function
2️⃣ Evaluate the
n
n
n
th derivative at
x
=
x =
x
=
0
0
0
3️⃣ Plug the derivatives into the formula
4️⃣ Express the series in sigma notation
The formula for the Maclaurin series is
f(x)
The term
n
!
n!
n
!
in the Maclaurin series formula represents the factorial of n
The Maclaurin series is a special case of the Taylor series centered at x_0 = 0</latex>
True
Match the component of the Maclaurin series with its description:
f
(
x
)
f(x)
f
(
x
)
↔️ Function to be represented
f
(
n
)
(
0
)
f^{(n)}(0)
f
(
n
)
(
0
)
↔️
n
n
n
th derivative at
x
=
x =
x
=
0
0
0
n
!
n!
n
!
↔️ Factorial of
n
n
n
x
x
x
↔️ Variable
The
n
n
n
th derivative of a function
f
(
x
)
f(x)
f
(
x
)
is denoted as
f
(
n
)
(
x
)
f^{(n)}(x)
f
(
n
)
(
x
)
The Maclaurin series is centered at
x
=
x =
x
=
0
0
0
True
The Maclaurin series for a function
f
(
x
)
f(x)
f
(
x
)
is given by
∑
n
=
0
∞
f
(
n
)
(
0
)
n
!
x
n
\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}
∑
n
=
0
∞
n
!
f
(
n
)
(
0
)
x
n
, where
f
(
n
)
(
0
)
f^{(n)}(0)
f
(
n
)
(
0
)
is the nth derivative of
f
f
f
evaluated at
x
=
x =
x
=
0
0
0
.
The Maclaurin series is a special case of the Taylor series centered at
x
0
=
x_{0} =
x
0
=
0
0
0
.
True
Match the
n
n
n
th derivative of
sin
(
x
)
\sin(x)
sin
(
x
)
at
x
=
x =
x
=
0
0
0
with its value.
f
′
(
0
)
f'(0)
f
′
(
0
)
↔️ 1
f
′
′
(
0
)
f''(0)
f
′′
(
0
)
↔️ 0
f
′
′
′
(
0
)
f'''(0)
f
′′′
(
0
)
↔️ -1
What is the
n
n
n
th derivative of a function denoted as?
f
(
n
)
(
x
)
f^{(n)}(x)
f
(
n
)
(
x
)
The
n
n
n
th derivative of
e
x
e^{x}
e
x
is
e
x
e^{x}
e
x
for all
n
n
n
.
True
What is the
n
n
n
th derivative of
e
x
e^{x}
e
x
?
e
x
e^{x}
e
x
Steps to calculate derivatives at
x
=
x =
x
=
0
0
0
1️⃣ Find the first few derivatives of
f
(
x
)
f(x)
f
(
x
)
2️⃣ Evaluate each derivative at
x
=
x =
x
=
0
0
0
3️⃣ Continue the process to recognize a pattern
The second derivative of
sin
(
x
)
\sin(x)
sin
(
x
)
evaluated at
x
=
x =
x
=
0
0
0
is 0.
True
What is the general formula for the Maclaurin series?
f
(
x
)
=
f(x) =
f
(
x
)
=
∑
n
=
0
∞
f
(
n
)
(
0
)
n
!
x
n
\sum_{n = 0}^{\infty} \frac{f^{(n)}(0)}{n!} x^{n}
∑
n
=
0
∞
n
!
f
(
n
)
(
0
)
x
n
What is the sigma notation for the series 1 + 4 + 9 + 16 + 25</latex>?
∑
n
=
1
5
n
2
\sum_{n = 1}^{5} n^{2}
∑
n
=
1
5
n
2
The Maclaurin series is a Taylor series centered at
x
0
=
x_{0} =
x
0
=
0
0
0
.
True
What is the Maclaurin series for
e
x
e^{x}
e
x
?
∑
n
=
0
∞
x
n
n
!
\sum_{n = 0}^{\infty} \frac{x^{n}}{n!}
∑
n
=
0
∞
n
!
x
n
The Maclaurin series for
f
(
x
)
=
f(x) =
f
(
x
)
=
e
x
e^{x}
e
x
is given by \sum_{n = 0}^{\infty} \frac{x^{n}}{n!}
The Maclaurin series is a Taylor series centered at
x
0
=
x_{0} =
x
0
=
0
0
0
.
True
The Maclaurin series formula includes the
n
n
n
th derivative evaluated at
x
=
x =
x
=
0
0
0
.
True
Steps to find the Maclaurin series for a function
f
(
x
)
f(x)
f
(
x
)
:
1️⃣ Find the
n
n
n
th derivative of
f
(
x
)
f(x)
f
(
x
)
2️⃣ Evaluate the derivative at
x
=
x =
x
=
0
0
0
3️⃣ Substitute into the Maclaurin series formula
The
n
n
n
th derivative of
f
(
x
)
=
f(x) =
f
(
x
)
=
x
3
x^{3}
x
3
is 0 for
n
>
3
n > 3
n
>
3
.
True
Steps to calculate the derivatives at
x
=
x =
x
=
0
0
0
for a function
f
(
x
)
f(x)
f
(
x
)
:
1️⃣ Find the derivatives of
f
(
x
)
f(x)
f
(
x
)
2️⃣ Evaluate each derivative at
x
=
x =
x
=
0
0
0
For
f
(
x
)
=
f(x) =
f
(
x
)
=
x
3
x^{3}
x
3
, the n</latex>th derivative
f
(
n
)
(
x
)
=
f^{(n)}(x) =
f
(
n
)
(
x
)
=
0
0
0
for
n
>
3
n > 3
n
>
3
.
True
Steps to calculate derivatives at
x
=
x =
x
=
0
0
0
1️⃣ Find the first few derivatives of
f
(
x
)
f(x)
f
(
x
)
2️⃣ Evaluate each derivative at
x
=
x =
x
=
0
0
0
3️⃣ Continue until a pattern emerges
The derivatives at
x
=
x =
x
=
0
0
0
for
sin
(
x
)
\sin(x)
sin
(
x
)
form the pattern 0, 1, 0, - 1, 0, 1, \ldots
What is the first non-zero term in the Maclaurin series for
sin
(
x
)
\sin(x)
sin
(
x
)
?
x
Steps to construct the Maclaurin series for
sin
(
x
)
\sin(x)
sin
(
x
)
1️⃣ Calculate the derivatives of
sin
(
x
)
\sin(x)
sin
(
x
)
at
x
=
x =
x
=
0
0
0
2️⃣ Plug the derivatives into the Maclaurin series formula
3️⃣ Simplify the series to recognize the pattern
The first-order derivative of
sin
(
x
)
\sin(x)
sin
(
x
)
at
x
=
x =
x
=
0
0
0
is 1
The general formula for the Maclaurin series is f(x) = f(0) + \frac{f'(0)}{1!}x + \frac{f''(0)}{
2!
}x^2 + \cdots</latex>
True
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