Learning Outcomes
- Determine an upper bound for the error in an approximation of a function via a Taylor polynomial.
Section 9.5 Taylor’s Theorem (PS5)
Subsection 9.5.1 Activities
Activity 9.5.1.
Recall that we can use a Taylor series for a function to approximate that function by using an \(k\)th degree Taylor polynomial.
(a)
Which of the following is the 3rd degree Taylor polynomial for \(f(x)=\sin x\) centered at 0.
- \(\displaystyle 1-\dfrac{x^2}{2}\)
- \(\displaystyle x-\dfrac{x^3}{3!}\)
- \(\displaystyle x+\dfrac{x^3}{3!}\)
- \(\displaystyle x-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!}\)
(b)
Use the 3rd degree Taylor polynomial for \(f(x)=\sin x\) to approximate \(\sin(1)\text{.}\)
(c)
Use technology to approximate \(\sin(1)\text{.}\)
Definition 9.5.2.
Given a infinitely differentiable function
\begin{equation*}
f(x)=\displaystyle\sum_{n=0}^\infty \dfrac{f^{(n)}(c)}{n!}(x-c)^n\text{,}
\end{equation*}
we define the remainder, denoted \(R_k(x)\text{,}\) to be the difference between the function \(f(x)\) and its \(k\)th degree Taylor polynomial \(T_k(x)\text{.}\) That is,
\begin{equation*}
R_k(x)=f(x)-T_k(x).
\end{equation*}
The error in the approximation \(f(x)\approx T_k(x)\) is given by \(|R_k(x)|\text{.}\)
Activity 9.5.3.
We saw in Fact 9.4.6, the Maclaurin series for \(f(x)=e^x\) is
\begin{equation*}
e^x=\displaystyle\sum_{n=0}^\infty \dfrac{1}{n!}x^n.
\end{equation*}
(a)
Compute \(R_2(4)\) using technology.
(b)
Compute \(R_3(4)\) using technology.
(c)
What do you expect from \(R_4(4)\text{?}\)
- There is not enough information.
- It will be greater than both \(R_2(4)\) and \(R_3(4)\text{.}\)
- It will be between \(R_2(4)\) and \(R_3(4)\text{.}\)
- It will be less than both \(R_2(4)\) and \(R_3(4)\text{.}\)
Fact 9.5.4.
Let \(f(x)\) be a function represented by a power series centered at \(x=c\)
\begin{equation*}
f(x)=\displaystyle\sum_{n=0}^\infty a_n(x-c)^n
\end{equation*}
with an interval of convergence \(I\text{.}\) Then for all \(x\) in \(I\text{,}\)
\begin{equation*}
\lim_{k\rightarrow\infty} R_k(x)=0.
\end{equation*}
Theorem 9.5.5. Taylor’s Theorem.
Let \(f(x)\) be an \((k+1)\) times differentiable function on an interval \(I\) of \(c\text{,}\) and let \(T_k(x)\) be its \(k\)th degree Taylor polynomial centered at \(x=c\text{.}\) Then for any \(x\) in the interval \(I\text{,}\) there exists \(p\) between \(c\) and \(x\) such that
\begin{equation*}
R_k(x)=\dfrac{f^{(k+1)}(p)}{(k+1)!}(x-c)^{k+1}.
\end{equation*}
If there exists \(M_k\) such that \(|f^{(k+1)}(x)|\leq M_k\) for all \(x\) in \(I\text{,}\) then the error in the approximation \(f(x)\approx T_k(x)\) has an upper bound:
\begin{equation*}
|R_k(x)|\leq \dfrac{M_k}{(k+1)!}|x-c|^{k+1}.
\end{equation*}
Remark 9.5.6. Using Taylor’s Theorem.
The trickiest part to using Taylor’s Theorem is calculating \(M_k\) to get a bound for the error \(|R_k(x)|\) for the approximation \(f(x)\approx T_k(x)\text{.}\)
Activity 9.5.7.
Consider the function \(f(x)=1/x\) defined on the interval \(I=[1,2]\text{.}\)
(a)
Calculate the derivatives \(f'(x)\text{,}\) \(f''(x)\text{,}\) \(f'''(x)\text{,}\) and \(f^{(4)}(x)\text{.}\)
Answer.
\(f'(x)=-1/x^2\text{,}\) \(f''(x)=2/x^3\text{,}\) \(f'''(x)=-6/x^4\text{,}\) \(f^{(4)}(x)=24/x^5\)
(b)
Which of the following can we say above the values of \(|f^{(k)}(x)|\) on \(I\) for \(k=1,2,3,4\text{?}\)
- \(|f'(x)|\) and \(|f'''(x)|\) are increasing, while \(|f''(x)|\) and \(|f^{(4)}(x)|\) are decreasing.
- All are decreasing.
- All are increasing.
- \(|f'(x)|\) and \(|f'''(x)|\) are decreasing, while \(|f''(x)|\) and \(|f^{(4)}(x)|\) are decreasing.
Answer.
B.
(c)
Calculate \(M_k\) for each \(k=1,2,3,4\) using your results from part (b).
Answer.
\(M_1=1, M_2=2, M_3=6, M_4=24\)
(d)
Use Taylor’s Theorem to calculate \(|R_k(1.5)|\) for each \(k=1,2,3,4\) to 3 decimal places. Use \(a=1\) as the center of the approximation.
Answer.
\(0.125, 0.042, 0.016, 0.006\)
(e)
Are the errors decreasing? Explain why or why not.
Activity 9.5.8.
Let \(f(x)=e^x\text{.}\) Your goal is to approximate \(f(1)=e\text{.}\)
(a)
Explain and demonstrate how to determine the upper bound \(M_k\) from Taylor’s Theorem for \(f(x)=e^x\text{.}\)
(b)
Use your value for \(M_k\) in part (a) to find an upper bound for the error \(|R_4(1)|\text{.}\)
(c)
Use your value for \(M_k\) in part (a) to find an upper bound for the error \(|R_8(1)|\text{.}\)
Subsection 9.5.2 Sample Problem
Example 9.5.9.
Here you are tasked with approximating the value of \(\cos(1)\text{.}\)
(a)
Calculate the 4th degree Taylor polynomial for \(f(x)=\cos x\) centered at \(\pi\text{,}\) then use it to approximate the value of \(\cos(1)\) to three decimal places.
(b)
Apply Taylor’s Theorem to find an upper bound for the error in this approximation.
(c)
Use technology to calculate \(|R_4(1)|\text{.}\) Is the error within the upper bound found in part (b)?
(d)
Explain whether the approximation error \(|R_{k}(1)|\) increases or decreases as \(k\rightarrow\infty\text{.}\)
