4.5: Taylor's Theorem
- Page ID
- 49117
In this section, we prove a result that lets us approximate differentiable functions by polynomials.
Let \(n\) be a positive integer. Suppose \(f:[a, b] \rightarrow \mathbb{R}\) is a function such that \(f^{(n)}\) is continuous on \([a, b]\), and \(f^{(n+1)}(x)\) exists for all \(x \in (a, b)\). Let \(\bar{x} \in [a, b]\). Then for any \(x \in [a, b]\) with \(x \neq \bar{x}\), there exists a number \(c\) in between \(\bar{x}\) and \(x\) such that \[f(x)=P_{n}(x)+\frac{f^{(n+1)}(c)}{(n+1) !}(x-\bar{x})^{n+1} ,\] where \[P_{n}(x)=\sum_{k=0}^{n} \frac{f^{(k)}(\bar{x})}{k !}(x-\bar{x})^{k} .\]
- Proof
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Let \(\bar{x}\) be as in the statement and let us fix \(x \neq \bar{x}\). SInce \(x-\bar{x} \neq 0\), there exists a number \(\lambda \in \mathbb{R}\) such that \[f(x)=P_{n}(x)+\frac{\lambda}{(n+1) !}(x-\bar{x})^{n+1} .\] We will now show that \[\lambda=f^{(n+1)}(c) ,\] for some \(c\) in between \(\bar{x}\) and \(x\).
Consider the function \[g(t)=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(t)}{k !}(x-t)^{k}-\frac{\lambda}{(n+1) !}(x-t)^{n+1} .\] Then \[g(\bar{x})=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(\bar{x})}{k !}(x-\bar{x})^{k}-\frac{\lambda}{(n+1) !}(x-\bar{x})^{n+1}=f(x)-P_{n}(x)-\frac{\lambda}{(n+1) !}(x-\bar{x})^{n+1}=0 .\] and \[g(x)=f(x)-\sum_{k=0}^{n} \frac{f^{(k)}(x)}{k !}(x-x)^{k}-\frac{\lambda}{(n+1) !}(x-x)^{n+1}=f(x)-f(x)=0 .\] By Rolle's theorem, there exists \(c\) in between \(\bar{x}\) and \(x\) such that \(g^{\prime}(c)=0\). Taking the derivative of \(g\) (keeping in mind that \(x\) is fixed and the independent variable is \(t\)) and using the product rule for derivatives, we have \[\begin{aligned}
g^{\prime}(c) &=-f^{\prime}(c)+\sum_{k=1}^{n}\left(-\frac{f^{(k+1)}(c)}{k !}(x-c)^{k}+\frac{f^{(k)}(c)}{(k-1) !}(x-c)^{k-1}\right)+\frac{\lambda}{n !}(x-c)^{n} \\
&=\frac{\lambda}{n !}(x-c)^{n}-\frac{1}{n !} f^{(n+1)}(c)(x-c)^{n} \\
&=0 \text {.}
\end{aligned}\] This implies \(\lambda=f^{(n+1)}(c)\). The proof is now complete. \(\square\)
The polynomial \(P_{n}(x)\) given in the theorem is called the \(n\)-th Taylor polynomial of \(f\) at \(\bar{x}\).
The conclusion of Taylor's theorem still holds true if \(x = \bar{x}\). In this case, \(c = x = \bar{x}\).
We will use Taylor's theorem to estimate the error in approximating the function \(f(x)=\sin x\) with it 3rd Taylor polynomial at \(\bar{x} = 0\) on the interval \([-\pi / 2, \pi / 2]\).
Solution
Since \(f^{\prime}(x)=\cos x\), \(f^{\prime \prime}(x)=-\sin x\) and \(f^{\prime \prime \prime}(x)=-\cos x\), a direct calculation shows that \[P_{3}(x)=x-\frac{x^{3}}{3 !} .\]
Moreover, for any \(c \in \mathbb{R}\) we have \(\left|f^{(4)}(c)\right|=|\sin c| \leq 1\). Therefore, for \(x \in[-\pi / 2, \pi / 2]\) we get (for some \(c\) between \(x\) and \(0\)), \[\left|\sin x-P_{3}(x)\right|=\frac{\left|f^{(4)}(c)\right|}{4 !}|x| \leq \frac{\pi / 2}{4 !} \leq 0.066 .\]
Let \(n\) be an even positive integer. Suppose \(f^{(n)}\) exists and continuous on \((a, b)\). Let \(\bar{x} \in (a, b)\). satisfy \[f^{\prime}(\bar{x})=\ldots=f^{(n-1)}(\bar{x})=0 \text { and } f^{(n)}(\bar{x}) \neq 0 .\] The following hold:
- \(f^{(n)}(\bar{x})>0\) if and only if \(f\) has a local minimum at \(\bar{x}\).
- \(f^{(n)}(\bar{x})<0\) if and only if \(f\) has a local maximum at \(\bar{x}\).
- Proof
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We will prove (a). Suppose \(f^{(n)}(\bar{x})>0\). Since \(f^{(n)}(\bar{x})>0\) and \(f^{(n)}\) is conitnuous at \(\bar{x}\), there exists \(\delta > 0\) such that \[f^{(n)}(t)>0 \text { for all } t \in B(\bar{x} ; \delta) \subset(a, b) .\] Fix any \(x \in B(\bar{x} ; \delta)\). By Taylor's theorem and the given assumption, there exists \(c\) in between \(\bar{x}\) and \(x\) such that \[f(x)=f(\bar{x})+\frac{f^{(n)}(c)}{n !}(x-\bar{x})^{n} .\] Since \(n\) is even and \(c \in B(\bar{x} ; \delta)\) we have \(f(x) \geq f(\bar{x})\). Thus, \(f\) has a local minimum at \(\bar{x}\).
Now, for the converse, suppose that \(f\) has a local minimum at \(\bar{x}\). Then there exists \(\delta > 0\) such that \[f(x) \geq f(\bar{x}) \text { for all } x \in B(\bar{x} ; \delta) \subset(a, b) .\] Fix a sequence \(\left\{x_{k}\right\}\) in \((a, b)\) that converges to \(\bar{x}\) with \(x_{k} \neq \bar{x}\) for every \(k\). By Taylor's theorem, there exists a sequence \(\left\{c_{k}\right\}\), with \(x_{k} \neq \bar{x}\) for every \(k\). By Taylor's theorem, there exists a sequence \(\left\{c_{k}\right\}\), with \(c_{k}\) between \(x_{k}\) and \(\bar{x}\) for each \(k\), such that \[f\left(x_{k}\right)=f(\bar{x})+\frac{f^{(n)}\left(c_{k}\right)}{n !}\left(x_{k}-\bar{x}\right)^{n} .\] Since \(x_{k} \in B(\bar{x} ; \delta)\) for sufficiently large \(k\), we have \[f\left(x_{k}\right) \geq f(\bar{x})\] for such \(k\). It follows that \[f\left(x_{k}\right)-f(\bar{x})=\frac{f^{(n)}\left(c_{k}\right)}{n !}\left(x_{k}-\bar{x}\right)^{n} \geq 0 .\] This implies \(f^{(n)}\left(c_{k}\right) \geq 0\) for such \(k\). Since \(\left\{c_{k}\right\}\) converges to \(\bar{x}\), \(f^{(n)}(\bar{x})=\lim _{k \rightarrow \infty} f^{(n)}\left(c_{k}\right) \geq 0\).
The proof of (b) is similar. \(\square\)
Consider the function \(f(x)=x^{2} \cos x\) defined on \(\mathbb{R}\).
Solution
Then \(f^{\prime}(x)=2 x \cos x-x^{2} \sin x\) and \(f^{\prime \prime}(x)=2 \cos x-4 x \sin x-x^{2} \cos x\). Then \(f(0)=f^{\prime}(0)=0\) and \(f^{\prime \prime}(0)=2>0\). It follows from the previous theorem that \(f\) has a local minimum at \(0\). Notice, by the way, that since \(f(0)=0\) and \(f(\pi)<0\), \(0\) is not a global minimum.
Consider the function \(f(x)=-x^{6}+2 x^{5}+x^{4}-4 x^{3}+x^{2}+2 x-3\) defined on \(\mathbb{R}\).
Solution
A direct calculations shows \(f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=f^{(4)}(1)=0\) and \(f^{(5)}(1)<0\). It follows from the previous theorem that \(f\) has a local maximum at \(1\).
Exercise \(\PageIndex{1}\)
Use Taylor's theorem to prove that \[e^{x}>\sum_{k=0}^{m} \frac{x^{k}}{k !}\] for all \(x >0\) and \(m \in \mathbb{N}\).
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Exercise \(\PageIndex{1}\)
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Exercise \(\PageIndex{2}\)
Find the 5th Taylor polynomial, \(P_{5}(x)\), at \(\bar{x} = 0\) for \(\cos x\). Determine an upper bound for the error \(\left|P_{5}(x)-\cos x\right|\) for \(x \in[-\pi / 2, \pi / 2]\).
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Exercise \(\PageIndex{3}\)
Use Theorem 4.5.3 to determine if the following functions have a local minimum or a local maximum at the indicated points.
- \(f(x)=x^{3} \sin x \text { at } \bar{x}=0\).
- \(f(x)=(1-x) \ln x \text { at } \bar{x}=1\).
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Exercise \(\PageIndex{4}\)
Suppose \(f\) is twice differentiable on \((a, b)\). Show that for every \(x \in (a, b)\), \[\lim _{h \rightarrow 0} \frac{f(x+h)+f(x-h)-2 f(x)}{h^{2}}=f^{\prime \prime}(x) .\]
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Exercise \(\PageIndex{5}\)
- Suppose \(f\) is three times differentiable on \((a, b)\) and \(\bar{x} \in (a, b)\). Prove that \[\lim _{h \rightarrow 0} \frac{f(\bar{x}+h)-f(\bar{x})-f^{\prime}(\bar{x}) \frac{h}{1 !}-f^{\prime \prime}(\bar{x}) \frac{h^{2}}{2 !}}{h^{3}}=\frac{f^{\prime \prime \prime}(\bar{x})}{3 !} . \]
- State and prove a more general result for the case where \(f\) is \(n\) times differentiable on \((a, b)\).
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Exercise \(\PageIndex{6}\)
Suppose \(f\) is \(n\) times differentiable on \((a, b)\) and \(\bar{x} \in (a, b)\). Define \[P_{n}(h)=\sum_{k=0}^{n} f^{(n)}(\bar{x}) \frac{h^{n}}{n !} \text { for } h \in \mathbb{R}\] Prove that \[\lim _{h \rightarrow 0} \frac{f(\bar{x}+h)-P_{n}(h)}{h^{n}}=0 .\] (Thus, we have \[f(\bar{x}+h)=P_{n}(h)+g(h) ,\] where \(g\) is a function that satisfies \(\lim _{h \rightarrow 0} \frac{g(h)}{h^{n}}=0\). This is called the Taylor expansion with Peano's remainder.)
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