
# 6.6: Taylor's Theorem


## 6.6.1 Derivatives of Higher Order

##### Definition

Suppose $$f$$ is differentiable on an open interval $$I$$ and $$f^{\prime}$$ is differentiable at $$a \in I .$$ We call the derivative of $$f^{\prime}$$ at $$a$$ the second derivative of $$f$$ at $$a,$$ which we denote $$f^{\prime \prime}(a)$$.

By continued differentiation, we may define the higher order derivatives $$f^{\prime \prime \prime},$$ $$f^{\prime \prime \prime \prime},$$ and so on. In general, for any integer $$n, n \geq 0,$$ we let $$f^{(n)}$$ denote the $$n$$th derivative of $$f,$$ where $$f^{(0)}$$ denotes $$f$$.

##### Exercise $$\PageIndex{1}$$

Suppose $$D \subset \mathbb{R}, a$$ is an interior point of $$D, f: D \rightarrow \mathbb{R},$$ and $$f^{\prime \prime}(a)$$ exists. Show that

$\lim _{h \rightarrow 0} \frac{f(a+h)+f(a-h)-2 f(a)}{h^{2}}=f^{\prime \prime}(a).$

Find an example to illustrate that this limit may exist even if $$f^{\prime \prime}(a)$$ does not exist.

For any open interval $$(a, b),$$ where $$a$$ and $$b$$ are extended real numbers, we let $$C^{(n)}(a, b),$$ where $$n \in \mathbb{Z}^{+},$$ denote the set of all functions $$f$$ with the property that each of $$f, f^{(1)}, f^{(2)}, \ldots, f^{(n)}$$ is defined and continuous on $$(a, b) .$$

### 6.6.2 Taylor's Theorem

##### Theorem $$\PageIndex{1}$$

(Taylor's Theorem).

Suppose $$f \in C^{(n)}(a, b)$$ and $$f^{(n)}$$ is differentiable on $$(a, b) .$$ Let $$\alpha, \beta \in(a, b)$$ with $$\alpha \neq \beta,$$ and let

\begin{aligned} P(x)=f(&\alpha)+f^{\prime}(\alpha)(x-\alpha)+\frac{f^{\prime \prime}(\alpha)}{2}(x-\alpha)^{2}+\cdots \\ &+\frac{f^{(n)}(\alpha)}{n !}(x-\alpha)^{n} \\=& \sum_{k=0}^{n} \frac{f^{(k)}(\alpha)}{k !}(x-\alpha)^{k}. \end{aligned}

Then there exists a point $$\gamma$$ between $$\alpha$$ and $$\beta$$ such that

$f(\beta)=P(\beta)+\frac{f^{(n+1)}(\gamma)}{(n+1) !}(\beta-\alpha)^{n+1}.$

Proof

First note that $$P^{(k)}(\alpha)=f^{(k)}(\alpha)$$ for $$k=0,1, \ldots, n .$$ Let

$M=\frac{f(\beta)-P(\beta)}{(\beta-\alpha)^{n+1}}.$

Then

$f(\beta)=P(\beta)+M(\beta-\alpha)^{n+1}.$

We need to show that

$M=\frac{f^{(n+1)}(\gamma)}{(n+1) !}$

for some $$\gamma$$ between $$\alpha$$ and $$\beta .$$ Let

$g(x)=f(x)-P(x)-M(x-\alpha)^{n+1}.$

Then, for $$k=0,1, \ldots, n$$,

$g^{(k)}(\alpha)=f^{(k)}(\alpha)-P^{(k)}(\alpha)=0.$

Now $$g(\beta)=0,$$ so, by Rolle's theorem, there exists $$\gamma_{1}$$ between $$\alpha$$ and $$\beta$$ such that $$g^{\prime}\left(\gamma_{1}\right)=0 .$$ Using Rolle's theorem again, we see that there exists $$\gamma_{2}$$ between $$\alpha$$ and $$\gamma_{1}$$ such that $$g^{\prime \prime}\left(\gamma_{2}\right)=0 .$$ Continuing for $$n+1$$ steps, we find $$\gamma_{n+1}$$ between $$\left.\alpha \text { and } \gamma_{n} \text { (and hence between } \alpha \text { and } \beta\right)$$ such that $$g^{(n+1)}\left(\gamma_{n+1}\right)=0 .$$ Hence

$0=g^{(n+1)}\left(\gamma_{n+1}\right)=f^{(n+1)}\left(\gamma_{n+1}\right)-(n+1) ! M.$

Letting $$\gamma=\gamma_{n+1},$$ we have

$M=\frac{f^{(n+1)}(\gamma)}{(n+1) !},$

as required. $$\quad$$ Q.E.D.

We call the polynomial $$P$$ in the statement of Taylor's theorem the Taylor polynomial of order $$n$$ for $$f$$ at $$\alpha .$$

##### Example $$\PageIndex{1}$$

Let $$f(x)=\sqrt{x} .$$ Then the 4th order Taylor polynomial for $$f$$ at 1 is

$P(x)=1+\frac{1}{2}(x-1)-\frac{1}{8}(x-1)^{2}+\frac{1}{16}(x-1)^{3}-\frac{5}{128}(x-1)^{4}.$

By Taylor's theorem, for any $$x>0$$ there exists $$\gamma$$ between 1 and $$x$$ such that

$\sqrt{x}=P(x)+\frac{105}{(32)(5 !) \gamma^{\frac{9}{2}}}(x-1)^{5}=P(x)+\frac{7}{256 \gamma^{\frac{9}{2}}}(x-1)^{5}.$

For example,

$\sqrt{1.2}=P(1.2)+\frac{7}{256 \gamma^{\frac{9}{2}}}(1.2-1)^{5}=P(1.2)+\frac{7}{256 \gamma^{\frac{9}{2}}}(0.2)^{5}=P(1.2)+\frac{7}{800000 \gamma^{\frac{9}{2}}},$

for some $$\gamma$$ with $$1<\gamma<1.2 .$$ Hence $$P(1.2)$$ underestimates $$\sqrt{1.2}$$ by a value which is no larger than $$\frac{7}{80000} .$$ Note that

$P(1.2)=\frac{17527}{16000}=1.0954375$

and

$\frac{7}{800000}=0.00000875.$

So $$\sqrt{1.2}$$ lies between 1.0954375 and 1.09544625.

##### Exercise $$\PageIndex{2}$$

Use the 5th order Taylor polynomial for $$f(x)=\sqrt{x}$$ at 1 to estimate $$\sqrt{1.2}$$. Is this an underestimate or an overestimate? Find an upper bound for the largest amount by which the estimate and $$\sqrt{1.2}$$ differ.

##### Exercise $$\PageIndex{3}$$

Find the 3rd order Taylor polynomial for $$f(x)=\sqrt[3]{1+x}$$ at 0 and use it to estimate $$\sqrt[3]{1.1}$$. Is this an underestimate or an overestimate? Find an upper bound for the largest amount by which the estimate and $$\sqrt[3]{1.1}$$ differ.

##### Exercise $$\PageIndex{4}$$

Suppose $$f \in C^{(2)}(a, b) .$$ Use Taylor's theorem to show that

$\lim _{h \rightarrow 0} \frac{f(c+h)+f(c-h)-2 f(c)}{h^{2}}=f^{\prime \prime}(c)$

for any $$c \in(a, b)$$.

##### Exercise $$\PageIndex{5}$$

Suppose $$f \in C^{(1)}(a, b), c \in(a, b), f^{\prime}(c)=0,$$ and $$f^{\prime \prime}$$ exists on $$(a, b)$$ and is continuous at $$c .$$ Show that $$f$$ has a local maximum at $$c$$ if $$f^{\prime \prime}(c)<0$$ and a local minimum at $$c$$ if $$f^{\prime \prime}(c)>0 .$$