Example 3. Write the Taylor polynomial of the third degree centered at the point $x=3$ for the function $f(x)=\sqrt{1+x}$.
Solution. We have $T_{3}(x)=c_{0}+c_{1}(x-3)+c_{2}(x-3)^{2}+c_{3}(x-3)^{3}$, where $c_{k}=\left.\frac{f^{(k)}(x)}{k!}\right|_{x=3}$. First, let's find the derivatives:

$$
f^{\prime}(x)=\frac{1}{2 \sqrt{1+x}}, \quad f^{\prime \prime}(x)=\frac{-1}{4 \sqrt{(1+x)^{3}}}, \quad f^{\prime \prime \prime}(x)=\frac{3}{8 \sqrt{(1+x)^{5}}}
$$

Now let's compute $c_{k}$.

$$
\begin{aligned}
& c_{0}=f(3)=2, c_{1}=f^{\prime}(3)=\frac{1}{4}, c_{2}=\frac{f^{\prime \prime}(3)}{2!}=\frac{1}{64}, c_{3}=\frac{f^{\prime \prime \prime}(3)}{3!}=\frac{1}{512} \\
& \text { Answer. } T_{3}(x)=2+\frac{1}{4}(x-3)-\frac{1}{64}(x-3)^{2}+\frac{1}{512}(x-3)^{3}
\end{aligned}
$$

## Exercises

Write the Taylor polynomial of degree $n$ centered at the point $x=a$ for the given function.

$$
\begin{aligned}
& \text { 1. } y=\sqrt{1+x}, n=4, a=0 . \quad \text { 2. } y=\sqrt[3]{1+x}, n=3, a=0 \text {. } \\
& \text { 3. } y=e^{x}, n=5, a=-1 . \quad \text { 4. } y=\ln x, n=5, a=1 \\
& \text { 5. } y=\operatorname{tg} x, n=3, a=0 . \text { 6. } y=\frac{1}{-3 x+2}, n=4, a=2
\end{aligned}
$$

## Answers

1. $1+\frac{x}{2}-\frac{x^{2}}{8}+\frac{x^{3}}{16}-\frac{5 x^{4}}{32} \cdot \mathbf{2} \cdot 1+\frac{x}{3}-\frac{x^{2}}{9}+\frac{5}{81} x^{3} \cdot 3 \cdot \frac{1}{e}+\frac{1}{e}(x+1)+$ $+\frac{(x+1)^{2}}{2 e}+\frac{(x+1)^{3}}{3!e}+\frac{(x+1)^{4}}{4!e}$. 4. $x-1-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}-$ $-\frac{(x-1)^{4}}{4}+\frac{(x-1)^{5}}{5}$. 5. $x-\frac{x^{3}}{3}$. 6. $-\frac{1}{4}+\frac{3}{16}(x-2)-\frac{9}{64}(x-2)^{2}+$ $+\frac{27}{256}(x-2)^{3}-\frac{91}{1024}(x-2)^{4}$.

## § 13. Function Analysis and Graph Construction

The construction of the graph of a given function $y=f(x)$ is preceded by a complete analysis of the function, which includes identifying the characteristic properties and peculiarities of the function. These include: the domain of definition $D=D(f)$, the range of values $E(f)$, continuity, differentiability, evenness, oddness, boundedness, periodicity of the function, its intervals of sign constancy, monotonicity, convexity/concavity, the presence of asymptotes (vertical, horizontal, oblique). In addition, it is necessary to determine characteristic points: points of discontinuity, points of intersection of the graph with the coordinate axes, points of extremum (maximum, minimum), points of inflection, etc.

First, let's recall the definitions of the concepts that were not covered earlier.

A function $y=f(x)$ is called:

- even, if $f(-x)=f(x), x \in D$;
- odd, if $f(-x)=-f(x), x \in D$;
- bounded above (below), if there exists a number $M(m)$ such that $f(x) \leqslant M(f(x) \geqslant m), x \in D$;
- bounded, if it is bounded both above and below;
- periodic, if there exists a number $T>0$ such that $f(x+T)=f(x), x \in D$.

The analysis of a function is carried out according to a certain scheme, the points of which we will establish as we go through the analysis. Note that if we do not have positive information on some criteria, the corresponding point can be omitted.

## Examples of Graph Construction