Example. Find the second-order derivative of the function given parametrically:

$$
\left\{\begin{array}{l}
x=\ln t \\
y=\operatorname{arctg} t
\end{array}\right.
$$
## Solution.

1. We compute

$$
\frac{d x}{d t}=\frac{1}{t}, \quad \frac{d y}{d t}=\frac{1}{1+t^{2}}
$$

and substitute these values into formula (1):

$$
\left\{\begin{array}{l}
x=\ln t \\
y^{\prime}=\frac{t}{1+t^{2}}
\end{array}\right.
$$

Differentiating $y^{\prime}$ with respect to $x$ as a composite function of $x$ and using the formula for the derivative of the inverse function, we get

$$
\left\{\begin{array}{l}
x=\ln t \\
y^{\prime \prime}=\left(\frac{t}{1+t^{2}}\right)^{\prime} \cdot t=\frac{t\left(1-t^{2}\right)}{\left(1+t^{2}\right)^{2}}
\end{array}\right.
$$

Answer. $\left\{\begin{array}{l}x=\ln t, \\ y^{\prime \prime}=\frac{t\left(1-t^{2}\right)}{\left(1+t^{2}\right)^{2}} .\end{array}\right.$

Conditions of the Problems. Find the second-order derivatives of functions given parametrically.

1. $\left\{\begin{array}{l}x=\cos ^{3} t, \\ y=\sin ^{3} t .\end{array}\right.$
2. $\left\{\begin{array}{l}x=t+\sin t, \\ y=3-\cos t .\end{array}\right.$
3. $\left\{\begin{array}{l}x=\ln \operatorname{tg} t, \\ y=1 / \sin ^{2} t .\end{array}\right.$
4. $\left\{\begin{array}{l}x=\ln \left(1+t^{2}\right), \\ y=\operatorname{arctg} t .\end{array}\right.$
5. $\left\{\begin{array}{l}x=\sqrt{1-t^{2}}, \\ y=\arcsin t .\end{array}\right.$
6. $\left\{\begin{array}{l}x=\sqrt{t}, \\ y=\ln t .\end{array}\right.$
7. $\left\{\begin{array}{l}x=\sin ^{2} t, \\ y=\ln \cos t .\end{array}\right.$
8. $\left\{\begin{array}{l}x=1 / t, \\ y=\operatorname{arctg} t .\end{array}\right.$
9. $\left\{\begin{array}{l}x=1 / t^{2}, \\ y=1 /\left(t^{2}+1\right) .\end{array}\right.$
10. $\left\{\begin{array}{l}x=\operatorname{arctg} t, \\ y=t^{2}+1 .\end{array}\right.$

Answers. (The corresponding expressions for $x(t)$ are omitted.)

1. $y^{\prime \prime}=1 /\left(3 \cos ^{4} t \sin t\right)$. 2. $\quad y^{\prime \prime}=1 /(1+\cos t)^{2}$.
3. $y^{\prime \prime}=4 \operatorname{ctg}^{2} t$.
4. $y^{\prime \prime}=-\left(1+t^{2}\right)\left(4 t^{3}\right)$.
5. $y^{\prime \prime}=-\sqrt{1-t^{2}} / t^{3}$.
6. $y^{\prime \prime}=-2 / t$.
7. $y^{\prime \prime}=-1 / \cos ^{4} t$.
8. $y^{\prime \prime}=2 t^{3} /\left(1+t^{2}\right)^{2}$.
9. $y^{\prime \prime}=-2 t^{6} /\left(1+t^{2}\right)^{3}$.
10. $y^{\prime \prime}=6 t^{4}+8 t^{2}+2$.

Chapter 5

## GRAPHS OF FUNCTIONS

When studying the topic of GRAPHS OF FUNCTIONS, you will learn to investigate the behavior of a function: find its domain, asymptotes, intervals of increase and decrease, points of extremum, intervals of concavity up and down, and points of inflection, and to represent the obtained results in the form of a sketch of the graph. Additionally, you will learn to find the smallest and largest values of a function continuous on an interval and to investigate the local behavior of a function using its higher-order derivatives.

Using the package SOLVER.VM, you can compute the derivatives of a function (of any order), solve equations to find points of possible extremum and inflection, find the values of the function at the required points, perform all numerical calculations, and verify the correctness of the results you have obtained.

## 5.1. General Scheme for Constructing the Graph of a Function

Problem Statement. Investigate the function $y=f(x)$ and construct its graph.

Plan for Solution. The results obtained in each step are sequentially recorded on the diagram as elements of the desired graph, and in the end, we obtain a sketch of the graph.

1. Find the domain $D$ of the function $f(x)$ and investigate its behavior at the boundary points $x=x_{1}, x_{2}, \ldots, x_{n}, \pm \infty$ of the domain $D$, including $x= \pm \infty$.

a) Let $x=x_{k} \quad(k=1,2, \ldots, n)$ be a finite boundary point of the domain $D$ (i.e., $f(x)$ is not defined at this point). Compute the one-sided limits

$$
\lim _{x \rightarrow x_{k}-0} f(x) \text { and/or } \lim _{x \rightarrow x_{k}+0} f(x) .
$$

If at least one of these limits is infinite, then $x=x_{k}$ is a vertical asymptote of the graph of $f(x)$.

b) Investigate the behavior of the function as $x \rightarrow+\infty$:

If there exist finite limits

$$
\lim _{x \rightarrow+\infty} \frac{f(x)}{x}=k \quad \text { and } \quad \lim _{x \rightarrow+\infty}[f(x)-k x]=b,
$$

then the line $y=k x+b$ is an oblique asymptote of the graph of the function $f(x)$ as $x \rightarrow+\infty$ (if $k=0$, i.e., $b=\lim _{x \rightarrow+\infty} f(x)$, then $y=b$ is a horizontal asymptote).

Similarly, investigate the behavior of the function as $x \rightarrow-\infty$.

Note that the asymptotes as $x \rightarrow+\infty$ and as $x \rightarrow-\infty$ may be different.

2. Determine the evenness and periodicity of the function.

If $f(-x)=f(x)$, then the function $f(x)$ is called even. The graphs of even functions are symmetric with respect to the $O Y$ axis. Therefore, it is sufficient to construct the graph of an even function for $x>0$ and then draw the entire graph by reflecting the obtained curve with respect to the $O Y$ axis.

If $f(-x)=-f(x)$, then the function $f(x)$ is called odd. The graphs of odd functions are symmetric with respect to the point $(0,0)$. Therefore, it is sufficient to construct the graph of an odd function for $x>0$ and then draw the entire graph by reflecting the obtained curve with respect to the point $(0,0)$.

If $f(x+T)=f(x)$ for some $T>0$, then the function $f(x)$ is called periodic. The graph of a periodic function has the same shape on each of the intervals $\ldots,[-2 T,-T],[-T, 0],[0, T]$, $[T, 2 T], \ldots$ Therefore, it is sufficient to construct the graph on any one such interval and then reproduce the obtained curve on the other intervals.

3. Find the points of intersection of the graph with the coordinate axes. For this, compute $f(0)$ and solve the equation $f(x)=0$.
4. Find the points of maximum and minimum of the function and the intervals of monotonicity. For this:

a) compute the derivative $f^{\prime}(x)$ and find the critical points of the function, i.e., points where $f^{\prime}(x)=0, \pm \infty$ or does not exist. Note that if $f^{\prime}(a)=0$, then the tangent to the graph at this point is horizontal, if $f^{\prime}(a)= \pm \infty$, then the tangent is vertical.

b) by determining the sign of the derivative, find the intervals of increase and decrease of the function: if $f^{\prime}(x)>0$, the function is increasing, if $f^{\prime}(x)<0$, the function is decreasing;

c) if the derivative changes sign at a critical point $a$, then $a$ is a point of extremum: if $f^{\prime}(x)>0$ for $x \in(a-\delta, a)$ and $f^{\prime}(x)<0$ for $x \in(a, a+\delta)$, then $a$ is a point of maximum; if $f^{\prime}(x)<0$ for $x \in(a-\delta, a)$ and $f^{\prime}(x)>0$ for $x \in(a, a+\delta)$, then $a$ is a point of minimum;

if the derivative retains its sign when passing through a critical point, then there is no extremum at this point.

5. Find the points of inflection of the function and the intervals of concavity up and down. For this:

a) compute the second derivative $f^{\prime \prime}(x)$ and find the points belonging to the domain of the function where $f^{\prime