# Instructions for mathematical analysis

## Review of basic instructions

Given the function $f\left(x\right)=\frac{2{x}^{3}-5{x}^{2}+4x+1}{2{x}^{2}-x-1}$, here's an exercise to find a series of properties that will allow you to represent it graphically by hand. Academically, it will help you review some basic commands related to the world of analysis.

To solve the first question, you must use the command `domain` and the icon you can find in the Calculus section of the Menu to calculate the limits. Notice how the points outside the domain of the function coincide with the roots of the denominator.

The second question is easier to solve with the tools seen in the previous question. This limit equals 0 guarantees that the line $y=x-2$ is an oblique asymptote of the initial function.

To solve the third question, you simply have to derive the initial function using the apostrophe. If you prefer it, you can also find it by using the action that you can find in the top bar or through the `differentiate` command. Note that if you want to calculate these derivatives by hand, you must use the derivation rule of a quotient.

You can solve the equations by equating the expression to 0 or using the `solve` command interchangeably. Naming the solutions allows you to save them in memory and use them in subsequent calculations using more decimal places than CalcMe shows on the screen.

The fifth question is solved by evaluating the previously found second derivative of the function at the critical points. If this is positive on the point, it's a relative minimum. On the contrary, if it's negative, it will be a relative maximum.

On the other hand, if you want to find the intervals of increase and decrease, you should see when the derived function is negative (decreasing function) and when it's positive (increasing function). These intervals are delimited by relative extremes and points outside the domain.

As for the inflection points, make sure the point nullifying the second derivative does not nullify the third one as well. Therefore, $c=0.27788$ will be an inflection point.

Finally, you need to represent the function and its asymptotes.

## Review of integrals

Given a set of functions, you can use CalcMe to find and draw the region's area between them. To do this, you will need to find the integration limits and see which function is more significant in the related range.

Given the functions $y=\mathrm{sin}\left(x\right)$ and $y=2·\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)$, you want to find the area of the region between them from the origin to the first point where they intersect with a positive abscissa. When looking for the cut-off points, you can see the first is $x=\frac{\mathrm{\pi }}{3}$. Therefore you will have to calculate the integral between 0 and $\frac{\mathrm{\pi }}{3}$ of the function that goes above (which is $y=2·\mathrm{sin}\left(x\right)·\mathrm{cos}\left(x\right)$) minus the one that goes below ($y=\mathrm{sin}\left(x\right)$).

Given this time the functions $y=-{x}^{2}+2x$ and $y={x}^{2}-2x+2$, you want to find the region area between the OY axis ($x=0$) and both functions. As before, you have to find its intersection and calculate the integral between 0 and this point. This time, $y={x}^{2}-2x+2$ goes over and $y=-{x}^{2}+2x$, above.

If you want to represent these regions, you have two choices. If the region you want to plot is encapsulated between two functions, you just have to use the `region` command and indicate the functions and the range you want to represent.

On the other hand, if the represented region is not correctly between two functions, you can directly intersect the necessary inequalities and draw it using the action.

## Optimization

Given any function with a wide variety of purposes, you may be interested in finding the maximum and minimum in a given interval. CalcMe can easily find them; you simply have to evaluate the function at its critical points and the extremes of the range of possible values.

Given a food price index model $I\left(t\right)=0.00009045·{t}^{5}$$+0.001438·{t}^{4}$$-0.0656·{t}^{3}$$+0.4598·{t}^{2}$$-0.6270·t$$+99.33$, you want to find when in the first six years the food has been cheaper and more expensive. When you look for the derivative and equate it to zero, you will see there are points outside our range that you will not have to consider. Therefore, it remains only to evaluate the function at the points in question and see when it is more extensive and diminutive.

Repeating the same exercise over a period of 12 years, you have to add $x=11.043$ to the list of optimal candidates. This way, you can see how the minimum is achieved later than in the first case.

## Taylor

Given any function, you can use CalcMe to find its Taylor polynomial of a specific degree centered on a given point. To do this, you can use the formula or the command `taylor`.

Given the function $g\left(x\right)=\mathrm{ln}\left(2-{x}^{2}\right)$, let's first calculate the Taylor polynomial of degree $2$ centered at $x=1$. You can use the formula, and CalcMe will directly find the coefficients.

Given the functions $g\left(x\right)=\mathrm{ln}\left(2-{x}^{2}\right)$, now let's calculate the Taylor polynomial of degree $7$ centered on the same point. Since the formula is very long, you can use the `taylor` command indicating the function, the variable, the point, and the degree.

After looking at the two approaches, you can consider representing them graphically to see how they relate to the initial function.

Note the higher the degree of the Taylor polynomial, the greater the interval at which the function approximates appropriately. In this example, the blue function adequately approximates the polynomial to the interval $\left(0.8,1.2\right)$ while the red one approaches the interval $\left(0.6,1.4\right)$.

## Improper integration

As you have seen before, given a set of functions, you can use the calculator to find the area of the region you are delineating. The usual process is to use the Barrow rule. That is, find a primitive of the function in question and evaluate it at the extremes of integration.

Sometimes, when these integration limits approach a non-real number ($+\infty$ and $-\infty$), we talk about an improper integral and calculate the limit of a definite integral.

Firstly, given the curve $y={\left(\sqrt{5}-\sqrt{x}\right)}^{2}$, calculate the cut-off point and the integral between 0 and the point of our function to find the area of the related region.

Secondly, given the curve $y=\frac{{x}^{2}-1}{{x}^{2}+1}$, calculate the integral with extremes of integration minus and plus infinity of the horizontal asymptote minus the function to find the area of the related function. Since integration bounds approach a specific non-real number ($+\infty$ and $-\infty$), we are talking about an improper integral.

Finally, you can calculate the area of the bounded region between the two curves above.

## Laplace transform

Lastly, CalcMe also lets you solve ODEs with or without initial conditions using the `solve` command. When writing the equation, you should consider entering the function using the icon in the Calculus section of the Menu. Otherwise, CalcMe will not understand you are talking about a differential equation.

Thus, CalcMe returns the variable equation that satisfies the ODE and the initial requirements in question once the initial conditions are given.