Math Application differential calculus Reconstruction of functions

# Reconstruction of functions

The reconstruction of functions deals with the establishment of functional equations. Some reconstruction tasks require differential calculus.

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### Remember

In the reconstruction of functions you look for a special function that satisfies given properties (e.g. type, points, slope, ...).
To do this, you set up equations and solve them using systems of equations.
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### Method

1. Function and derivation
2. Set up equations
3. Solve equations
4. Specify function equation

### Example

We are looking for a second degree function that has an intersection with the y axis at $(0|-3)$ and a maximum point at $H(3|2)$.

1. #### Function and derivation

A second degree function is a quadratic function. This looks like this:
$f(x)=ax^2+bx+c$

The derivative is also needed:
$f'(x)=2ax+b$

The goal is now to find the variables $a$, $b$ and $c$ with the given points.
2. #### Set up equations

The other information is now used to build equations.

The intersection with the y-axis $S_y(0|-3)$ is inserted into the function $f(x)=ax^2+bx+c$:
$f(0)=-3$
$a\cdot0^2+b\cdot0+c=-3$
$c=-3$

The same happens with the maximum point at $H(3|2)$
$f(3)=2$
$a\cdot3^2+b\cdot3+c=2$
$9a+3b+c=2$

The derivative is zero at maximum points.
$f'(3)=0$
$2a\cdot3+b=0$
$6a+b=0$
3. #### Solve equations

The equations can be solved with a system of linear equations.
1. $c=-3$
2. $9a+3b+c=2$
3. $6a+b=0$
First of all, the insertion method lends itself by inserting the I. equation into II.
1. $9a+3b-3=2$
2. $6a+b=0$
There are now several possibilities, but the insertion process makes sense here. First change and then use.
1. $9a+3b-3=2$
2. $6a+b=0\quad|-6a$
$b=-6a$
II in I
$9a-18a-3=2\quad|+3$
$-9a=5\quad|:(-9)$
$a=-\frac59$
4. #### Specify function equation

The following variables are already known:
$a=-\frac59$ and $c=-3$
$b$ can be calculated with one of the equations:
$b=-6a$ $=-6\cdot(-\frac59)$ $=\frac{10}3$

The variables are used and we get the function we are looking for.
$f(x)=ax^2+bx+c$
$f(x)=-\frac59x^2+\frac{10}3x-3$