Math Application integral calculus Reconstruction

# Reconstruction of quantities

Sometimes you know the derivative or the rate of change, but not the parent function.

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### Remember

For the reconstruction of an quantity function $f$ you need the rate of change $f'$ and a function value.

One can then integrate $f'$ and use the function value to determine the constant of integration $C$.

### Example

Find the function equation of $f$ with the rate of change $f'(x)=\frac12x$ and the value $f(2)=-1$.

1. #### Integration

$f'$ is the rate of change of $f$. By integrating we get all the antiderivatives of f '. Our wanted function is exactly one of these antiderivatives.

$\int \frac12x\,\mathrm{d}x$ $=\frac14x^2\color{red}{+C}$

2. #### Calculate C

Now only C has to be determined to get our final function. For this we use the second information, namely the function value.

$f_C(x)=\frac14x^2\color{red}{+C}$

$f(2)=-1$

The function value is now used and the equation is converted to C.

$-1=\frac14\cdot2^2+C$
$-1=1+C\quad|-1$
$C=-2$

3. #### Specify function

Insert the calculated $C$ to get our desired function.

$f(x)=\frac14x^2-2$

## Applications

There are many possible examples and applications for reconstruction tasks. Here is a listing of some.

$f=\int f'$ $f'$
Quantity function Rate of change
Distance $s$ Speed $v=s'$
Work $W$ Force $F=W'$
Work $W$ Power $P=W'$
Man days Number of workers