Math Application integral calculus Reconstruction

Reconstruction of quantities

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



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 $C$.


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.



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


  3. Specify function

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



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