Reconstruction of quantities

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

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.