\[E(v) = \frac {mc^2}{\sqrt {1-\frac {v^2}{c^2}}}\]
where \(E(v)\) is the energy of an object with “Rest mass” \(m\) and velocity \(v\).
Let us analyze this more closely.
First find the linear approximation to the function \(f(u) = \frac {mc^2}{\sqrt {1-u}}\) at \(u=0\).
Using this approximation, and substituting \(u = \frac {v^2}{c^2}\), we can obtain an approximation for \(E(v)\) which is valid for small velocities \(v\).
The approximation you obtain should have two terms. One of which is the famous \(E = mc^2\) (representing the resting energy) and the other should be the classical kinetic energy of the object.
The local linearization of \(\frac {m c^2}{\sqrt {1-u}}\) at \(u=0\) is \(\answer {m c^2 +\frac {m c^2}{2}u}\).
So we get that \(E(v) \approx \answer {mc^2+\frac {1}{2}mv^2}\).