A waveform is a function that repeats itself or cycles after a
specified period of time. This time, denoted *T*, is
called the period of the waveform.

The diagram to the right shows an electronic circuit that takes
as input a voltage waveform and gives as output an approximation
of the derivative of that waveform. The approximation can be made
quite good by using small enough values of *R* and
*C* or restricting ourselves to slowly varying input
voltages.

To calculate the derivative of a waveform it is sufficient to
calculate the derivative of just the first cycle. This is because
**the derivative of a waveform is another
waveform**. The derivative of all the other cycles will
*look the same* as that of the first cycle. If *f*
(*t*) is the formula for the first cycle then *f
′* (*t*), its derivative, is the formula for
the derivative of the first cycle. If required, the formula for
the derivative of any other cycle can be found by simply applying
a shift to the formula for the derivative of the first
cycle.

Calculate and sketch the derivative of the waveform shown to the right, whose period is 2 seconds and whose first cycle is defined as:

We begin by calculating the derivative,
*F′*(*t*), of the above formula:

This is the formula for the first cycle of the derivative. Any other cycle looks the same as the first one. A sketch of this derivative is shown below.

A power supply can be represented by an ideal voltage source
of *E* volts in series with an internal resistance
* Rint*. If a load is connected to the power supply, show that
the maximum power that can be supplied to the load is achieved
when the resistance *R* of the load is chosen to equal the
internal resistance * Rint* of the power supply.

We will express the power *P* dissipated in the
resistor *R* as a function of *R* and then find
where this function has its maximum. The power *P*
dissipated in a resistor *R* is given by the formula
*P* = *I* 2 *R*, where *I*
is the current flowing through the resistor. Expressing the
current *I* as a function of *R* (using
Ohm’s law) gives:

Now we can express the power *P* as a function of
*R*:

Notice that this function is zero when *R* is zero, is
positive for all positive values of *R*, and decreases
like 1/*R* when *R* is very large. Clearly
*P* must have a maximum. Now calculate the derivative
*dP*/*dR*:

This derivative is equal to zero when *R* =
*Rint*. From the above discussion it is clear
that the derivative is zero here because this is the value of the
resistance *R* at which the power *P* is a
**maximum** (as opposed to say a minimum).