# Differentiation Applications

Applications of differentiation take advantage of the stationary points calculations. And below I shall be showing such differentiation applications. There are three types of stationary points as you must know, these are the maximum, minimum and points of inflexion points. This is a good lesson about stationary point and it is very important that you’re very familiar with stationary points for this maths revision.

## Example

A ball has been thrown upwards. It;s height at time **t** seconds can be given by the following formula.

^{2}

Find the maximum height it reaches.

If you’re to sketch the problem on a graph you would get something like this.

That is the ball goes up in the air until it reaches a maximum point and then falls to the ground.

From learning stationary points you know that stationary points can be found where the gradient **f ‘ (x)** is equal to zero as expressed here:

This means we have to differentiate the given equation above.

where **f ‘ (x)** is equal to zero.

…now we know at **t = 3.5** we have a stationary point when **t = 3.5** the height is:

This gives us the maximum height it reaches at **61.25** You can actually check whether it’s the maximum as shown below.

we know that when:

the point is at a maximum height.

## Example 2

Farmer John wants to make a pen for his sheep. What is the largest are of the largest rectangular pen he can make?

It might be a good idea yo illustrate the problem in otherwise draw the rectangular pen he’s after. The rectangle is shown below.

I have named the sides x and y. We know that the formula to find the area is **A =x x y** We have the perimeter as the given value; 1200m . We need to get reed of one of the letters in our area formula. Make all the letters in the equation the same. The perimeter formula is **P = 2x + 2y** This can be expressed as below since we have the perimeter value;

Simplify…

this means…

If we substitute this into the formula to find the area it becomes.

^{2}

using the stationary points calculations the largest rectangular pen can be found where…

Now use the x value in the area formula to find the required area…

The area of the largest rectangular pen that farmer John can make is 90,0000m^{2}

You can check this as the in the following;

when

the point is a maximum so our area is correct.