# 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.h = 35t – 5t

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 ^{2}**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;2x + 2y = 1200

Simplify…x + y = 600

this means…y = 600 – x

If we substitute this into the formula to find the area it becomes.A = x(600 – x)

A = 600x – x

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}^{2}You can check this as the in the following; when the point is a maximum so our area is correct. LINK TO PAGE

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