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.
ExampleA ball has been thrown upwards. It;s height at time t seconds can be given by the following formula.
h = 35t – 5t2Find 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 2Farmer 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 = 1200Simplify…
x + y = 600this means…
y = 600 – xIf we substitute this into the formula to find the area it becomes.
A = x(600 – x)
A = 600x – x2using 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 90,0000m2 You can check this as the in the following; when the point is a maximum so our area is correct.