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)
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