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• Level: GCSE
• Subject: Maths
• Word count: 4626

# I have been given the equation y = axn to investigate the gradient function for varied values of &quot;a&quot; and &quot;n&quot;.

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Introduction

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and Q(x+?x,x+?y) be two near by points on the curve of y = 2x then y + ?y = 2x+?x ??y = 2x+?x-y =2x+?x-2x Therefore ?y=2x(2?x-1) This gives : ?y= 2x(2?x-1) ?x ?x Now dy= limit (?x tends to 0) [dy/dx] dx Therefore we have : dy=lim(?x tends to 0) [2x(2?x-1)] dx ?x Taking small values for ?x and using the calculator it appears that Lim(?x tends to 0) (2?x-1) ? 0.693 ?x Therefore d_ (3x) = 3x X lim(3?x-1) dx (?x tends to 0) ?x Taking small values for ?x and again using the calculator it appears that Lim (3?x-1) ?1.099 (?x tends to 0) ?x Therefore d_ (3x) ? 1.099 X 3 dx From the above calculations we can see that in the case of y=2x the gradient of the curve y=2x the gradient of the curve is less than y=2x whereas in case of y=3x the gradient of the curve is greater than y=3x.In fact there exists an exponential function y=ex such that the gradient function equals y=ex From my investigation above I cans see that 2<a<3.In fact e = 2.71828 correct to 5 decimal places.The value of e is irrational. The function y=ex is called the exponential function and dy = ex dx The graph of y = ex is shown below : In general : d(eax)_= aeax dx I started with drawing the graphs and then calculating the gradient by drawing the tangents using a capillary tube after which I had to work out relations for between the gradient function and the point at which we draw the tangent which was a very tedious and time consuming process. I then proceeded using the small increment method which was an effective method but again was a very tedious process. After constant reference to advance level math books I came into the topic of differentiation which was a little complicated but was more effective as I was able to find out the derivative which simplifies everything. I have thus come to the conclusion that calculus is the best way to investigate the gradient function of any graph. ...read more.

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