The Gradient Function
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The Gradient Function Aim: To find the gradient function of curves of the form y=axn. To begin with, I should investigate how the gradient changes, in relation to the value of x. Following this, I plan to expand my investigation to see how the gradient changes, and as a result how a changes in relation to this. Method: At the very start of the investigation, I shall investigate the gradient at the values of y=xn. To start with, I shall put the results in a table, but later on, as I attempt to find the gradient through advanced methods, a table may be unnecessary. As I plot the values of y=x2, this should allow me to plot a line of best fit and analyze, and otherwise evaluate, the relationship between the gradient and x in this equation. I have begun with n=2. After analyzing this, I shall carry on using a constant value of "a" until further on in the investigation, and keep on increasing n by 1 each time. I shall plot on the graphs the relative x values and determine a gradient between n and the gradients. Perhaps further on in the investigation, I shall modify the value of a, and perhaps make n a fractional or negative power. Method to find the gradient: These methods would perhaps be better if I demonstrated them using an example, so I will illustrate this using y=x2. This is the graph of y=x2. I will find out the gradient of this curve, by using the three methods - drawing a tangent to the curve, using an increment method, and proving it via Binomial Expansion. Tangent on curve method - The tangent method is shown on the graph, when x=2, touching the very edge of the curve. Once the line has been drawn, I shall draw a triangle structure, and look at the change in y and the change in x.
However, thus far I have not perhaps produced enough evidence to support this theory, and therefore I shall attempt to prove one last n value in this section, where n=-2. Since I am nearly convinced that this formula is proven for any value, where "a" is a constant 1, I will just binomially prove y = x -2. In accordance to the formula nx n-1, I predict that the gradient function for this = -2x-3. y=x-2. 1 . x² - (x+h)² . 1* = x² - (x²+2hx+h²)** (x + h) ² x² (x+h)² x² x²(x²+2hx+h²) x²(x²+2hx+h²) x + h - x h = x² - (x²+2hx+h²) = -2hx - h² = -2x - h = -2 - h/x = -2 - 0/x hx²(x²+2hx+h²) hx²(x²+2hx+h²) x²(x²+2hx+h²) x(x²+2hx+h²) x(x²+((2)(0x))+0²) = -2 = -2x-3 - here we assume x, is not 0. x-3 * IN all the past equations I have rationalized the numerator to make them easier to work with and thus solve. Otherwise, they are thus irrational and impossible to solve, with all the knowledge of this topic I possess. ** h tends to 0. I now move onto the second and final section of this piece of coursework - where we now change a new variable in this investigation - the value of a. I will try to combine this with different value of n, to see how it affects the end result. axn In this section, I will change the values of a by an increase of 1 in each example, and furthermore do the same for the value of n. While it is bad practice to change 2 variables at once as a rule, I have found the effect the changing n in the previous investigation, so is therefore valid. Firstly, I shall try to investigate 2x², and if successful, move onto greater values of a. y=2x² x y second value x second value y gradient 4 32 4.1 33.62 16.4 4 32 4.01 32.1602 16.04 4 32 4.001 32.016002 16.004 3
144.7209 4 192 4.001 192.144036 144.072 3 81 3.1 89.373 86.49 3 81 3.01 81.812703 81.5409 3 81 3.001 81.081027 81.05401 2 24 2.1 27.783 39.69 2 24 2.01 24.361803 36.3609 2 24 2.001 24.036018 36.03601 1 3 1.1 3.993 10.89 1 3 1.01 3.090903 9.1809 1 3 1.001 3.009009003 9.018009 x y second value x second value y gradient 4 128 4.1 137.842 100.86 4 128 4.01 128.962402 96.4806 4 128 4.001 128.096024 96.04801 3 54 3.1 59.582 57.66 3 54 3.01 54.541802 54.3606 3 54 3.001 54.054018 54.03601 2 16 2.1 18.522 26.46 2 16 2.01 16.241202 24.2406 2 16 2.001 16.024012 24.02401 1 2 1.1 2.662 7.26 1 2 1.01 2.060602 6.1206 1 2 1.001 2.006006002 6.012006 Gradient + Gradient = Gradient 3x3 + Gradient 2x3 Gradient (3x3) Gradient (2x3) a + b 151.29 100.86 252.15 144.7209 96.4806 241.2015 144.072 96.04801 240.12001 86.49 57.66 144.15 81.5409 54.3606 135.9015 81.05401 54.03601 135.09002 39.69 26.46 66.15 36.3609 24.2406 60.6015 36.03601 24.02401 60.06002 10.89 7.26 18.15 9.1809 6.1206 15.3015 9.018009 6.012006 15.030015 From these results, I can figure out that the overall gradient function of each value added together is : 9x2 + 6x2 = x(9x + 6x) This proves quite a clear pattern. Now it is clear that both functions correspond to the gradient function nax^n-1. However, the formula naxn-1 + naxn-1 is not valid : this is because we are assuming that the a and x values are not equal to each other. Therefore, I shall use a different letter to use in these equations. I shall substitute "n" with "m" and "a" with "l". Therefore the formula for adding powers together is : nxn + lxm = naxn-1 + lmxm-1. This is the overall formula for working out the basic functions of any curve of axn + lxm. If the powers are being subtracted from each other, the + is simply replaced by a -. Now I have proved this, the coursework is complete. ?? ?? ?? ?? Jamie Warner 11M
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Here's what a teacher thought of this essay
Generally an excellent piece of work with some sophisticated results. A good conversational style describes clearly why he is following particular lines of investigation.
Marked by teacher Mick Macve 18/03/2012
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