8.0802-8/2.01-2 = 8.02
From this answer I could speculate that the gradient of the curve at this point could be 8. Just to ensure this, I could ‘zoom in’ to the graph further and apply the same method above to the point 2.001:
8.008002-8/2.001-2 = 8.002
Bearing in mind that as you ‘zoom in’ to the graph further you get a more accurate estimate as to what the gradient is, you can see that from the above example, as you zoom in further, the .2 gets smaller and smaller. If I zoomed into the graph further, I am sure that the .2 would get smaller and smaller. This shows that the gradient is actually 8.
Therefore the small increment method is an accurate estimate
I will now use the small increment method to work out the gradients for other curves in the hope of spotting a pattern.
Y=x2
X=1, y=1
Gradient = 1.21-1/1.1-1
= 2.1
Gradient = 1.0201-1/ 1.01-1
= 2.01
Gradient = 1.002001-1/1.001-1
= 2.001
X=2, Y = 4
Gradient = 4.41-4/2.1-2
Gradient = 4.1
Gradient = 4.0401-4/2.01-2
Gradient = 4.01
Gradient = 4.004001-4/2.001-2
Gradient = 4.001
X=3, Y = 9
Gradient =9.61-9/3.1-3
=6.1
Gradient = 9.0601-9/3.01-3
= 6.01
Gradient = 9.006001-9/3.001-3
= 6.001
X = 4, Y = 16
Gradient = 16.81-16/4.1-4
= 8.1
Gradient = 16.0801-16/4.01-4
= 8.01
Gradient = 16.008001-16/4.001-4
= 8.001
Y = 2x2
X=1, Y=2
Gradient = 2.42-2/1.1-1
= 4.2
Gradient = 2.0402-2/1.01-1
= 4.02
Gradient = 2.004002-2/1.001-1
= 4.002
X=2, Y=8
Gradient = 8.82-2/2.1-2
= 8.2
Gradient = 8.0802-8/2.01-2
= 8.02
Gradient = 8.008002-8/2.001-2
= 8.002
X=3, Y=18
Gradient = 19.22-18/3.1-3
= 12.2
Gradient = 18.1202-18/3.01-3
= 12.02
Gradient = 18.012002-18/3.001-3
= 12.002
X=4, Y=32
Gradient = 33.62-32/4.1-4
= 16.2
Gradient =32.1602-32/4.01-4
= 16.02
Gradient = 32.016002-32/4.001-4
= 16.002
Y = 3x2
X=1, Y=3
Gradient = 3.63-3/1.1-1
= 6.3
Gradient = 3.0603-3/1.01-1
= 6.03
Gradient = 3.006003-3/1.001-1
= 6.003
X=2, =12
Gradient = 13.23-12/2.1-2
= 12.3
Gradient = 12.1203-12/2.01-2
= 12.03
Gradient = 12.012003-12/2.001-2
= 12.003
X=3, Y=27
Gradient = 28.83–27/3.1-3
= 18.3
Gradient =27.1803-27/3.01-3
= 18.03
Gradient = 27.018003-27/3.001-3
=18.003
X=4, Y=48
Gradient = 50.43-48/4.1-4
= 24.3
Gradient = 48.2403-48/4.01-4
=24.03
Gradient = 48.024003-48/4.001-4
=24.003
Y = x3
X = 1, Y=1
Gradient = 1.331-1/1.1-1
= 3.31
Gradient = 1.030301-1/1.01-1
= 3.0301
Gradient = 1.003003001-1/1.001-1
= 3.003001
X = 2, Y=8
Gradient = 9.261-8/2.1-2
= 12.61
Gradient = 8.120601-8/2.01-2
=12.0601
Gradient = 8.012006001-8/2.001-2
= 12.006001
X=3, Y=27
Gradient = 29.791-27/3.1-3
=27.91
Gradient = 27.270901-27/3.01-3
= 27.0901
Gradient = 27.027009-27/3.001-3
= 27.009001
X=4, Y=64
Gradient = 68.921-64/4.1-4
= 49.21
Gradient = 64.481201-64/4.01-4
= 48.1201
Gradient =64.048012-64/4.001-4
=48.012001
Y = 2x3
X =1, Y=2
Gradient = 2.662-2/1.1-1
= 6.62
Gradient = 2060602-2/1.1-1
= 6.0602
Gradient = 2.006006002-2/1.001-1
= 6.006002
X=2, Y=16
Gradient = 18.522-16/2.1-2
= 25.22
Gradient = 16.241202-16/2.01-2
=24.1202
Gradient = 16.024012-16/2.001-2
= 24.012002
X=3, Y=54
Gradient =59.582-54/3.1-3
= 55.82
Gradient = 54.541802-54/3.01-3
= 54.1802
Gradient = 54.054018-54/3.001-3
=54.018002
Y= 3x3
X = 1, Y=3
Gradient = 3.993-3/1.1-1
= 9.93
Gradient = 3.090903-3/1.01-1
= 9.0903
Gradient = 3.009009003-3/1.001-1
=9.009003
X=2, Y=24
Gradient =27.783-24/2.1-2
=37.83
Gradient = 24.361803-24/2.01-2
=36.1803
Gradient = 24.036018-24/2.001-2
=36.0180
X=3, Y=81
Gradient = 89.373-81/3.1-3
= 83.73
Gradient = 81.812703-81/3.01-3
= 81.2703
Gradient = 81.081027-81/3.001-3
= 81.027
X=4, Y=192
Gradient =206.763-192/4.1-4
= 147.63
Gradient = 193.443603-192/4.01-4
= 144.3603
Gradient = 192.144036-192/4.001-4
=144.036003
I can now calculate the gradient function of the curves to see if a pattern occurs.
There are a few patterns in the above table:
1) If I take the lines to the power of x2, I can see that the co-efficient of x in the gradient function is always double the co-efficient of x in the equation of the line, i.e. the line y=2x2, the co efficient of the gradient function is 4. The co-efficient of x in the equation of the line is 2; the co-efficient of x in the gradient function is 4, double. This pattern occurs with all the lines to the power of two.
2) If I compare the gradient functions with the equations of the lines, the power of x is always one less in the gradient function than in the equation of the line.
From these findings I can speculate that the gradient function’s power of x is always one less than that of the equation of the line. Therefore if I am trying to find a general formula to calculate the equation of the line for the set of graphs y=axn, I could say that the equation may have something to do with xn-1 .
I now believe that if I calculate the gradient function for the line y=x4 the answer would be nx3 where n is an unknown number.
Y=x4
X=1, Y=1
Gradient = 1.4641-1/1.1-1
=4.641
Gradient = 1.04060401-1/1.01-1
=4.060401
Gradient = 1.004006004-1/1.001-1
=4.006004
X= 2, Y=16
Gradient = 19.4481-16/2.1-2
= 34.481
Gradient = 16.32240801-16/2.01-2
= 32.240801
Gradient = 16.03202401-16/2.001-2
=32.024008
X = 3, Y=81
Gradient = 92.3521-81/3.1-3
= 113.521
Gradient = 82.08541201-81/3.01-3
= 108.541201
Gradient = 81.10805401-81/3.001-3
= 108.054012
X= 4, Y=256
Gradient = 282.5761-256/4.1-4
=265.761
Gradient = 258.569616-256/4.01-4
= 256.961601
Gradient = 256.256096-256/4.001-4
= 256.096016
I have worked out the gradient function for the graph y=x4 to be 4x3. This shows that I correctly predicted the power of x to be 3.
I have decided to construct another table using lines which have the co-efficient of x to be 1.
I can see from this table that:
- The power of x in the gradient function is always 1 less than the coefficient of x.
- The power of x in the equation of the line is always equal to the co-efficient of x in the gradient function.
- The power of x in the gradient function is always one less than the power of x in the equation of the line.
This tells me that if n is the power of x and a is the digit before the x in the equation of the line, then to work out the gradient of the curve at any point you apply the following equation:
Naxn-1
Therefore for curve y=x3, to find the gradient function you take away one from the power of x, so it is x2 and find the product of the n and a which in this case in 3 x 1 = 3 therefore the gradient is 3x2.
I can prove this gradient function is correct by working through this equation using algebra.
Graph of y=x3, where I will attempt to find the gradient at the point s (x,y)
(x+h)3 –x3/h
x2+h2+hx+hx
(x2+h2 +2x)(x+h)
x3+xh2+2hx2+2xh2+h3+hx2
x3+h3+3xh2+3hx2-x3/h
h(h2+3xh+3x2)/h
h2+3xh+3x2
Therefore if h=0
Gradient = 3x2
The gradient of the curve y=x3 is 3x2 as I predicted.
I am now certain that this equation is correct for the set of graphs y=axn. I am not sure that this equation will be correct for other sets of graphs. To test my formula I will now find out the gradient of various unexplored curves. I will do this by finding out the gradient function using the small increment method and then testing that against the results using the formula.
Small increment method to find the graph of y=x1/2
Where x=4, y=2
Gradient =2.0248457-2/4.1-4
=0.248457
Gradient =2.002498439-2/4.01-4
=0.2498439
Gradient =2.00025-2/4.001-4
=0.25
Gradient using small increment method of curve y=x1/2 at point x=4, y=2
=0.25
I will then check this using my equation:
½ x (41/2-1)
=0.25
I can now conclude that this formula is correct for graphs where the power of x is a fraction.
Finally I will check if the formula works for a compound graph.
I will take the graph y=x2+4x. To calculate the gradient of this graph I will use algebra in the hope of
y=x2+4x
Using the formula I have discovered I can prediction that the gradient function of this curve will be= 2x+4
Gradient = y-y/x-x
(x+h)2+4(x+h)-(x2+4x)/x+h-x
(x+H)(x+H)+4x+4h-(x2+4x)/h
x2+2hx+h2+4x+4h-x2-4x/h
2hx +h2+4h/h
2x+4+h
Therefore h=0
Gradient =2x+4
I can therefore say that my prediction is correct and can conclude that the equation works for compound graphs.
Overall I can say that for all the different types of graphs I have tried, my equation has been correct. I can put forward the suggestion that the equation naxn-1 gives the gradient of any curve. However I would need to extend this investigation to insure this is conclusive and try it with many more graphs to check there are no anomalies.