3.0001-3
= 0.00810027
0.0001
= 81.00270003
The gradient is approaching 81
Tabulating The Results For Graph → y=3x3
The formula for this graph is:
Gradient= 9(x2)
Graph → y=x3
Drawing tangents
Point x= (1,1)
Point y= (2,8)
Point z= (3,27)
Triangle A (point x)= 2
0.5
Triangle B (point y)= 6.5
0.5
Triangle C (point z)= 14
0.5
Gradient x= 4
Gradient y= 13
Gradient z= 28
Graph → y=x3
Small increase method
Fixed point A= (1,1)
B1= (1.1,1.13)
B2= (1.01,1.013)
B3= (1.001,1.0013)
AB1 Gradient = 1.13-1
1.1-1
= 0.331
0.1
= 3.31
AB2 Gradient = 1.013-1
1.01-1
= 0.030301
0.01
= 3.0301
AB3 Gradient = 1.0013-1
1.001-1
= 0.003003
0.001
= 3.003
The Gradient is approaching 3
Graph → y=x3
Small increase method
Fixed point A= (2,8)
B1= (2.1,2.13)
B2= (2.01,2.013)
B3= (2.001,2.0013)
AB1 Gradient = 2.13-2
2.1-2
= 1.261
0.1
= 12.61
AB2 Gradient = 2.013-2
2.01-2
= 0.120601
0.01
= 12.0601
AB3 Gradient = 2.0013-2
2.001-2
= 0.012006
0.001
= 12.006
The Gradient is approaching 12.
Graph → y=x3
Small increase method
Fixed point A= (3,27)
B1= (3.1,3.13)
B2= (3.01,3.013)
B3= (3.001,3.0013)
AB1 Gradient = 3.13-3
3.1-3
= 2.791
0.1
= 27.91
AB2 Gradient = 3.013-3
3.01-3
= 0.2709
0.01
= 27.09
AB3 Gradient = 3.0013-3
3.001-3
= 0.027009
0.001
= 27.009
The gradient is approaching 27
Tabulating results for the graph → y=x3
As you can see, from this table and the results from the tangents of y=x3, that the tangent method is fairly inaccurate. The formula for this graph is:
Gradient = 3(x2)
Graph → y=x2
Tangent method
Point x= (1,1)
Point y= (2,4)
Point z= (3,9)
Triangle A = 1
0.5
Triangle B = 2
0.5
Triangle C = 3
0.5
Gradient x= 2
Gradient y= 4
Gradient z= 6
Graph → y=x2
Small increase method
Fixed point A= (1,1)
B1= (1.1,1.12)
B2= (1.01,1.012)
B3= (1.001,1.0012)
AB1 Gradient = 1.12-1
1.1-1
= 0.21
0.1
= 2.1
AB2 Gradient = 1.012-1
1.01-1
= 0.0201
0.01
= 2.01
AB3 Gradient = 1.0012-1
1.001-1
= 0.002001
0.001
= 2.001
The gradient is approaching 2.
Graph → y=x2
Small increase method
Fixed point A = (2,4)
B1= (2.1,2.12)
B2= (2.01,2.012)
B3= (2.001,2.0012)
AB1 Gradient = 2.12-4
2.1-2
= 0.41
0.1
= 4.1
AB2 Gradient = 2.012-4
2.01-2
= 0.0401
0.01
= 4.01
AB3 Gradient = 2.0012-4
0.001-2
= 0.004001
0.01
= 4.001
The gradient is approaching 4.
Graph → y=x2
Small increase method
Point A= (3,9)
B1= (3.1,3.12)
B2= (3.01,3.012)
B3=(3.001,3.0012)
AB1 Gradient = 3.12-9
3.1-3
= 0.61
0.1
= 6.1
AB2 Gradient = 3.12-9
3.01-3
= 0.0601
0.01
= 6.01
AB3 Gradient = 3.0012-9
3.001-3
= 0.006001
0.001
= 6.001
The gradient is approaching 6.
Tabulating The Results For The Graph → y=x2
The formula for this graph is:
Gradient = 2x
Tabulating The Results For The Graph → y=3x2
The formula for this graph is:
Gradient = 6x
Graph → y=3x2
Small increase method
Fixed point A= (3,27)
B1= (3.01,27.1803)
B2= (3.001,27.018003)
B3= (3.0001,27.00180003)
AB1 Gradient = 27.1803-27
3.01-3
= 0.1803
0.01
= 18.03
AB2 Gradient = 27.018003-27
3.001-3
= 0.018003
0.001
= 18.003
AB3 Gradient = 27.00180003-27
3.0001-3
= 0.00180003
0.0001
= 18.0003
The gradient is approaching 18.
Graph → y=2x2
Small increase method
Fixed point A= (1,2)
B1= (1.1,2.42)
B2= (1.01,2.0402)
B3= (1.001,2.004002)
AB1 Gradient = 2.42-2
1.1-1
= 0.42
0.1
= 4.2
AB2 Gradient = 2.0402-2
1.01
= 0.0402
0.01
= 4.02
AB3 Gradient = 2.004002-2
1.001-1
= 0.004002
0.001
= 4.002
The gradient is approaching 4
Graph → y=2x2
Small increase method
Fixed point A= (2,8)
B1= (2.1,8.82)
B2= (2.01,8.0802)
B3= (2.001,8.008002)
AB1 Gradient = 8.82-8
2.1-2
= 0.82
0.1
= 8.2
AB2 Gradient = 8.0802-8
2.01-2
= 0.0802
0.01
= 8.02
AB3 Gradient = 8.008002-8
2.001-2
= 0.008002
0.001
= 8.002
The gradient is approaching 8
Graph → y=2x2
Small increase method
Fixed point A= (3,18)
B1= (3.01,18.1202)
B2= (3.001, 18.012002)
B3= (3.0001,18.00120002)
AB1 Gradient = 18.1202-18
3.01-3
= 0.1202
0.01
= 12.02
AB2 Gradient = 18.012002-18
3.001-3
= 0.012002
0.001
= 12.002
AB3 Gradient = 18.00120002-18
3.0001-3
= 0.00120002
0.0001
= 12.0002
The Gradient is approaching 12
Tabulating The Results For The Graph → y=2x2
The formula for this graph is:
Gradient= 4x
Graph → y=2x3
Small increase method
Fixed point A= (1,2)
B1= (1.1,2.662)
B2= (1.01,2.060602)
B3= (1.001,2.006006002)
AB1 Gradient = 2.662-2
1.1-1
= 0.662
0.1
= 6.62
AB1 Gradient = 2.060602-2
1.01-1
= 0.060602
0.01
= 6.0602
AB3 Gradient = 2.006006002-2
1.001
= 0.006006002
0.001
= 6.006002
The gradient is approaching 6
Graph → y=2x3
Small increase method
Fixed point A= (2,16)
B1= (2.01,16.241202)
B2= (2.001,16.024012)
B3= (2.001,16.00240012)
AB1 Gradient = 16.241202-16
2.01-2
= 0.241202
0.01
= 24.1202
AB2 Gradient = 16.024012-16
2.001-2
= 0.024012
0.001
= 24.012
AB3 Gradient = 16.00240012-16
2.0001-2
= 0.00240012
0.0001
= 24.0012
The gradient is approaching 24.
Graph → y=2x3
Small increase method
Fixed point A= (3,54)
B1= (3.01,54.541802)
B2= (3.001,54.054018)
B3= (3.0001,54.00540018)
AB1 Gradient = 54.541802-54
3.01-3
= 0.541802
0.01
= 54.1802
AB2 Gradient = 54.054018-54
3.001-3
= 0.054018
0.001
= 54.018
AB3 Gradient = 54.00540018-54
3.0001-3
= 0.00540018
0.0001
= 54.0018
The gradient is approaching 54.
Tabulating The Results Fro The Graph → y=2x3
The formula for this graph is:
Gradient = 6(x2)
y=axn – The Formula
Once this formula is found, you can work out any gradient function gradient. The formula I put forward is:
Gradient = nx(n-1)
This is when:
N= the power of x (e.g. x3)
X= the x coordinate on the graph that relates to the gradient.
I shall test this formula with y=x2 and y=x3 curves.
Y=x2 → 2x(2-1) →1 x 2x1 → 2x → y=2X
Y=x3 → 3x(3-1) → 1 x 3x2 → 3x2 → y=3x2
When we look back at the result tables for y=x2 and y=x3 the formulas match. Therefore I conclude that the formula:
Gradient= nx(n-1)
Is the correct formula and can be applied for all gradient functions.
The Gradient Function
