# In this project I will be using 2 methods to find the gradients of curves with the formula: y=axn

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Introduction

## Introduction

In this project I will be using 2 methods to find the gradients of curves with the formula:

y=axn

The two methods I will be using are the tangent method and the small increase method.

The tangent method involves drawing a line, by hand, which reflects the gradient.

The graph shown above is the curve y=x2. The line drawn on the graph is the gradient at the point (3,9). The Gradient is worked out by using this formula:

## QN

MN

In this example the gradient is:

9

1.5

= 6

This method is inaccurate because it is drawn by eye and no real mathematics was involved to find the gradient.

## The small increasemethod is a more precise method of finding the gradient of a curve. It uses 2 points on the same curve, which are very close to each other, and uses the straight line between the points as the tangent. As this increase becomes gradually smaller, the line reflects the gradient more and more.Graph → y=3x3

### Small increase method

Fixed point A (1,3)

B1 = (1.1,3.993)

B2 = (1.01,3.090903)

B3 = (1.01,3.009009003)

AB1 Gradient = 3.993-3

1.1-1

= 0.993

0.1

= 9.93

Middle

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

X | Y | Gradient |

1 | 1 | 3 |

2 | 8 | 12 |

3 | 27 | 27 |

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

x | y | Gradient |

1 | 1 | 2 |

2 | 4 | 4 |

3 | 9 | 6 |

Conclusion

## 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

x | y | Gradient |

1 | 2 | 6 |

2 | 16 | 24 |

3 | 54 | 54 |

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

This student written piece of work is one of many that can be found in our GCSE Gradient Function section.

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