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  • Level: GCSE
  • Subject: Maths
  • Word count: 1677

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

Extracts from this document...

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.

image08.pngimage06.pngimage04.pngimage05.pngimage02.pngimage03.pngimage00.pngimage01.png

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

...read more.

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

image10.pngGraph → 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

...read more.

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 3x2y=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

image11.pngimage07.png

...read more.

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

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