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  • Level: GCSE
  • Subject: Maths
  • Word count: 3331

Gradient Fucntion

Extracts from this document...

Introduction

The Gradient Function

In this coursework I am going to try and find out what the gradient function is for various curved graphs. I will do this by drawing graphs and find the gradient of the tangent of a point.

1.        I am going to draw a graph of y=x². I will use values from 0-5, I will obtain the gradient of the tangent at different points.

This is the table of the results of y=x².

X

0

1

2

3

4

5

Y=x²

0

1

4

9

16

25

This is the results of the gradient of the tangent:

Gradient of  y=x².

At (1,1)

1.8

1

1.8 = 2(to 1sf)

At (2.4)

2

0.5

4

At (3,9)

4.7

0.8

5.875 = 6(to 1sf)

At (4,16)

3

0.3

10

I have noticed that the gradient of the tangent doubles each time. Except for at (4,16) so I fink that this is an anomalious result, which would mean that the tangent must be drawn incorrectly.

I am now going to draw a graph of y=x³. I will use values from 0-5, I will

obtain the gradient of the tangent at different points.

This is the table of the results of y=x³.

X

0

1

2

3

4

5

Y=x³

0

1

8

27

64

125

This is the results of the gradient of the tangent:

Gradient of  y=x³.

At (1,1)

3

0.9

3.33333 = 3(to 1sf)

At (2.8)

15

1.3

11.538462 = 12(to 1sf)

At (3,27)

20

0.8

25

At (4,64)

22

0.5

44

...read more.

Middle

=

0.40527009

       0.001

=405.27009

At (4,1024)

1025.28064016 – 1024

       4.001 – 4

=

1.28064016

       0.001

=1280.64016

At (5,3125)

3128.12625025 – 3125

       5.001 - 5

=

3.12625025

       0.001

= 3126.2525

These are quite accurate but when it gets to at (4,1024) and (5,3125) it starts to get inaccurate again so I will go to 0.0001 instead of 0.001.

I will do (4,1034) and (5,3125) again, going to 0.0001.

At (4,1024)

1024.1280064 – 1024

         4.0001 – 4

=

0.1280064

    0.0001

=1280.064

At (5,3125)

3125.3125125 – 3125

         5.0001 - 5

=

0.3125125

   0.0001

= 3125.125

These results are more accurate.

I will also use this method for y=x1, though I know that the answer is one all the time, just to prove that this method is more accurate.

At (1,1)

1.001 – 1

1.001 – 1

=

0.001

0.001

1

At (2,2)

2.001 – 2

2.001 – 2

=

0.001

0.001

1

At (3,3)

3.001 – 3

3.001 – 3

=

0.001

0.001

1

At (4,4)

4.001 – 4

4.001 – 4

=

0.001

0.001

1

At (5,5)

5.001 – 5

5.001 – 5

=

0.001

0.001

1

These are very accurate results.

Overall results.

These are the results of the small increments method. The results will be to 1 the nearest whole integer.

Y= x1

Y= x2

Y= x3

Y= x4

Y=x5

X=1

1

2

3

4

5

X=2

1

4

12

32

80

X=3

1

6

27

108

405

X=4

1

8

48

256

1280

X=5

1

10

75

500

3126

I will now compare the results of the tangent of the gradient method and the small increments method. (TG= gradient method, S.I small increments method)

Y= x1

Y= x2

Y= x3

Y= x4

Y=x5

TG

S.I

TG

S.I

TG

S.I

TG

S.I

TG

S.I

X=1

1

1

2

2

3

3

10

4

5

X=2

1

1

4

4

12

12

30

32

80

X=3

1

1

6

6

25

27

65

108

405

X=4

1

1

10

8

44

48

286

256

1280

X=5

1

10

75

500

3126

The highlighted ones are the anomalous results. This shows me that the small increments method is more accurate. I did not use the gradient method for x=5 or y=x5.

There must be an easier way to find out what the results should be. I will now try and find a formula for each one using the small increment results.

Y = x1.

1        1        1        1        1                n= x0

image02.pngimage02.pngimage02.pngimage02.png

    1               1               1              1

Y=x2

2        4        6        8        10                n=2x1

image02.pngimage02.pngimage02.pngimage02.png

      2              2              2              2

Y=x3

3        12        27        48        75

image02.pngimage02.pngimage02.pngimage02.png

      9              15              21              27                        n=3x2

image02.pngimage02.pngimage02.png

        6        6        6

Y=x4

4        32        108        256        500

image02.pngimage02.pngimage02.pngimage02.png

      32               76        148       244                        n=4x3

image02.pngimage02.pngimage02.png

        48        72          96image02.pngimage04.png

            24                24

Y=x5

5        80        405        1280          3125

image02.pngimage02.pngimage02.pngimage02.png

      75                325      875         1845

image02.pngimage02.pngimage02.png

          250      550       970                                n=5x4

image02.pngimage02.png

  1. 420image02.png

      120

This is the table of results for the formulae.

Y= x1

Y= x2

Y= x3

Y= x4

Y=x5

X=1

1

2

3

4

5

X=2

1

4

12

32

80

X=3

1

6

27

108

405

X=4

1

8

48

256

1280

X=5

1

10

75

500

3126

X

X0

2x1

3x2

4x3

5x4

...read more.

Conclusion

Small Increments Method – this method proves to be more accurate than the tangent method, but it still has a level of inaccuracy when the numbers start to get bigger. So from this method I could find formulas and use the…Formula Method – this method is very accurate and once you know the overall formula you can work out any equation of a curved graph. The overall formula is Y=nxn-1. This formula works for positive, negative and fractional indices.

Throughout the investigation I have used two of the three methods for each equation. This is so that I can check the results of both methods and compare them to double-check it and to find a more accurate method. My overall conclusion of the methods is that the tangent method is very inaccurate as there are many factors that can make it inaccurate (the curve, the tangent drawn etc), and the formula method is very accurate as it makes it easier to find the correct answer and does not cause inaccuracies unless the formula is worked out the wrong way.

I have found out from the positive indices the numbers increase, whereas the negative indices decrease, as well as the fractional indices.

Maths Coursework: The Gradient Function

...read more.

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