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Gradient Function Maths Investigation

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Introduction

Gradient Function

My task

My task is to investigate the relationship between the gradients of tangents on the curves of graphs when y=axn

Where a is a constant and is not 0, n is equal to 0, 1, 2, 3………

Definition

Gradient of the curve between x1 and x2 is defined as:

image00.png

When x2 is getting close to x1 the gradient becomes the gradient of the curve at x1.

The gradient to a curve, at a particular point, is given by the gradient of the tangent to the curve at that point.

Method

  1. I will draw the curves, y=2x2 and y=ax2 by hand on 5mm graph papers. Next I will draw the tangents and find the gradient of the tangents on the curve when x= -3, -2, -1, 0, 1, 2, 3.
...read more.

Middle

Δx

y

y+Δy

Δy/Δx

(x a)

(x a)

(x a)

y=ax

-3

0.001

-3

-2.9990000000

1.0000000000

-2

0.001

-2

-1.9990000000

1.0000000000

-1

0.001

-1

-0.9990000000

1.0000000000

0

0.001

0

0.0010000000

1.0000000000

1

0.001

1

1.0010000000

1.0000000000

2

0.001

2

2.0010000000

1.0000000000

3

0.001

3

3.0010000000

1.0000000000

Observation: When n=1, y=ax, the gradient is equal to a for all values of x

x

Δx

y

y+Δy

Δy/Δx

(x a)

(x a)

(x a)

y=ax2

-3

0.001

9

8.9940010000

-5.9990000000

-2

0.001

4

3.9960010000

-3.9990000000

-1

0.001

1

0.9980010000

-1.9990000000

0

0.001

0

0.0000010000

0.0010000000

1

0.001

1

1.0020010000

2.0010000000

2

0.001

4

4.0040010000

4.0010000000

3

0.001

9

9.0060010000

6.0010000000

Observation: When n=2, y=ax2, the gradient is approximately equal to 2ax for all values of x

x

Δx

y

y+Δy

...read more.

Conclusion

>

3.0030010000

2

0.001

8

8.0120060010

12.0060010000

3

0.001

27

27.0270090010

27.0090010000

Observation: When n=3, y=ax3, the gradient is approximately equal to 3ax2 for all values of x

x

Δx

y

y+Δy

Δy/Δx

(x a)

(x a)

(x a)

y=ax4

-3

0.001

81

80.8920539880

-107.9460119990

-2

0.001

16

15.9680239920

-31.9760079990

-1

0.001

1

0.9960059960

-3.9940039990

0

0.001

0

0.0000000000

0.0000000010

1

0.001

1

1.0040060040

4.0060040010

2

0.001

16

16.0320240080

32.0240080010

3

0.001

81

81.1080540120

108.0540120010

Observation: When n=4, y=ax4, the gradient is approximately equal to 4ax3 for all values of x

Pattern observed: When y=axn the gradient of the curve is naxn-1

x

Δx

y

y+Δy

Δy/Δx

gradient

(x a)

(x a)

(x a)

10ax9

y=ax10

-3

0.001

59049.0000000000

58852.4649827130

-196535.0172869510

-196830.0000000000

-2

0.001

1024.0000000000

1018.8915046534

-5108.4953465673

-5120.0000000000

-1

0.001

1.0000000000

0.9900448802

-9.9551197903

-10.0000000000

0

0.001

0.0000000000

0.0000000000

0.0000000000

0.0000000000

1

0.001

1.0000000000

1.0100451202

10.0451202103

10.0000000000

2

0.001

1024.0000000000

1029.1315353735

5131.5353734478

5120.0000000000

3

0.001

59049.0000000000

59246.1255075931

197125.5075931400

196830.0000000000

To prove this I have done a table using the small increment method to calculate the gradients of the curves y=ax10 and also using my formula: naxn-1 the two results show above are very close.

Conclusion: The gradient function for y=axn is naxn-1.

...read more.

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