The Gradient Function

(14.06250075 - 14.0625) / (3.7500001 - 3.75)

7.5

When investigating the results the gradient is clearly visible.

x

Gradient

st Step

2

2

4

2

3

6

2

4

8

2

When looking at the first step, there is a constant difference between the consecutive gradients. In this case the difference is 2. The 2 in the 1st step tell us that for the gradient to be found x needs to multiplication of 2. In this case the formula 2x works.

Take three as an example. Three times by two will give you the answer 6. Hence the gradient.

Gradient of x =

x

x change from

y changes from

Change in y / change in x

Gradient

2 to 2.01

8 to 9.261

(9.261 - 8) / (2.1 - 2)

2.61

2 to 2.01

8 to 8.120601

(8.120601 - 8) / (2.01-2)

2.0601

2 to 2.001

8 to 8.012006001

(8.012006001 - 8) / (2.001 -2)

2.006001

3 to 3.1

27 to 29.791

(29.791 - 27) / (3.1 - 3)

27.31

3 to 3.01

27 to 27.270901

(27.270901 - 27) / (3.01 - 3)

27.0901

3 to 3.001

27 to 27.027009

(27.027009 - 27) / (3.001 - 3)

27.009

3.75 to 3.751

52.734 to 52.77657

(52.77657 - 52.734) - (3.751 - 3.75)

42.57

3.75 to 3.7501

52.734 to 52.73859386

(52.73859386 - 52.734) - (3.7501 - 3.75)

45.9386

When looking at my results no obvious observations was found. Due to that, I decided to investigate further into the gradient differences.

x

Gradient

st Step

2nd Step

2

2

3

27

5

4

48

21

6

5

75

27

6

6

08

33

6

When looking at the table (left) it is now much more clear. On the second steps the difference between the consecutive gradients are the same. Because of this, it states that a squared power is involved. So to make it easier and clearer I have written a new table to show x squared.

x

Gradient

x squared

2

2

4

3

27

9

4

48

6

5

75

25

6

08

36

Although x is now squared, it still doesn't match the gradient. But the table shows that if three were multiplied to the squared numbers the gradient would be found. So we now know the exact formula for x cubed. This is x squared, times by three. Take for example three: three squared is nine. Nine multiplied by three is twenty-seven. Hence the gradient.

Gradient of x =

x

x change from

y changes from

change in y / change in x

Gradient

2 to 2.1

6 to 19.4481

(19.4481 - 16) / (2.1-2)

34.481

2 to 2.01

6 to 16.3224

(16.3224 - 16) / (2.01 - 2)

32.2408

2 to 2.001

6 to 16.032

(16.032 - 16) / (2.001 - 2)

32.024

3 to 3.1

81 to 92.3521

(92.3521 - 81) / (3.1-3)

13.521

3 to 3.01

81 to 82.085

(82.085 - 81) / (3.01 - 3)

08.5

3 to 3.001

81 to 81.108

(81.108 - 81) / (3.001 - 3)

08.054

3 to 3.0001

81 to 81.01080054

(81.01080054 - 81) / (3.0001 - 3)

08.0053

3 to 3.00001

81 to 81.00108

(81.00108 - 81) / (3.00001 - 3)

08

When looking at my results no obvious observations was found. Due to that, a further investigation on the gradient is required.

A similar method of finding x is produced when finding x .

x

Gradient

st Step

2nd Step

3rd Step

2

32

3

08

76

4

256

48

72

5

500

244

96

24

6

864

364

20

24

When looking at the table, unlike x , on the third steps, the difference between the consecutive gradients are the same. So this shows that a cubed is involved in finding a formula for x . To make it clearer another table will be drawn showing the gradient along side x to the power of 3.

x

Gradient

x cubed

Gradient / x cubed

2

32

8

4

3

08

27

4

4

256

64

4

5

500

25

4

6

864

216

4

After analysing the table (right) another table can be added. I have noticed that when the gradient is divided by x cubed the answer 4 is repeated. Due to the recurrence of 4, the opposite function of is processed. In this case, the opposite function would be multiplication.

All the information is collected up and a formula can now be drawn up. That is x cubed, times by four. Take 2 for example. Three cubed is twenty-seven and twenty-seven multiplied by four is one hundred and eight. Hence the Gradient.

Gradient of x =

x

x

X

When comparing the formulas a few observations could be made.

* That the multiplication of x, went up one every consecutive power. For example for x , x is multiplied by 2 and for x , x is multiplied by 3.

* That the power of each formula is the original power of x minus one.

A formula for a gradient of a curve could be now made;

nx

n is the power

Examples:

a) Finding the Gradient for x at the point x=2

nx

5(2)

b) Finding the Gradient for x at the point x=2

nx

6(2)