# The Gradient Function

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Introduction

The Gradient Function A curve does not have a constant gradient because its direction is constantly changing. The gradient of a continuous curve is y = f (x) at any point on the curve is defined as the gradient of the tangent to the curve at this point. Investigating the Gradient Function for y=x I have gained the following results for x x change from y changes from Change in y / change in x Gradient 1 to 1.1 1 to 1.21 (1.21 - 1) / (1.1 - 1) 2.1 1 to 1.01 1 to 1.0201 (1.0201 - 1) / (1.01 - 1) 2.01 1 to 1.001 1 to 1.002001 (1.002001 - 1) / (1.001 - 1) 2.001 2 to 2.1 4 to 4.41 (4.41 - 4) / (2.1 - 2) 4.1 2 to 2.01 4 to 4.0401 (4.0401 - 4) / (2.01-2) 4.01 2 to 2.001 4 to 4.004001 (4.004001 -4) / (2.001-2) 4.001 3 to 3.1 9 to 9.61 (9.61 - 9) ...read more.

Middle

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 1st Step 2nd Step 2 12 3 27 15 4 48 21 6 5 75 27 6 6 108 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 12 4 3 27 9 4 48 16 5 75 25 6 108 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. ...read more.

Conclusion

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) ...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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