• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

The Gradient Function

Extracts from this document...

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.

The above preview is unformatted text

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Gradient Function essays

  1. Peer reviewed

    The Gradient Function Coursework

    5 star(s)

    The gradient of the chord PQ would thus be: (y+ ?x) -y ?y (x+ ?x) +x ?x Because the equation of the curve is y=f(x), the coordinates of P can also be written as [x, f(x)] and the coordinates of Q as [(x+?x), f(x+?x)].

  2. Curves and Gradients Investigation

    This is illustrated by Graph G. A General Rule? I now want to see if I can find a general rule to find the gradient function for a curve or the form y = xn . This can hopefully be achieved by further investigation of the gradient functions of other curves using the 'Small Increments of Size "h" Method'.

  1. The Gradient Function

    I am therefore predicting, that the gradient function of y=3x2 will equal 6x. Below is the table of values for the curve y=3x2: x -2 -1 0 1 2 x2 4 1 0 1 4 3x2 12 3 0 3 12 y=3x2 12 3 0 3 12 Once again, below

  2. The Gradient Function Investigation

    y = x� Gradient = (x + h)� - x� (x + h) - x = (x + h)(x� + 2xh + h�) - x� (expand brackets) h = (x� + 2x�h + xh� + hx� + 2xh� + h�)

  1. Gradient function

    0.999 174.919 4 256 175 1 175 Power: 1 Coefficient: 4 Fixed point: 3 My second fixed point: 5, 625 x y increase in y increase in x gradient 4 256 -369 -1 369 4.1 282.5761 -342.4239 -0.9 380.471 4.2 311.1696 -313.8304 -0.8 392.288 4.3 341.8801 -283.1199 -0.7 404.457 4.4

  2. Gradient Function

    0.3 70.59 4.8 110.592 14.408 0.2 72.04 4.9 117.649 7.351 0.1 73.51 4.99 124.2515 0.748501 0.01 74.8501 4.999 124.925 0.074985001 0.001 74.985 5 125 5.001 125.075 -0.075015 -0.001 75.015 5.01 125.7515 -0.751501 -0.01 75.1501 5.1 132.651 -7.651 -0.1 76.51 5.2 140.608 -15.608 -0.2 78.04 5.3 148.877 -23.877 -0.3 79.59 5.4

  1. The Gradient Function

    a=10, x=1 2 x 10 x 1 = 20 a=10, x=1.5 2 x 10 x 1.5 = 30 a=10, x=2 2 x 10 x 2 = 40 a=10, x=3 2 x 10 x 3 = 60 a=10, x=-2 2 x 10 x -2 = -40 I checked this on Omnigraph

  2. The Gradient Function

    = 4 1-(-1) Graph: 0.5x1 The table below shows the points I have used to plot the graph: X Y 0 0 1 0.5 2 1 -1 -0.5 The gradient of the line is: Y2-Y1 X2-X1 = 1.5-0.5 = 0.5 3-1 Graph: Y= -3x1 The table below shows the points, which I have

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work