• Join over 1.2 million students every month
• Accelerate your learning by 29%
• Unlimited access from just £6.99 per month   # Investigate gradients of functions by considering tangents and also by considering chords of the graph and using algebra.

Extracts from this document...

Introduction

Introduction

Aim: to investigate gradients of functions by considering tangents and also by considering chords of the graph and using algebra.

Research: What does “gradient” mean? Generally, the “steepness” of a curve is measured by its gradient.

Firstly in my coursework I will investigate gradients using tangents and find out the bests way to use tangents. I will start of by investigating the gradients of y=x, y=x2, y=x3 because they are likely to be simpler to solve and so I can understand at first then I will move onto more complex equations later.

Then I will look at chords and finally algebra.  I hope I will learn a lot during my time of doing this coursework, as I am sure you will too.

In vertical and horizontal graph lines it is easy to work out the gradient. For a graph of a horizontal line the line has no steepness and so the gradient is zero also for a vertical graph line the graph line is infinite so the gradient is infinite.

Gradients are just as important in other subjects as they are in maths. In physics acceleration is the gradient of a velocity time graph and velocity is the gradient of an instance time graph. Also gradients can be used in radioactivity and decay in physics.

In biology and chemistry population growth is thought of as gradient.

...read more.

Middle

Things are not solved yet; I have to look at the formula for these three times to find any possible relationship. In y=x, g=1=x0; in y=x2, g=2x; in y=x3, g=3x2; as we can see, in the three gradient formulas, the number before x and the indices are both increasing as the indices of the equation get higher, and the difference between the number before x and the index is 1 (the index number is smaller than the number before x). So I predict that the gradient formula for y=x4 is: g=4x3.

Let’s try it out.

The co-ordinates:

 x 1 2 3 4 Y 1 16 81 256

The gradients in y=x4:

 x g (The Gradient)Increment Method 1 4.040601 2 32.240801 3 108.541201 4 256.96101

g1=4=4×1=4x13

g2=32=4×2×2×2=4x23

g3=108=4×3×3×3=4x33

g4=256=4×4×4×4=4x43

So my prediction works this time! Look at the table of the previous steps below:

 Equation y=x y=x2 y=x3 y=x4 y=xn Gradient Function g=x0 g=2x g=3x2 g=4x3 g=nxn-1

I got the gradient function. It can be written in this way:

When y=xn, g=nxn-1

4

Just to be certain, I will try y=x5:

 x 1 2 3 4 y 1 32 243 1024
 x g (The Gradient)Increment Method 1 5.10100501 2 80.80401001 3 407.70902 4 1286.41602

5×14=3
5×2
4=80
5×3
4=405
5×4
4=1280

My formula does work! And notice that because of the high number, increment method is now not so perfect.

...read more.

Conclusion

Therefore dy/dx =

lim 2x + dx

dx --> 0

When dx becomes zero, dy/dx = 2x.

Therefore the gradient of y = x² is 2x.

For example, at the point (2, 4), the gradient is 2x = 4 .

10

### Conclusion

During this coursework I have learnt a great deal about straight lines, curves, algebra and in particular gradients. I have been able to set out my results in a table and I noticed a general pattern. From this pattern I was able to produce a general formula for the gradient of a curve i.e.

if y=xn, G=nxn-1

This result enabled me to find the gradient of more complicated curves, such as the coursework curve i.e.

if y=4x3+2x-5

G=49(3x2)+2-0

=12x2+2

I can now find the gradient at any point on the coursework curve just by substituting the x value at that point into the formula for the gradient i.e.

if at the point P (0, -5), G=2

I have no doubt that the mathematics I have learned will be useful to me at GCSE level and in more advanced work in other subjects in the future.

If I had more time I would like to investigate other curves to see if my result still holds

e.g.  y=1/x2, y=1/xn, y=sin x, y=cos x, y=tan x

I predict that the formula would work for the first two groups but I am not sure if it would work for the trigonometry. This could be investigated in a further coursework project.

11

...read more.

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

# Related GCSE Gradient Function essays

1.  ## The Gradient Function Coursework

5 star(s)

As I have mentioned earlier calculus was first created by Newton and also Leibniz, although some of the ideas were already used by Fermat and even Archimedes. Calculus is divided into two parts. The part of calculus which I dealt with on the previous two pages is called differential calculus and the other part is called integral calculus.

2. ## I have been given the equation y = axn to investigate the gradient function ...

gradient of the tangent at A Taking for example the equation y = x2 where the gradient function = 2x, I would like to calculate the gradient function at x = 2 using the small increament method using my calculator x

1. ## Maths Coursework - The Open Box Problem

10 1000 I have now found out that in a 10 by 10 square the cut out which gives the largest volume is between 5 and 6 but more towards 5 since 5 gives a larger volume. To obtain a more accurate result I will zoom in the shaded region.

2. ## Aim: To find out where the tangent lines at the average of any two ...

Plug the point, (-2.25, 4.21875) into the function (After finding out the function of the tangent line, draw this linear function with the original cubic function together into the graph calculator, then we can use this tool in the calculator to find out their intersecting point.)

1. ## The Gradient Function

I have put all the gradient functions that I have found through my investigation together to try and find the gradient function which applies to all y=anx(n-1). Graph Gradient Function ax2 2ax ax3 3ax2 ax4 4ax3 xn nx(n-1) axn anx(n-1)

2. ## The Gradient Function

the quadratic curves, the cubic curves and the hyperbolas. I will take 4 increments for all the tangents, which I have drawn. I will show the first calculation as an example for how I obtained the values in the table.

1. ## The Gradient Function

on graph y=2x6: PREDICTIONS X=1 2 x 6 x 15 = 12 X=2 2 x 6 x 25 = 384 X=3 2 x 6 x 35 = 2916 The graph I have of y=2x6 proves that this equation is true.

2. ## Investigate the gradients of the graphs Y=AXN

Y=X-2 X Predicted gradient with formula (3dp) X Co-ordinates Y Co-ordinates Increment formula Increment gradient 1 -2*1(-2-1) = -2 1,1.001 1,0.998002996 -0.001997004/0.001 -1.997004 2 -2*2(-2-1)= -0.25 2,2.001 0.25,0.249750187 -0.00024913/0.001 -0.24913 3 -2*3(-2-1) = -0.074 3,3.001 0.1111r, 0.111037074 -0.000074037/0.001 -0.074037 4 -2*4(-2-1) = -0.031 4,4.001 0.0625,0.062468761 -0.000031239/0.001 -0.031239 This table shows that my earlier formula ANXN-1 still works for negative powers. • Over 160,000 pieces
of student written work
• Annotated by
experienced teachers
• Ideas and feedback to
improve your own work 