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  • Level: GCSE
  • Subject: Maths
  • Word count: 3186

The Gradient Function Coursework

Extracts from this document...

Introduction

The Gradient Function Coursework


In this piece of coursework I am going to do research on the gradient of various graphs at various points, in order to find a function, which will determine the gradient of these points without drawing or using approximations. I will only need to know the coordinates of the point as well as the type of graph I am considering, to submit them into the gradient function and determine the gradient at this point. The formulae I will use and produce will have particular parameters. Now I am going to explain them.

a: this letter will stand for the coefficient of x in the function y=ax^n and

    determines how steep the graph will be.

n: this letter will be the power to which x is raised in the function y=ax^n and

    determines the shape of the graph.

m: this letter will stand for the gradient at any point of any graph. I can say for

     example the gradient at the point P(1;1) of the graph y=x is 1. Therefore here m=1.

The first range of graphs I am going to investigate will have the function y=ax. I will draw three graphs on the next pages and hope to see a pattern between the gradient and the function of the graphs. I do not need to consider the coordinates of the points at which I will determine the gradient, as the gradient is the same at any point on the graph y=ax.

From these three graphs I clearly recognise a pattern. I will show how I noticed it, by presenting the graphs and their gradients:

y=x        m=1

y=2x      m=2

y=3x      m=3

...read more.

Middle

1

2

6

y=2x³

2

16

24

y=2x³

3

54

54

When I looked at this, it took a while until I found the relationship, but in the end I noticed that for this range of graphs y=ax³ the gradient function will be m=3ax². I think that this is enough research to find a relationship between these three ranges of graphs and their gradients. I will draw up a table with the results and try to generalise the rule.

Type of Graph

Formula for Gradient (m)

y=ax

1axº

y=ax²

2ax¹

y=ax³

3ax²

y=ax^n

nax^n-1

The term in the box should determine the gradient of any point on any graph. I suggest that the gradient function for any graph would be m= nax^n-1. But this may not be true for negative or fractional powers, so I will have to test this function on these ranges of graphs. I will start with y=ax^-1. For this purpose I have to calculate some gradients of random points on two graphs with -1 as their power. Near the calculation I will present the result of my formula. If it equals approximately the calculated gradient, my formula is correct. If it is not, then I will have to change my formula. The result of my formula for the first example below was calculated like this: m=nax^n-1 => m=-1*1*1^-2 => m= -1

image05.png

I typed it in as a formula, so that Excel will automatically calculate the result of the formula for me. Here the gradient according to my formula is correct, but I will have to test it more than one time to be sure. The gradient calculated by my formula should be the exact number, and the formula calculated by using the chord method is only an approximation of the gradient at this point. So both numbers must be very close in order for the formula to be correct. But the number calculated by using the chord method in Excel is only an approximation itself, so it could be that both numbers are equal, which also proves my formula to be correct. I will also work out the gradient for the calculation on the next page: m= nax^n-1 => m=-1*1*2.5^-2 => m= -0.16.

image06.png

image07.png

image08.png

image09.png

The function proves to be correct for negative powers. This function may be the gradient function for any graph y=ax^n at any point, but I do not know if the function will determine the correct gradient for graphs with fractional powers. To ensure this, I will test the function on the range of graphs with formula y=ax^0.5. For the example below I will show how the gradient is worked out by using the formula: m= nax^n-1 => m=0.5*1*1^0.5 => m=0.5

image10.png

The function appears to be correct so far, but I will have to use many more examples in order to show that the gradient function I developed is generally correct.

image11.png

image13.png

I will show working out the gradient for the calculation below by using the formula: m=nax^n-1 => m=0.5*3.31*1^-0.5 => m=1.655

image14.png

image15.png

image16.png

I think that this is enough evidence to show that the gradient function is m= nax^n-1.

Now I will use this formula to predict the gradient of two complex functions by combining simpler ones. My first graph to consider will be y=x²+3x+5. To work out the gradient function for this graph I will need to look at each summand separately and use the rule for it. Then I will need to add the gradients together:

Graph:      y=    x²    +         3x     +      5

Gradient:          2x    +         3       +     0

So the gradient for the graph y=x²+3x+5 will be 2x+3. It is as simple as that. The following two calculations will prove that this is correct. Both calculations are based on the graph y=x²+3x+5:

x1 value

y1 value

x2 value

y2 value

Gradient

1

9

1.1

9.51

5.1

1

9

1.01

9.0501

5.01

1

9

1.001

9.005001

5.001

1

9

1.0001

9.00050001

5.0001

1

9

1.00001

9.00005

5.00001

1

9

1.000001

9.000005

5.000001

...read more.

Conclusion

Conclusion

In this assignment I wrote about a function that I wanted to develop by using the chord and tangent method. I finally found this function to be m=nax^n-1, which proved to be correct for both fractional and negative powers and also of course positive powers. I also showed how you can work out the gradient of complex functions by combining the results of my formula on simpler parts of the function. In the last part I researched into calculus and in particular differentiation, as this relates to my assignment. It showed that the gradient function I developed was actually used in calculus, which proves my findings to be correct. Overall I created a significant piece of coursework, which gave me an initial view of calculus and differentiation, and what I can expect when I attend an A-level Mathematics course in 6th form.

By Kassra Rashidian, 11u

...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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Here's what a star student thought of this essay

5 star(s)

Response to the question

This piece of coursework superbly explores the gradient of basic polynomial curves. The argument posed is logical, and diagrams and screenshots of excel are used to support this argument.

Read full review

Response to the question

This piece of coursework superbly explores the gradient of basic polynomial curves. The argument posed is logical, and diagrams and screenshots of excel are used to support this argument.

Level of analysis

The analysis in this piece is strong. I liked how they begin by trying to work out the gradient of a function at a point, and then showing the understanding that this isn't accurate enough. They use plenty of excel screenshots to give evidence for their hypothesis, but what I really liked was the proof by first principles (shown under the more about calculus section). Using and understanding limits at GCSE level shows high level analysis and I imagine showing this technique will help secure top marks in any piece of coursework. This piece of coursework even goes further to understanding how differential equations and integration may be useful in a real-life application. If I were to make one suggestion, it would be to explore the notation a bit further.

Quality of writing

This piece of coursework is structured well, and spelling and grammar are strongly utilised. Although pieces of formal coursework are becoming less common in mathematics at GCSE, I would note that this piece could be made more sophisticated by removing the frequent use of the first person. Other than that, it should be admired!


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