Maths Gradients Investigation
Maths Coursework 2001 This coursework is all about finding different gradients. The gradient of y = x² or y= x³. The first graph I will do is y = x². I have found the gradients for all of the numbers shown, I did this by drawing a triangle as close to the graph line as I could. I then find the height and the width of the triangle. Then I would divide the height by the width to find the gradient. The following is an example.... If the height of the triangle 3 = 12 And the width of triangle 3 = 3 Therefore.......12/3 = 4 The gradient of point (3,9) is 3. This table shows my results The difference between each one is obviously 2, so my formula must be.. Gradient = 2x I will now show the gradients of chords starting at the point (2,4) and finishing at various other points along the plotted line for example, 2 to 5 and 2 to 4 etc. The pattern is seen. There is a formula is the A level textbook 'Essential Pure Mathematics' The formula is RQ = NQ - NR = (a + h)² - a² = 2ah + h² and also PR = h So the gradient of the chord PQ is RQ = 2ah + h² PR h = h(2a + h) h = 2a + h As h gets smaller it would be stupid to add it onto the formula, so this proves that my original formula was correct as 2a is now left in the formula. Katie Curtis 5L Page
Gradient Graphs Investigation
I have been set a task to find the gradient of different graphs ie. The gradient of y = x² or y= x³. I will start with the graph of y = x². (This is shown below) I have found the gradients for all of the numbers shown, I did this by drawing a triangle as close to the graph line as I could. I would then find the height and the width of the triangle. Then I would divide the height by the width. Here is an example:- Height of triangle 3 = 12 Width of triangle 3 = 2 So. 12 = 6 2 The gradient of point (3,9) is therefore 6. Here is a table of all of my results:- Diff 1 2 2 2 2 2 This means that my formula should be:- Gradient = 2x I will now show the gradients of chords starting at point (2,4) and finishing at various other points along the plotted line ie. 2 to 5 and 2 to 4 etc. You can see the pattern that is emerging. I shall attempt to prove this by using a formula that is provided in the 'Essential Pure Mathematics' A-level textbook. The formula is:- RQ = NQ - NR = (x + h)² - x² = 2xh + h² and PR = h The gradient of the chord PQ is:- RQ = 2xh + h² PR h = h(2x + h) h = 2x + h As I rotate the chord clockwise about point P, Q approaches P along the curve and the gradient of the chord PQ is the gradient of the tangent at P, and h will be 0. So when h will be 0, the gradient of the PQ chord is 2x +
Investigate gradients of functions by considering tangents and also by considering chords of the graph and using algebra.
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. So moving on to curves now I need to find out the best way to find the gradient of any curve. Tangents
The Gradient Function Coursework
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
My aim is to investigate the gradient function for all kinds of curves.
Aim: to investigate the gradient function for all kinds of curves. Research: What does "gradient" mean? Generally, the "steepness" of a curve is measured by its gradient. We can look at the figure below: This is the curve of y=x2. The point P (3, 9) has been marked and the tangent QPM drawn. The gradient of the tangent is QN/MN. So we can use the "tangent method" to obtain the gradients of graphs of different functions. First Step: I am going to investigate the gradient of y=x, y=x2, y=x3 first because they are likely to be the simplest equations to solve, and after getting these results easily, by looking at them, the more complex equations will seem easier to discover. I am going to look at y=x first because it is the easiest. x 2 3 4 y 2 3 4 Please see graph on separate pieces of paper As we see, the gradients of y=x is very simple, a=1. We even do not need to draw any tangents to obtain the gradients. So the relationship between a and x can be shown in the table below: x a (The Gradient) 2 3 4 So it is extremely obvious that in the graph of y=x, whatever x is, the gradient a stays 1. Then let us try the graph of y=x2. x 2 3 4 y 4 9 6 By drawing the tangents of each point, we can calculate the gradients. However, as the graph is not always accurately drawn, there must be some error between the results. In order to avoid this, I am going to use
Gradient function
My coursework on the gradient function is to investigate the gradient on different points on the line and curves. I will start of my investigation with y = x2. This will be a parabolic curve and the gradient will move from point to point. I will need to start of with a fixed point. I have chosen (2, 4). I will use a table to get close to the point. My table will have five columns. The first column will be x, which will have any numbers between 1and 3. The second column will be y, which will be the result of squaring an x numeral. The third column will be the increase in y, where squared value gets subtracted from 4. The fourth column will be the increase in x, this is where the x values get subtracted from 2. The fifth and last column will be the gradient, where the change in y divided by the change in x, gives me the results. I will do different fixed points so the numbers will vary. I will then do other functions such as Y=4x2 Next I will move to a function y = x4 and investigate the gradient at different points. I will use the same method used in the equation y = x2 but instead of squaring the numbers I will cube them. After my investigating finishes I will come up with a conclusion which will summarize my investigation. y=x2 My first fixed point is 2, 4 x y change in y change in x gradient 3 3 .1 .21 2.79 0.9 3.1 .2 .44 2.56 0.8 3.2 .3 .69
Gradient Function.
Gradient Function Coursework Gradient Function Coursework My task is to investigate the relationship between the gradients of tangents on the curves of graphs, such as y=x2. To do this, I will first find the gradient of tangents on the graph y=x2 by drawing the graph. I have labelled the tangents a-g. They go from x=-3 to x=4. Below are the calculations for their gradients. (I am using the formula (y2-y1)/(x2-x1)to calculate the gradient of the line. a= (12-6)/(-3.5--2.5) = 6/-1 = -6 b= (6-2)/(-2.5--1.5) = 4/-1 = -4 c= (2-0)/(-1.5--0.5) = 2/-1 = -2 d= (2-0)/(1.5-0.5) = 2/1 = 2 e= (6-2)/(2.5-1.5) = 4/-1 = 4 f= (12-6)/(3.5-2.5) = 6/1 = 6 g= (20-12)/(4.5-3.5) = 8/1 = 8 As you can see, the results I have obtained are good round numbers. These results however are not accurate to the tangents I drew on the graph. There is always going to be an inaccuracy in a graph, even if that inaccuracy is 0.25 of a millimetre. Therefore, I can only accept these results as estimates. As you can see, they are all twice their x value. Therefore, for the graph of y=x2, the formula for the gradient of a tangent is g=2x. I am left with the question of accuracy. I cannot get total accuracy, but there are ways I can get very close to an accurate answer. One such way is to use the method of using a line inside the curve. (4,16) (3.5,12.25) (3.01,9.0601) (3,9) Judging by this trend, I can
The Gradient Function
The Gradient Function Part 1 Investigate the gradient function for the set of graphs: - y=axn where a and n are constant. For Part 1 of my investigation I will be looking at many different types of y=axn graph such as: y = x2, y = 2x2, y = 3x2, y = 4x2, y = x3, y = 2x3, y = 3x3, y = 4x3 y = x4, y = 2x4, y = 3x4, y = 4x4 y = x2, y = x3, y= x4, y= x5 y = 2x2 y = 2x3 y =2x4 y = 2x5 I will be trying to find the gradients of certain points on the graph which can lead me to the gradient function for that curve. Gradient: measures the steepness of a curve. The gradient of any particular point on a curve is defined as the gradient of the tangent drawn to the curve at that point. Tangent: is a straight line which touches but does not cut the given curve at a particular point. The gradient of any tangent is vertical horizontal Gradient function: is a common function for the gradients of a set of tangents on a particular curve. When this function is obtained, tangents do not need to be drawn to work out the gradient. Prediction: I predict that there will be a gradient function to every graph that is y=axn then from those gradient function a common function can be found. This gradient function will apply to all graphs of y=axn. I also predict that there will be different ways to work the gradients or gradient function.
The Gradient Function.
The Gradient Function The Gradient Function I am trying to find a formula that will work out the gradient of any line (the gradient function) I am going to start with the simplest cases, e.g. g=c², g=c, g=c3 etc. as they are probably going to be the easiest equations to solve as they are likely to be less complex and hopefully the formulas to the more complex equations will be easier to discover by looking at the previous formulas. I am going to look at the line g=c² first. g=c² c 2 3 4 g 4 9 6 Please see graph on separate piece of paper One of the most obvious things I notice is that as the co-ordinates increase so does the gradient. Not only can you see that from the results below, but also on the graph you can that the line gets steeper and steeper. This makes sense as the higher the number c is the larger the difference between c² and c. Another thing that I have noticed is that the larger the co-ordinates the smaller the increase in gradient. Point Gradient (tangent) Gradient (Small Increment Method) (1,1) 2 2.01 (2,4) 3.3 3.01 (3,9) 6.36 (2dp) 6.01 (3.5,12.3) 6.4 7.001 As the table above shows there are two methods that I am using for calculating the gradient of line. The first being drawing a tangent at the point, working out the distances on the tangent using the scale on the graph and then using this formula: dg/dc However there
The Gradient Function
The Gradient Function Aim: To investigate the gradient of different curves I will investigate the gradient of different equations for which the general formula is Y = a?n In this equation I will investigate the gradient by varying the values of 'a' and 'n'. Gradient: The property possessed by a line or surface that departs from the horizontal is called the gradient of the line. In mathematical terms, the gradient of the line simply tells us how steep a line is. The gradient for all lines parallel to the X-axis is 0. Gradient Function: Gradient function is the name of a rule, specific to a graph ex: Y = X3 which can be used to find the gradient at any point of the graph. The 'X' value is substituted in the equation and this gives the exact gradient for that specific graph. I plan to find the gradient using these two methods: Tangent Method: The tangent method involves making a tangent at a point on the graph on which the gradient is to be found. The figure below shows a tangent: To find the gradient of a straight line, we use the equation: Y2-Y1 X2-X1 This equation would give us the gradient. [X1, Y1], [X2, Y2] can be any two points on the graph. This equation can be used only on straight-line graphs. To find the gradient of curves is much more difficult than the straight-line graphs. For the gradient of curves, a tangent is drawn at a point. A tangent is a line,