I then joined up the points with a flexi-curve. This instrument enabled me to obtain an accurate curve hence enabling me to draw accurate tangents to obtain accurate gradients. I then checked the points again once I had plotted the curve to make sure that I had not misplaced any points because this could affect my tangents. I then drew the tangents to do this I first took a ruler and a sharp pencil, I then aligned the ruler along the curve making sure that the tangent that was about to be drawn was equally balanced along the line and that the two closest points were the same distance away from the tangent. We then used the various methods available to calculate the gradient of the tangent, one such method is shown in Figure 1 above. We first had to take a point from the x axis which I took as a whole number each time for example 3 which when followed up to the tangent and in turn followed across to the y axis showed 9. We then took another point along the x axis and then followed that up to the tangent and then followed that across to the y axis. I then recorded the results. The method is shown in greater detail on the next page. Here is an example of how to find the gradient.
The other method to calculate the gradient via the tangent is;
The results for the gradients using the tangent method are shown below by showing the points on the x axis where the tangents were drawn and the gradients obtained by these points.
Even though I used an accurate method, which was the tangent method I felt that I could have, improved and found the gradients using a method, which could check and maybe even improve values of the gradients. This method is called the small increment method and is shown below. This method is calculated first by taking a point on the x axis and then square rooting it. You then take another point that is close to the original point and you then square root the x axis point and you follow the method below. Though one rule is that the original point must always remain the same.
Using the small increment method the results for my calculations were;
These results were the same as the results found using the tangent method. The calculations used to find the gradients in the small increment method can be found on the next page and the graph for y = x2 can also be found.
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Choose two other graphs of the type y=xn. Obtain the gradient of the tangent at different points. Record the results.
The two graphs I have chosen are y = x1 and y = x3 these graphs both carry on the sequence of y = xn and both will enable me two draw conclusions and observe patterns in the gradients. I first had to discover the x values for each graph.
I then used these values and I plotted a graph. I then joined the points up to create a curve. I then drew the tangents as accurately as possible and worked out the gradients.
y = x1
y = x3
I then used the small increment method for the calculation of the gradient and I acquired the results;
y = x1
y = x3
