-4 < α > -3
-1 < β > 0
0 < γ > 1
For one root, I worked out where the change of sign took place to 5 sig.fig. The first step to complete this process was to start with entering -4 to -3 (in 0.1’s) into Excel and the formula that I was using into the column next to it to show me where the change of sign was – between -3.5 and -3.4.
The second step was to move closer by entering into Excel by entering -3.5 to -3.4 (in 0.01’s) with the same formula in the next column. This gave me the answer as to where the change of sign was – between -3.5 and -3.49.
After that, I moved in closer, entering -3.5 to -3.49 (in 0.001’s) with the formula in the column next to it. This gave me the answer as to where the change of sign was – between -3.491 and -3.49.
Finally, I entered the numbers between -3.491 and -3.49 (in 0.0001’s) into Excel. This showed me that the change of sign was between -3.4909 and -3.4908.
This is now up to 5 sig.fig. meaning that I have found the answer with the error bounds on the left hand side. The error bounds show that this works as there is a change of sign.
FAILURES
This method fails in two ways – when there is a repeated root or there are two roots within consecutive integers. To show how this method fails, I am using the equation x3+x2-5x+3=0. To do this, I will be drawing the function of f(x)= x3+x2-5x+3.
Method 2: The Newton-Raphson Method
Root α
Root β
For the second method, the Newton-Raphson Method, I will be using a different function. The equation I will be using is x3-7x2+x+3=0. This means that I will be drawing the function of f(x)=x3-7x2+x+3.Root γ
Firstly, I started by entering the equation into Autograph to get a sketch of the graph. This allowed me to work out that the three real roots of the function were between the integers of -1 and 7. These are:-1 < α > 0 0 < β > 1 6 < γ > 7
The first root I have chosen to use is Root α. Below is the Excel spread sheet with f(xn) and its derivative, f’(x) shown.
Root α
On the right is the equation that shows the sliding tangent of the Newton-Raphson Method in a graphical form for Root α. The red line is the function of f(x)=x3-7x2+x+3 and the pink lines are the sliding tangents.The second root I have chosen to use is Root β. Below is the Excel spread sheet with f(xn) and its derivative, f’(x) shown.
Root β
On the left is the equation that shows the sliding tangent of the Newton-Raphson Method in a graphical form for Root β. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents. The third root I have chosen to use is Root γ. Below is the Excel spread sheet with f(xn) and its derivative, f’(x) shown.
Root γ
On the left is the equation that shows the sliding tangent of the Newton-Raphson Method in a graphical form for Root y. The red line is the function of f(x)=x3-7x2+x+3 and the blue lines are the sliding tangents. Root γ = 2.4023 (5 sig.fig.).This root has error bounds
FAILURES
y=2x3-3x2-6x+4
Root γ
Method 3: The Rearrangement Method
The third method that I am using is the Rearrangement Method. For this method, I will be using a different function to the other two. The equation I will be using for the Rearrangement method is y=x3+2x2-2x-2. This means that the function I will be using is f(x)= x3+2x2-2x-2.