In the introduction of the exercise, it mentions that the fixed point of the function g(x) is the point where the y = x line intersects the y = g(x) curve. And since it was given that g(x) = cos x, a substitution can be done to say that the fixed point, is the point where the y = x line intersects the g(x) = cos x curve. When a close up is made, as shown below, it can be seen that the fixed point is between x = 0 and x = 2 as suggested. Therefore, the fixed point is (0. 739, 0. 739).
b) Using the iteration xn+1 = g(xn) and the initial value x1 = 1, find the first five iterations and explain graphically why the iteration is approaching the fixed point of the function g(x). Here, several substitutions must be made in order to graph the iteration. First, we know that when n = 0, x1 = 1. When we increase the value of n by 1, x2= g(x1), but from part (a), we also know that g(x) = cos x. So from this, we can also state that that is the same thing as g(x1) = cos x1. Hence, if we make the substitution, it would result to x2= cos x1, where we can further substitute the x1 to 1, making the whole equation
x2= cos 1. This would then be the first iteration. Subsequently, this entire substitution process would be repeated another four times, each time using the previous answer as a substitute. The calculations below will provide a better understanding of the concept. From these calculations, we can see that each x value is used to find the following x value by being multiplied by cosine. When shown graphically, we start using the given x1 = 1. From that point on the x-axis, travel vertically up until the line touches the g(x) = cos x line. Then, travel horizontally, and where it crosses the y-axis is where g(x2) is.
From there, go horizontally to the left, and it will cross the y = x line. When a straight line is drawn directly from that point to the x-axis, the new point on the x-axis will be the value of x2. This can then be repeated continuously until it touches the fixed point. When the graph is drawn without including the horizontal lines back to the y-axis, a diagram known as the cobweb diagram is formed, where a spiral has points which get closer to the fixed point. From the graphs, I discovered that the x values get closer and closer to the fixed point, or converge.
c) Repeat the process, but this time let g(x) = 2cosx. Comment on your results and interpret them graphically. Here, the calculations are still the same, and the fact that x1 = 1 still applies. Therefore, we do the same steps and get the results as shown below. When this is graphed, it can be seen that instead of converging closer into the fixed point, it diverges out. This is interesting because it is caused just by the coefficient added to the cosine, and also because it shows that the coefficient, apart from changing where cosine crosses the y-axis, also has a different role to play.
At this point, only the hypothesis that the steepness/coefficient of the cosine line is what causes the difference between whether the function converges to or diverges against the fixed point. d) Repeat this process for g(x) = a cos x, using some other values between a = 1 and a = 2 to discover which converge to a positive fixed point for g(x). Suggest why this only happens for some values of a. For this, the value of x1 was still assumed to equal 1, while the value of a that was chosen to begin with was a = 1. 1. Then, it was decided that the value of a would keep on increasing until 0. 1 until the point where a divergence happened.