JSFHMAT 155Lab 8 Newton's MethodIt is very useful to have methods for solving f(x)=0. (I know that Maple has the fsolve command but it is useful to know how Maple finds the solution.)
Example: Find a solution to x^2-2=0. In other words find LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1zcXJ0R0YkNiMtSSNtbkdGJDYkUSIyRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnRjhGK0Y4. The first step is to give this function a 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print(); # input placeholderNext we find an approximate solution.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMUYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPkZARj4=print(); # input placeholderLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUkjbW5HRiQ2JFEiMkYnL0YzUSdub3JtYWxGJy8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnRj5GPkZARj4=print(); # input placeholderThe function changes from negative to positive as x goes from 1 to 2. There must be a solution to f(x) =0 between 1 and 2. So our first approximation is 2.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEjYTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjJGJ0Y5LyUrZXhlY3V0YWJsZUdGPUY5print(); # input placeholderThe next step is to find the equation of the tangent line to the graph of f(x) at x=a1. The slope is f'(a1) and the point (a1,f(a1)) in on the line. This means that the equation is 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(); # input placeholderNow let's graph the tangent line and the graph together.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%;What we really want is the place where the parabola crosses the x-axis. The place where the tangent line crosses the x-axis is easy to compute and is an improvement on our first approximation. So that will be second approximation. It is easy to calculate the value of x where the tangent line crosses the axis.To solve L(x) = 0 we just plug in to the formulaf'(a1)*(x-a1)+f(a1) =0 gives x = a1 -f(a1)/f'(a1)So this gives our second approximation - note that I added an evalf( ) command to have a decimal answer.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(); # input placeholderNow we just repeat this to get the next tangent line and 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print(); # input 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%;LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=We can see that the graph and the tangent line cross the x-axis very close now so we expect this to be a good 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print(); # input placeholderLet's do it one more timeLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY3LUkjbWlHRiQ2JVEjTDNGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GLDYlUSJ4RidGL0YyLUY2Ni1RKCZzcmFycjtGJ0Y5RjtGPkZARkJGREZGRkhGSkZNRjUtRiw2JVEiZkYnRi9GMi1GNjYtUSInRidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjExMTExMTFlbUYnRk0tSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUSNhM0YnRi9GMi8lK2V4ZWN1dGFibGVHRj1GOUY5RjUtRjY2LVEnJnNkb3Q7RidGOUY7Rj5GQEZCRkRGRkZIRkpGTUY1LUZebzYkLUYjNidGVS1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORmJwRmJvRmVvRjlGOUY1LUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGYXBGY3BGNUZlbkZdb0Zlb0Y5print(); # input 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%;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print(); # input placeholderJust to be careful we will do one more approximation without finding the tangent line or plotting the graphLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2JVEjYTVGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSJ+RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSYwLjBlbUYnL0ZORlMtRiw2JVEmZXZhbGZGJ0YvRjItSShtZmVuY2VkR0YkNiQtRiM2Jy1GLDYlUSNhNEYnRi9GMi1GNjYtUSgmbWludXM7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjIyMjIyMjJlbUYnL0ZORl5vLUkmbWZyYWNHRiQ2KC1GIzYmLUYsNiVRImZGJ0YvRjItRlk2JC1GIzYlRmduLyUrZXhlY3V0YWJsZUdGPUY5RjlGXHBGOS1GIzYnRmVvLUY2Ni1RIidGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMTExMTExMWVtRidGVEZob0ZccEY5LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZqcC8lKWJldmVsbGVkR0Y9RlxwRjlGOUZccEY5print(); # input placeholdera5 and a4 agree on 1.41421 so it seems safe to say that 1.4142 is good to 4 decimal places. We can check by using the fsolve( ) 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print(); # input placeholderIn fact a5 is good to 9 decimal places. Here is another example:PkkiZkc2ImYqNiNJInhHRiRGJDYkSSlvcGVyYXRvckdGJEkmYXJyb3dHRiRGJCwoKiQ5JCIiJCIiIiokRi0iIiMhIiRGL0YvRiRGJEYkprint(); # input placeholderLUklcGxvdEc2IjYjLUkiZkdGJDYjSSJ4R0Yk%;LUklcGxvdEc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkLUkiZkdGJzYjSSJ4R0YnL0YsOyEiJCIiJQ==%;Now we have to choose which of the roots we wish to find because there are obviouslu 3 of them. It depends on our initial point. Let us start with x=3JSFHPkkjYTFHNiIiIiQ=print(); # input placeholderPkkjTDFHNiJmKjYjSSJ4R0YkRiQ2JEkpb3BlcmF0b3JHRiRJJmFycm93R0YkRiQsJiomLUklZXZhbEclKnByb3RlY3RlZEc2JC1JJWRpZmZHRi82JC1JImZHRiQ2IzkkRjcvRjdJI2ExR0YkIiIiLCZGN0Y6RjkhIiJGOkY6LUY1NiNGOUY6RiRGJEYkprint(); # input placeholderJSFHJSFHQyU+SSNhMkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCZJI2ExR0YlIiIiKiYtSSJmR0YlNiNGK0YsLUklZXZhbEdGKDYkLUklZGlmZkdGKDYkLUYvNiNJInhHRiVGOS9GOUYrISIiRjtGLC1GLzYjRiQ=print(); # input placeholderQyU+SSNhM0c2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCZJI2EyR0YlIiIiKiYtSSJmR0YlNiNGK0YsLUklZXZhbEdGKDYkLUklZGlmZkdGKDYkLUYvNiNJInhHRiVGOS9GOUYrISIiRjtGLC1GLzYjRiQ=print(); # input placeholderQyU+SSNhNEc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCZJI2EzR0YlIiIiKiYtSSJmR0YlNiNGK0YsLUklZXZhbEdGKDYkLUklZGlmZkdGKDYkLUYvNiNJInhHRiVGOS9GOUYrISIiRjtGLC1GLzYjRiQ=print(); # input placeholderNow let's start with b1=1PkkjYjFHNiIiIiI=print(); # input placeholderQyU+SSNiMkc2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCZJI2IxR0YlIiIiKiYtSSJmR0YlNiNGK0YsLUklZXZhbEdGKDYkLUklZGlmZkdGKDYkLUYvNiNJInhHRiVGOS9GOUYrISIiRjtGLC1GLzYjRiQ=print(); # input placeholderQyU+SSNiM0c2Ii1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLCZJI2IyR0YlIiIiKiYtSSJmR0YlNiNGK0YsLUklZXZhbEdGKDYkLUklZGlmZkdGKDYkLUYvNiNJInhHRiVGOS9GOUYrISIiRjtGLC1GLzYjRiQ=print(); # input placeholderSo we are getting the middle solution.This is a good method because we can write a little Maple program to do this over and over again or use the existing Maple program We can do this by going to Tools on the menu bar, choosing Tutors and Calculus Single Variable and then Newton's Method QyQtJiZJKFN0dWRlbnRHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2I0kqQ2FsY3VsdXMxR0YqNiNJM05ld3RvbnNNZXRob2RUdXRvckdGKkYqIiIiJSFHAssignment #1 Use Newton's Method to find an approximate solution for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictSSZtc3FydEdGJDYjLUkjbW5HRiQ2JVEjMThGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKzJEfkNvbW1lbnRGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRidGK0Y5 starting with an integer that you think is a good first approximation. For the first three steps in the process draw the graph of f(x) and the tangent line approximation. For steps after that you need only compute the next approximation. Continue until you get two solutions in a row that differ by less than .000001. Assignment #2 Let f(x)= 3*x^5 + 12*x^3 - x^2 -100. Use Newton's method to find a solution to f(x)= 0 starting with an integer that you think is a good first approximation. For the first three steps in the process draw the graph of f(x) and the tangent line approximation. For steps after that you need only compute the next approximation. Continue until you get two solutions in a row that differ by less than .000001. JSFH