LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Solutions to Nonlinear EquationsExample 1.QyQ+SSRlcTFHNiIvLCoqJEkieEdGJSIiJCIiIiokRikiIiMiIiVGKSEiJCEjPUYrIiIhRis=print(); # input placeholderQyQtSSZzb2x2ZUc2IjYkSSRlcTFHRiVJInhHRiUiIiI=print(); # input placeholderQyQ+SSNmMUc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwqKiQ5JCIiJCIiIiokRi4iIiMiIiVGLiEiJCEjPUYwRiVGJUYlRjA=print(); # input placeholderQyQtSSVwbG90RzYiNiQtSSNmMUdGJTYjSSJ4R0YlL0YqOyEjNSIjNSIiIg==%;Once we see the plot we can zoom on the interval in which we think the roots can be found.QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSNmMUdGKDYjSSJ4R0YoL0YtOyEiJiIiJiIiIg==%;Can we factor the expression? QyQtSSdmYWN0b3JHNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHNiI2Iy1JI2YxR0YoNiNJInhHRigiIiI=print(); # input placeholderWow, it works!What about writing the equation as the equality of two functions? The points at which the two graphs intersect are the solutions of the original equation.QyQ+SSJnRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiQ5JCIiJEYlRiVGJSIiIg==print(); # input placeholderQyQ+SSJoRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJDkkIiIjISIlRi4iIiQiIz0iIiJGJUYlRiVGMw==print(); # input placeholderQyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JC1JImdHRig2I0kieEdGKC1JImhHRihGLS9GLjshIiYiIiYiIiI=%;Again we can see that the two graphs intersect at -3 and 2. Bisection Method We will try to approximate the solution at 2 using the bisection method. We start with two values on one side and the other of the real root such that the function changes its sign exactly once on the interval between the two values. We takeQyg+SSNhMUc2IiIiIkYmPkkjYjFHRiUkIiNEISIiRiY+SSN4MUdGJS1JIipHJSpwcm90ZWN0ZWRHNiQsJkYkRiZGKEYmI0YmIiIjRiY=print(); # input placeholderQygtSSNmMUc2IjYjSSNhMUdGJSIiIi1GJDYjSSNiMUdGJUYoLUYkNiNJI3gxR0YlRig=print(); # input placeholderSince 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 it means that the zero we are looking for is between LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn and LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjYjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn. So we continue the iterations definingQyg+SSNhMkc2IkkjeDFHRiUiIiI+SSNiMkdGJUkjYjFHRiVGJz5JI3gyR0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJw==print(); # input placeholderQyQtSSNmMUc2IjYjSSN4MkdGJSIiIg==print(); # input placeholderNotice that LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEjZjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSN4MkYnRi9GMi9GM1Enbm9ybWFsRidGPS1JI21vR0YkNi1RIj5GJ0Y9LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZFLyUpc3RyZXRjaHlHRkUvJSpzeW1tZXRyaWNHRkUvJShsYXJnZW9wR0ZFLyUubW92YWJsZWxpbWl0c0dGRS8lJ2FjY2VudEdGRS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlQtSSNtbkdGJDYkUSIwRidGPUY9 has the same sign as LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjZjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSNiMkYnRi9GMi9GM1Enbm9ybWFsRidGPUY9 so we will continue to bisect the interval LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkobWZlbmNlZEdGJDYkLUYjNiYtSSNtaUdGJDYlUSNhMkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIixGJy9GOFEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjYvJSlzdHJldGNoeUdGQi8lKnN5bW1ldHJpY0dGQi8lKGxhcmdlb3BHRkIvJS5tb3ZhYmxlbGltaXRzR0ZCLyUnYWNjZW50R0ZCLyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMzMzMzMzM2VtRictRjE2JVEjeDJGJ0Y0RjdGPkY+Rj4=. Therefore the next iteration is Qyg+SSNhM0c2IkkjYTJHRiUiIiI+SSNiM0dGJUkjeDJHRiVGJz5JI3gzR0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJw==print(); # input placeholderQyQtSSNmMUc2IjYjSSN4M0dGJSIiIg==print(); # input placeholderWe will continue until we get LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= such that 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Qyg+SSNhNEc2IkkjeDNHRiUiIiI+SSNiNEdGJUkjYjNHRiVGJz5JI3g0R0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJw==print(); # input placeholderQyQtSSNmMUc2IjYjSSN4NEdGJSIiIg==print(); # input placeholderQyg+SSNhNUc2IkkjYTRHRiUiIiI+SSNiNUdGJUkjeDRHRiVGJz5JI3g1R0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJw==print(); # input placeholderQyQtSSNmMUc2IjYjSSN4NUdGJSIiIg==print(); # input placeholderQyo+SSNhNkc2IkkjeDVHRiUiIiI+SSNiNkdGJUkjYjVHRiVGJz5JI3g2R0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJy1JI2YxR0YlNiNGLEYnprint(); # input placeholderQyo+SSNhN0c2IkkjYTZHRiUiIiI+SSNiN0dGJUkjeDZHRiVGJz5JI3g3R0YlLUkiKkclKnByb3RlY3RlZEc2JCwmRiRGJ0YpRicjRiciIiNGJy1JI2YxR0YlNiNGLEYnprint(); # input placeholderSince 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we stop and therefore the approximate root is LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJzEuOTk2MUYnRjktRjY2LVEiLkYnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZXRjk=Notice that we cannot use this method to approximate the double root LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjNGJ0Y5LUY2Ni1RIn5GJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GV0Y5since the function does not change sign near this root.Newton-Raphson MethodWe will consider the same equation from example 1 but we will apply Newton-Raphson method to find an approximate root. we use the same value for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnJiM5NDk7RicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JI21vR0YkNi1RIj1GJ0YyLyUmZmVuY2VHRjEvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGMS8lKnN5bW1ldHJpY0dGMS8lKGxhcmdlb3BHRjEvJS5tb3ZhYmxlbGltaXRzR0YxLyUnYWNjZW50R0YxLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGSS1JI21uR0YkNiRRJDAuMUYnRjItRjY2LVEiLkYnRjJGOUY7Rj1GP0ZBRkNGRS9GSFEmMC4wZW1GJy9GS0ZURjI= Let us start with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSI9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRkwtSSNtbkdGJDYkUSIxRidGOUY5 and we construct the tangent line to the graph of the function LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEjZjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RidGL0YyL0YzUSdub3JtYWxGJ0Y9Rj0= at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIzEuRidGOUY5 Its equation is 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. Below you see the derivative computed by Maple. The command "unapply" is used in order to work with the derivative as a function. Now LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEoZjFwcmltZUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GM1Enbm9ybWFsRic= is the derivative of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjZjFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn but we can work with it in Maple as a function. QyQ+SSRmcDFHNiItSSVkaWZmRyUqcHJvdGVjdGVkRzYkLUkjZjFHRiU2I0kieEdGJUYtIiIiprint(); # input placeholderQyQ+SShmMXByaW1lRzYiLUkodW5hcHBseUdGJTYkSSRmcDFHRiVJInhHRiUiIiI=print(); # input placeholderNow we are ready to continue with the Newton method. We are goin to plot the function together with the tangent drawn at LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSN4MUYnRi9GMkY1LUkjbW5HRiQ2JFEiMUYnRjlGOQ==. First we define the equation of the tangent line.QyQ+SSh0YW5nZW50RzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYqJiwmOSQiIiIhIiJGMEYwLUkoZjFwcmltZUdGJTYjRjBGMEYwLUkjZjFHRiVGNEYwRiVGJUYlRjA=print(); # input placeholderJSFHQyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JC1JI2YxR0YoNiNJInhHRigtSSh0YW5nZW50R0YoRi0vRi47IiIhIiImIiIi%;Next we define LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEjeDJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn as the intersection of the tangent line with the x-axis and we continue with the algorithm.QyY+SSN4Mkc2IiwmIiIiRicqJi1JI2YxR0YlNiNGJ0YnLUkoZjFwcmltZUdGJUYrISIiRi5GJy1GKjYjRiRGJw==print(); # input placeholderQyY+SSN4M0c2IiwmSSN4MkdGJSIiIiomLUkjZjFHRiU2I0YnRigtSShmMXByaW1lR0YlRiwhIiJGL0YoLUYrNiNGJEYoprint(); # input placeholderQyY+SSN4NEc2IiwmSSN4M0dGJSIiIiomLUkjZjFHRiU2I0YnRigtSShmMXByaW1lR0YlRiwhIiJGL0YoLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtRis2I0YkRig=print(); # input placeholderWe used the command "evalf" just to have the value expressed as a decimal rather than fraction.QyY+SSN4NUc2IiwmSSN4NEdGJSIiIiomLUkjZjFHRiU2I0YnRigtSShmMXByaW1lR0YlRiwhIiJGL0YoLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtRis2I0YkRig=print(); # input placeholderAt this moment we can stop since 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. Therefore we get the approximate solution 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. Notice that the method required fewer iterations than the bisection method.Example 2. We try to approximate the root at 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We start with L0kjeTFHNiIkISIjIiIhQyg+SSN5MUc2IiEiIyIiIj5JI3kyR0YlLCZGJEYnKiYtSSNmMUdGJTYjRiRGJy1JKGYxcHJpbWVHRiVGLiEiIkYxRictSSZldmFsZkclKnByb3RlY3RlZEc2Iy1GLTYjRilGJw==print(); # input placeholderQyY+SSN5M0c2IiwmSSN5MkdGJSIiIiomLUkjZjFHRiU2I0YnRigtSShmMXByaW1lR0YlRiwhIiJGL0YoLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtRis2I0YkRig=print(); # input placeholderQyY+SSN5NEc2IiwmSSN5M0dGJSIiIiomLUkjZjFHRiU2I0YnRigtSShmMXByaW1lR0YlRiwhIiJGL0YoLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtRis2I0YkRig=print(); # input placeholderQyQtSSZldmFsZkclKnByb3RlY3RlZEc2I0kjeTRHNiIiIiI=print(); # input placeholderYou can see that the approximation of the root is not as good as it was in Example 1. Remember that -3 is a double root of the equation.Example 3. We will see how Newton-Raphson method works for an equation having a root of order three.QyQ+SSNmMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSokLCY5JCIiIiEiIkYvIiIkRiVGJUYlRi8=print(); # input placeholderQyQ+SSRmMnBHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJCokLCY5JCIiIiEiIkYwIiIjIiIkRiVGJUYlRjA=print(); # input placeholderQyg+SSN4MUc2IiIiISIiIj5JI3gyR0YlLCZGJEYnKiYtSSNmMkdGJTYjRiRGJy1JJGYycEdGJUYuISIiRjFGJy1JJmV2YWxmRyUqcHJvdGVjdGVkRzYjLUYtNiNGKUYnprint(); # input placeholderQyY+SSN4M0c2IiwmSSN4MkdGJSIiIiomLUkjZjJHRiU2I0YnRigtSSRmMnBHRiVGLCEiIkYvRigtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1GKzYjRiRGKA==print(); # input placeholderIf we keep the same value for 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 get a pretty bad approximation for the zero: 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, quite far from the true root LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRIjFGJ0Y5Rjk=Back to Example 1. We use the Secant Method to estimate the root at 2.JSFHQyo+SSN4MUc2IiIiJCIiIj5JI3gyR0YlIiIlRic+SSN4M0dGJSwmRilGJyooLUkjZjFHRiU2I0YpRicsJkYpRidGJCEiIkYnLCZGL0YnLUYwNiNGJEYzRjNGM0YnLUkmZXZhbGZHJSpwcm90ZWN0ZWRHNiMtRjA2I0YsRic=print(); # input placeholderQyQ+SSdzZWNhbnRHNiJmKjYjSSJ4R0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUsJi1JI2YxR0YlNiNJI3gyR0YlIiIiKigsJjkkRjFGMCEiIkYxLCZGLUYxLUYuNiNJI3gxR0YlRjVGMSwmRjBGMUY5RjVGNUYxRiVGJUYlRjE=print(); # input placeholderBelow is a plot of the graph of the function together with the first secant in the iterative procedure.JSFHQyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQ8JC1JI2YxR0YoNiNJInhHRigtSSdzZWNhbnRHRihGLS9GLjsiIiEiIiYiIiI=%;JSFHQyY+SSN4NEc2IiwmSSN4M0dGJSIiIiooLUkjZjFHRiU2I0YnRigsJkYnRihJI3gyR0YlISIiRigsJkYqRigtRis2I0YuRi9GL0YvRigtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1GKzYjRiRGKA==print(); # input placeholderQyY+SSN4NUc2IiwmSSN4NEdGJSIiIiooLUkjZjFHRiU2I0YnRigsJkYnRihJI3gzR0YlISIiRigsJkYqRigtRis2I0YuRi9GL0YvRigtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1GKzYjRiRGKA==print(); # input placeholderQyY+SSN4Nkc2IiwmSSN4NUdGJSIiIiooLUkjZjFHRiU2I0YnRigsJkYnRihJI3g0R0YlISIiRigsJkYqRigtRis2I0YuRi9GL0YvRigtSSZldmFsZkclKnByb3RlY3RlZEc2Iy1GKzYjRiRGKA==print(); # input placeholderQyQtSSZldmFsZkclKnByb3RlY3RlZEc2I0kjeDZHNiIiIiI=print(); # input placeholderSo the approximate solution, LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1JI21uR0YkNiRRJjIuMDAyRidGOUY5 was found at the sixth iteration. It requierd more steps than the Newton-Raphson methods but fewer than the Bisection method to reach the approximate solution.Example 4.QyQ+SSRlcTJHNiIvLCgqJEkieEdGJSIiJSIiIkYpISMhKSIkPyJGKyIiIUYrprint(); # input placeholderQyQtSSZzb2x2ZUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2IjYkSSRlcTJHRihJInhHRigiIiI=print(); # input placeholderThis hasn't been very helpful. We plot the function. QyQ+SSNmMkc2ImYqNiNJInhHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJSwoKiQ5JCIiJSIiIkYuISMhKSIkPyJGMEYlRiVGJUYwprint(); # input placeholderQyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSNmMkdGKDYjSSJ4R0YoL0YtOyEjOiIjOiIiIg==%;Does the equation have any real solutions? How can we find out? Let us zoom on the graph.QyQtSSVwbG90RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliRzYiNiQtSSNmMkdGKDYjSSJ4R0YoL0YtOyEiJiIiJiIiIg==Aha, it seems that there are two real roots! One is somewhere between 1 and 2 and one is between 3 and 4. Approximate them using the methods learned in this module.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=