I have been informed that on March 7th from 6:00am to 6:00pm Central Time Lamar University will be doing some maintenance to replace a faulty UPS component and to do this they will be completely powering down their data center.
Unfortunately, this means that the site will be down during this time. I apologize for any inconvenience this might cause.
Paul
February 18, 2026
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 4.13 : Newton's Method
For problems 1 – 3 use Newton’s Method to determine \({x_{\,2}}\) for the given function and given value of \({x_{\,0}}\).
- \(f\left( x \right) = 7{x^3} - 8x + 4\), \({x_{\,0}} = - 1\)
- \(f\left( x \right) = \cos \left( {3x} \right) - \sin \left( x \right)\), \({x_{\,0}} = 0\)
- \(f\left( x \right) = 7 - {{\bf{e}}^{2x - 3}}\), \({x_0} = 5\)
For problems 4 – 8 use Newton’s Method to find the root of the given equation, accurate to six decimal places, that lies in the given interval.
- \({x^5} = 6\) in \(\left[ {1,2} \right]\)
- \(2{x^3} - 9{x^2} + 17x + 20 = 0\) in \(\left[ { - 1,1} \right]\)
- \(3 - 12x - 4{x^3} - 3{x^4} = 0\) in \(\left[ { - 3, - 1} \right]\)
- \({{\bf{e}}^x} = 4\cos \left( x \right)\) in \(\left[ { - 1,1} \right]\)
- \({x^2} = {{\bf{e}}^{2 - {x^{\,2}}}}\)in \(\left[ {0,2} \right]\)
For problems 9 – 12 use Newton’s Method to find all the roots of the given equation accurate to six decimal places.
- \(2{x^3} + 5{x^2} - 10x - 4 = 0\)
- \({x^4} + 4{x^3} - 54{x^2} - 92x + 105 = 0\)
- \(\displaystyle \frac{3}{2} - {{\bf{e}}^{ - {x^{\,2}}}} = \cos \left( x \right)\)
- \(\ln \left( x \right) = 2\cos \left( x \right)\)
- Suppose that we want to find the root to \({x^3} - 7{x^2} + 8x - 3 = 0\). Is it possible to use \({x_{\,0}} = 4\) as the initial point? What can you conclude about using Newton’s Method to approximate roots from this example?
- Use the function \(f\left( x \right) = {\cos ^2}\left( x \right) - \sin \left( x \right)\) for this problem.
- Plot the function on the interval \(\left[ {0,9} \right]\).
- Use \({x_{\,0}} = 4\) to find one of the roots of this function to six decimal places. Did you get the root you expected to?
- Use \({x_{\,0}} = 5\) to find one of the roots of this function to six decimal places. Did you get the root you expected to?
- Use \({x_{\,0}} = 6\) to find one of the roots of this function to six decimal places. Did you get the root you expected to?
- What can you conclude about choosing values of \({x_{\,0}}\) to find roots of equations using Newton’s Method.
- Use \({x_{\,0}} = 0\) to find one of the roots of \(2{x^5} - 7{x^3} + 3x - 1 = 0\) accurate to six decimal places. Did we choose a good value of \({x_{\,0}}\) for this problem?