Section 4.8 ñ Newton’s Method out of Class Project
Due April 18
This project covers Section 4.8 in your textbook: Newton’s Method for approximating the roots of functions, another application of derivatives.
High Level of Calculus in a Math Project
You must read Section 4.8 in your textbook before attempting this project. You are welcome to research the topic online if you would like, but this is not required (if you use online resources, site them).
This assignment must be typed (double spaced) and stapled. The project rubric is posted separately and must be printed and stapled as a cover page. Graphs may be hand-drawn or done on the computer, but may not be copied from any source. Do not plagiarize (copy) from the book, any online sources, or a classmate.
High Level of Calculus in a Math Project Part I
- In your own words, explain the purpose of Newton’s Method and how the method approximates the roots of a function. State the Newton’s Method Formula; include a graph that illustrates the method. You may use a generic graph or a specific function
f (x) .
- How does Newton’s Method relate to the linearization of a function?
- Provide two examples in which the method does not converge to a root and explain.
High Level of Calculus in a Math Project Part II
( ) 5 2
f x ? x ? x ? , use the initial approximation values given below and Newton’s Method to estimate the roots of f (x) to 6 decimal places. Write out the first iteration of the formula for each approximation. Record the calculator outputs for the rest of the iterations. This part may be hand written.
x1 ? ?2
x1 ? ?0.2
x1 ? 3
? See the Calculator Help sheet for completing this on your calculator. Excel also works well.
? You have your approximation when you see the same 6 digits repeated in successive iterations.
- How many real roots does
( ) 5 2
f x ? x ? x ? have? Use the calculator to find the roots of
f (x) accurate to 6 decimal places (see the Calculator Help sheet if needed). List these roots.
High Level of Calculus in a Math Project Part III
- Sketch the graph of the polynomial function from Part II and the tangent lines at each of the three initial (??1
) approximation values (you can use the calculator to find the tangent lines ñ see the
Calculator Help sheet. Desmos or some other program can be use as well).
- Did the initial approximations in Part II, number 1 provide all of the function’s roots? Based on your graph from Part III, number 1, why do you think this happens? What could the initial guesses be to provide all the roots?
- What advice would you give someone trying to use Newton’s Method (what should they think about when choosing their initial values)?