7.2.3 Newton's Method

Question 1

Newton’s Method

Answer 1

• Newton’s method iterates the approximation process and thereby finds successively better approximations for the solution to a function.

Question 2

note

Answer 2

• The tangent line can be used to approximate solutions of functions.
• Graphically, a solution to a function is the x-value where the graph of the function crosses the x-axis. Finding these solutions is actually very complicated. Formulas exist for second- and third-order polynomials, but there are many more functions that you might want to solve.
• However, it is easy to find the point where a line crosses the x-axis. Notice that the line tangent to a curve can be a decent approximation.
• More importantly, if you repeat the process of finding the tangent line, you can refine your guess and get closer and closer to the actual solution.
• Sir Isaac Newton noticed that tangent lines could be used to approximate the solutions to functions. More importantly, he noticed that repeating the process could refine that approximation. This process is known as Newton’s method.
• Newton’s method starts with a guess. Once you have selected the guess, find the equation of the line tangent to the curve at that point.
• Setting the y-value of that equation equal to 0 solves for where the line crosses the x-axis.
• The value where the curve crosses the x-axis is often a better guess than the original.
• Once you have found that second guess, you can repeat the process with that new point and find an even better guess.
• However, Newton’s method does require that the curve
  behave a specific way. In some cases, Newton’s method will produce successively worse guesses.

Question 3

Complete two iterations of Newton’s method for the given function and indicated initial guess.
f(x)=x^2−7,x1=3

Which of the following is equal to x3?

Answer 3

127/48

Question 4

Given the following equation and initial guess, Newton’s method fails to approximate a solution.
(x−2)^3+4,x1=2

Why did Newton’s method fail?

Answer 4

The slope of the function was equal to zero at the initial guess.

Question 5

Apply two iterations of Newton’s method to approximate the x-value of a point of intersection of these two functions using the given initial guess.
f(x)=x^2−3
g(x)=3x+2,
x1=0
What is the value of x3 ?

Answer 5

−70/57

Question 6

Complete two iterations of Newton’s method for the function and indicated initial guess.
f(x)=2x^2−3,x1=1

Which of the following is equal to x3?

Answer 6

49/40