10.6
Newton’s Method
Objectives
Use Newton’s Method to approximate the
zeros of functions.
Understand the situations in which
Newton’s Method may not converge.
1. Newton’s Method
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Zeros of a Function
Def : A real zero of a function is a real number
that makes the value of the function equal to zero.
Graphically, the real zero
of a function is where the
graph of the function
Crosses the x‐axis;
that is, the real zero
of a function is the
The x-intercepts of the function are
x‐intercept(s) of
(x1, 0), (x2, 0), (x3, 0), and (x4, 0).
This means that for the graph shown
the graph
above, its real zeros are
of the function.
{x1, x2, x3, x4}.
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Newton’s Method
Finding the zeros of a function f (x) is
Solving the equation f (x) = 0
that a fundamental problem in algebra and in calculus.
How to find the zeros of a function?
polynomial function
rational function
other functions
In some cases, you can use various approximation methods
to find the zeros. One such method, called Newton’s
Method, is described in this section.
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Newton’s Method
Consider a function f, such as the one whose graph is
shown in Figure.
Staring x1
The x-intercept (x2,0) of the tangent line approximates the
zero of f.
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Newton’s Method
An actual zero of the function is x = c. To approximate this
zero, choose x1 close to c and form (the first-degree Taylor
polynomial centered at x1)=equation of tangent line thru
(x1, f (x1))
Graphically, you can interpret this polynomial as the
equation of the tangent line to the graph of f at the point
(x1, f(x1)).
Newton’s Method is based on the assumption that this
tangent line will cross the x-axis at about the some point
more near c.
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Newton’s Method
With this assumption, set S1(x) equal to zero, solve for x,
and use the resulting value as a new, approximation of the
actual zero c.
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Newton’s Method
So, from the approximation x1, you form a second
approximation,
To obtain an even
better approximation,
you can use x2 to calculate x3,
Repeated application of this
process is called Newton’s
Method.
.
Staring x1
First estimate
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Newton’s Method
牛頓求根法 是一個演算法 , 需要跑軟體幫我們做複雜的計算
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Example 1 – Using Newton’s Method
Calculate three iterations of Newton’s Method to approximate
a zero of
Use x1 = 1 as the initial guess.
Solution:
The first derivative of f is f(x) = 2x. So, the iterative formula
for Newton’s Method is
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Ex 1
– Solution cont’d
The calculations for three iterations are shown in the table.
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Example 1 – Solution
cont’d
The first iteration is depicted graphically.
So, the approximation is 1.414216.
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Example 1 – Solution
cont’d
For this particular function, you can easily determine the
exact zero to be
So, after only three iterations of Newton’s Method, you can
obtain an approximation that is within 0.000002 of an actual
root.
x3 = 1.414216
- 1.414214…
error 0.000002
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Example 3 – Finding a Point of Intersection
Use Newton’s Method to estimate the x-value of the point
of intersection of the graphs of y = e–x and y = x. Continue
the iterations until two successive approximations differ by
less than 0.001.
Solution:
y = e–x
解
之 x-value
y=x
which implies that
hence
To use Newton’s Method, let
iterative formula
and form the
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Ex 3
– Solution
cont’d
The table shows two iterations of Newton’s Method
beginning with an initial approximation of x1 = 0.5.
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Example 3 – Solution
cont’d
From the table, you can estimate that the point of intersection
occurs when x  0.56715, as shown in Figure.
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2. Convergence of
Newton’s Method
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Convergence of Newton’s Method
When, the approximations approach a zero of a function,
Newton’s Method is said to converge.
You should know, however, that Newton’s Method does not
always converge.
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Convergence of Newton’s Method
Two situations in which it may not converge are
1. when f(xn) = 0 for some n. 切線無法交 x-axis.
n =1
n=2
Figure 1 :Newton’s Method fails to converge when f (xn) = 0.
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Convergence of Newton’s Method
2. when
does not exist. (See Figure 2.)
Newton’s Method fails to converge for every x-value other than the actual zero of f.
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Convergence of Newton’s Method
The type of problem illustrated in Figure 1 can usually be
overcome with a better choice of x1.
The problem illustrated in Figure 2, however, is usually
more serious.
For instance, Newton’s Method does not converge for any
choice of x1 (other than the actual zero) for the function
See 手稿
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10.6 Homework
上完課會公告 Exercises 10.6 的題目
請交第 1, 19, 20
共3題
其他數字太雜需要計算機幫忙故址挑3題
基本上此節交一個演算法求函數零值
或說是 解方程式的根
本身就是用來寫程式跑電腦的
我們儘做些介紹無法在此實際操作電腦
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Review : series
The nth-Term Test for Divergence
Geometric series Test
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Review : series

Thm :
then
a
n 1
n
Integral Test :
p-series Test :
+ …..
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Comparison Test
欲知
已猜
去找來比較的
證明完成 猜測正確
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欲知
去找來比較的
趨近0的階一樣大
趨近0的階an 比較大
趨近0的階bn 比較大
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絕收必收
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Ratio Test
倒數就是無窮小階
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