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4.6 Newton’s method
y = f(x)
x2
x1
x1
x1
x1 is initial guess at root (or zero)
The equation of tangent line to graph of f(x) at (x1, f(x1)) is
Let (x2, 0) denote the place where the tangent line crosses the x-axis. So (x2, 0) is on the tangent line
and thus we have
Solving for x2, we get
Repeat and get x3
In general
xn 1  xn 
f (xn )
f (xn )
EXAMPLE
Use Newton’s method with initial guess of x1 = 1 to approximate a zero of f (x)  x 3  x 2  1 .
EXAMPLE
Use Newton’s method to find
EXAMPLE
Find, correct to six decimal places, the root of the equation cos x = x.
6
2 correct to eight decimal places.