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Use the Newton’s method to approximate the solution of the equation: 1-x-tan x = 0 starting at x0 = 0.5 (the initial value). i.e. Find x1,x2,x3.. .

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Use the Newton’s method to approximate the solution of the equation: 1-x-tan x = 0 starting at x0 = 0.5 (the initial value). i.e. Find x1,x2,x3.. .

Oct 28, 2014

#1
+117762
+5

Newton's method

You are probably supposed to know the formula but I can't remember formulas so I work it out ever time.

$$Remember, the gradient of the tangent at x=x_1  is given by f'(x_0)\\ so\\\\ f'(x_0)=\frac{f(x_0)-0}{x_0-x_1}\\\\ f'(x_0)=\frac{f(x_0)}{x_0-x_1}\\\\ x_0-x_1=\frac{f(x_0)}{f'(x_0)}\\\\ -x_1=-x_0+\frac{f(x_0)}{f'(x_0)}\\\\ x_1=x_0-\frac{f(x_0)}{f'(x_0)}\\\\ Now you have the formula\\\\ \boxed{x_1=x_0-\frac{f(x_0)}{f'(x_0)}}\\\\$$

1-x-tan x = 0 starting at x0 = 0.5

Let

$$\\f(x)=1-x-tanx\\ f'(x)=-1-sec^2x\\\\ f(0.5)=1-0.5-tan0.5\\ f(0.5)=0.5-0.546\\ f(0.5)=-0.046\\\\ f'(0.5)=-1-sec^2(0.5)\\ f'(0.5)=-1-(1/cos^2(0.5))\\ f'(0.5)=-2.298\\ so\\ x_1=0.5-\frac{f(0.5)}{f'(0.5)}\\\\ x_1=0.5-\frac{-0.046}{-2.298}\\\\ x_1=0.480$$

I have applied newtons method just once.

If you want more accuracy you can apply it again and again until you are happy. :)

Oct 28, 2014

#1
+117762
+5

Newton's method

You are probably supposed to know the formula but I can't remember formulas so I work it out ever time.

$$Remember, the gradient of the tangent at x=x_1  is given by f'(x_0)\\ so\\\\ f'(x_0)=\frac{f(x_0)-0}{x_0-x_1}\\\\ f'(x_0)=\frac{f(x_0)}{x_0-x_1}\\\\ x_0-x_1=\frac{f(x_0)}{f'(x_0)}\\\\ -x_1=-x_0+\frac{f(x_0)}{f'(x_0)}\\\\ x_1=x_0-\frac{f(x_0)}{f'(x_0)}\\\\ Now you have the formula\\\\ \boxed{x_1=x_0-\frac{f(x_0)}{f'(x_0)}}\\\\$$

1-x-tan x = 0 starting at x0 = 0.5

Let

$$\\f(x)=1-x-tanx\\ f'(x)=-1-sec^2x\\\\ f(0.5)=1-0.5-tan0.5\\ f(0.5)=0.5-0.546\\ f(0.5)=-0.046\\\\ f'(0.5)=-1-sec^2(0.5)\\ f'(0.5)=-1-(1/cos^2(0.5))\\ f'(0.5)=-2.298\\ so\\ x_1=0.5-\frac{f(0.5)}{f'(0.5)}\\\\ x_1=0.5-\frac{-0.046}{-2.298}\\\\ x_1=0.480$$

I have applied newtons method just once.

If you want more accuracy you can apply it again and again until you are happy. :)

Melody Oct 28, 2014