Use newtons method to find the second and third approximation

Question

Use newtons method to find the second and third approximation of a root of cos(x^2+5)=x^3 starting with x1=1 as the initial approximation.

 

The second approximation is x2:_______?

 

The third approximation is x3:_______?

Answer 1

The function is f(x) = cos(x^2 + 5) - x^3

and we want to know where y=0 using NEwton's method.

 

We need the derivative. It is

 

-sin(x^2 + 5) (2x) - 3x^2  <--- derivative of cosine is -sin, chain rule applies, and power rule too

 

Here's the website that has the NEwton's Method calculator:

 http://keisan.casio.com/exec/system/1244946907

 

The second and third approximations are 0.9 and the first decimal place is correct.

The error for them is in the second decimal place.

 

After 50 iterations, the approximation is

0.9831508933

correct to 8 decimal places.

 

Examining the graph on desmos.com verifies this solution