This exercise explores how to use Newton’s method to evaluate an inverse function. To explain, given  = ( ), then the inverse function satisfies  = −1 ( ). The problem is, given , what is the value of...

This exercise explores how to use Newton’s method to evaluate an inverse function. To explain, given
 =
(), then the inverse function satisfies
 =

⁻¹(). The problem is, given
, what is the value of
?

(a) Assuming
 is given, and setting() =
 −
(), show that Newton’s method (2.10) gives

(b) Use the result from part (a) to evaluate

²
and

⁻³
using the ln() function.

(c) Use the result from part (a) to evaluate arccos(1/2) and arccos(1/3).

(d) The error function is defined as

The inverse error function is denoted as erf⁻¹(). Use the result from part (a) to evaluate erf⁻¹(1/2) and erf⁻¹(1/3). In doing this, you can use MATLAB’s erf command to evaluate the error function.

(e) The complete elliptic integral of the first kind is defined as

The inverse function is denoted as

⁻¹(). Use the result from part (a) to evaluate

⁻¹(2) and

⁻¹(4). It is useful to know that

where() is the complete elliptic integral of the second kind. In doing this, you can use MATLAB’s ellipke command to evaluate
 and
.