Chapter 4.8, Problem 44E

(a)

To determine

To verify: The 0 is a root of multiplicity 2 of the function $f\left(x\right)=e^{2\sin x}-2x-1$.

(b)

To determine

To find: The third approximation $x_3$ for the root $x_0=0$ using the Newton’s method and the modified Newton’s method using initial approximation $x_0=0.1$ for the function $f\left(x\right)=e^{2\sin x}-2x-1$.

(c)

To determine

To find: The third approximation $x_3$ for the root of the function $f\left(x\right)=\frac{8x^2}{3x^2+1}$ using the modified Newton’s method and using initial approximation $x_0=0.15$.