^{3}- x - 1 = 0 between 1 and 2 by using the bisection method.

2. Find the root of f(x) = e

^{x}- 4x = 0 using the Regula-Falsi method.

3. Use the R-F method to find the roots of the equation e

^{-x}- x = 0. Assume the two initial guess values as 0 and 1.

4. Using Newton-Raphson method, solve the equation x - e

^{-x}= 0.

5. Develop a C program to evaluate the root of a function of the form f(x) = 0 using the Newton-Raphson method.

6. Derive the relation E = 1 + Δ

7. Derive the relation E

^{-1}= 1 - ∇

8. Show that E

^{n}f(x) = f(x + nh)

9. Show that E

^{-n}f(x) = f(x - nh)

10. Show that EE

^{-1}= 1

11. Show that the nth difference of a polynomial of degree n is constant.

12. Find the polynomial of degree 3 which takes the values as shown below:

f(0) = 1, f(1) = 1, f(2) = 2, and f(4) = 5.

[Hint: Since points are equally spaced, use Lagrange's Interpolation technique.]