1. Find the real root of the equation x3 - x - 1 = 0 between 1 and 2 by using the bisection method.
2. Find the root of f(x) = ex - 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 Enf(x) = f(x + nh)
9. Show that E-nf(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.]