1. (10 pts) Write the decimal representation for the following floating point number in binary

2. (10 pts) If the bisection method is used to find the solution for f(x) =0 starting at the interval [a,b], how many iterations are necessary so that the error /x_k-x_k-1/ < epsilon? Consider the interval [a,b] as the first iteration.

3. (10 pts.) Use the false position method to compute c₀, c₁, c_2, c₃ using the function e^x - 2 - x = 0 starting at the interval [a₀,b₀]=[-2.4, -1.6]. Also, describe the type of converge/divergence.

4. (15 pts) The Newthon method works by approximating the function f(x) = 0 using a line at the point x_k-1 and the first derivative. Write an improved method to determine x_k that approximates f(x) using a parabola that passes through the points x_k-1 and x _k-2, and uses the first and second derivative. If the iteration equation gives more than one root, choose x_k to be the closest root to x_k-1. Write the iteration equation for this method.

5. (15 pts) Solve the following system of linear equations using gaussian elimination.

6. (10 pts.) Use both the Gauss-Seidel and the Jacobi iteration methods to find (x_k,y_k) for k = 1, 2, 3. Start at the point (x₀,y₀)=(0,0). Choose iteration equations that will make the method converge.

-x +3y = 1
6x -2y = 2

7. (10 pts.) Let f(x)= 2sin(3.14x/6), where x is in radians. Use quadratic Lagrange interpolation based on the nodes x₀=0, x ₁=1, and x₂=3 to approximate f(4) 

8. (10 pts.) Given the following table, compute the divided-difference table for the function x^1/2. Also, write down the Newthon polynomials P1(x), P2(x), and P3(x).

9. (10 pts.) Using Pade' approximations, find R_2,2(x) for f(x)=arctan(x^1/2)/x^1/2. Start with the Maclaurin expansion: 

f(x)=1 - x/3 + x²/5 - x³/7 + x⁴/9