1. Prove 2 i=21- and show the upper bound for relative error on rounding is 2 t when t is the number of bits used to represent the mantissa in IEEE 754 standard. 2. Implement the taylor series. 3. Let's assume that we are given the 3rd order Taylor approximation of a function f(x) . Find out Co, C1, C2 and C3 such that, f ( x , ) = T 3 ( X r ) f ( 1 ) ( X r ) = T ( 1) 3 ( Xr ) f ( 2) ( x r ) = T ( 2 ) 3 ( x r ) f ( 3 ) ( X r ) = T ( 3 ) 3 ( X r ) T3 (X) = Co + C1 ( X - Xr) + C2(X - X)2 + C3(X - Xr) 3 4. Find the first 4 terms of the Taylor series for the following functions : a. In x centered at a = 1 b. sin x centered at a = 4 c. (x - 1) ex near x=1. 5. Find the first 3 terms of the Taylor series for the function sin (x) centered at a = 0.5. Use your answer to find an approximate value to sin(x/2 + x/10) . 6. Consider the following function f(x) = e-3x a. Find the 2nd order Taylor approximation around x = 1. b. Also find the 2nd order Taylor approximation around x = 0. Note: The Taylor series or polynomial approximation around reference point 0 is called the Maclaurin series approximation