MATH 5610/6860PRACTICE MIDTERM EXAMProblem 1. Let αn→ 0, xn= O(αn) and yn= O(αn).Show that xnyn= o(αn).Problem 2.(a) Write the Taylor expansion of ln(1+x) about x = 0 with the Lagrangeform of the residual term.(b) Assume the Taylor series for ln(1 + x) is truncated after the terminvolving x10and is used to approximate the number ln 2. What boundon the error can be given?Problem 3. Consider the number x = 26+ 2−16+ 2−19.(a) Write x in scientific base 2 (binary) notation of the form (1.b1b2b3. . . bn)×2S.(b) If IEEE single precision is used, the number of bits above is limited to23. What is x+(floating point number immediately above x) and x−(floating point number immediately below x).(c) What is fl(x) (the floating point representation of x, assuming roundto nearest)(d) What is machine precision  in this system?(e) Verify that the relative error between x and fl(x) is less than machineprecision.Problem 4. Halley’s method for solving f(x) = 0 uses the iteration formulaxn+1= xn−fnf0n(f0n)2− (fnf00n)/2,where fn= f(xn) and so on. Show that this formula results from applyingNewton’s method to the function f /√f0.Problem 5. Consider the polynomial p(z) = z4+ 2x3+ 3x2+ 3z + 2.(a) Compute p(2) using Horner’s method(b) Compute p0(2) using Horner’s method(c) Write p(z) in the form p(z) = (z − 2)q(z) + r, specifying q(z) and r.Problem 6. Consider a smooth function f with the following values.x -1 0 1 2f(x) -1 0 3 -1(a) Write the polynomial interpolating f (x) at the first three nodes inLagrange form.(b) Use divided differences to find the polynomial p(x) interpolating f(x)in Newton form.(c) Assuming all derivatives of f are available, give an expression of theinterpolation error f(t) − p(t) for some t ∈ [−1, 2].12 MATH 5610/6860 PRACTICE MIDTERM EXAMProblem 7. Let f (x) be a function of x and x0, . . . , xnbe n + 1 distinctnodes. For k = 0, . . . , n, let pkbe the polynomial interpolating f at thenodes x0, x1, . . . , xk. Let q be the polynomial interpolating f at the nodesx1, . . . , xn. Show that:pn(x) = q(x) +x − xnxn− x0(q(x) −