Ways to prove that the equation

Show at least two different ways to prove that the equation x = 2−x has
exactly one real solution.

  1. (10 points) Suppose f ∈ C[a, b], that x1 ≤ x2 . . . ≤ xn are in [a, b]. Show that there
    exists a number ξ between x1 and xn with, f(ξ) = 1
    n
    Xn
    i=1
    f(xi).
  2. (10 points) Suppose function f has a continuous third derivative. Show that:





    −3f(x) + 4f(x + h) − f(x + 2h)
    2h
    − f
    0
    (x)





    ≤ ch2
    .
  3. (10 points) As h → 0, find the rate of convergence of the function
    F(h) =
    sin h − h +
    h
    3
    6
    h
    5
    .
  4. (25 points) Consider the function f(x) = ln(x).
    (a) Find the Taylor polynomial of degree n about x0 = 1. Write the simplified
    expressions for the polynomial approximation Pn(x) and the remainder Rn(x).
    Write a computer program (in MATLAB or PYTHON) to approximate f(x) by
    the polynomial approximation for n terms. Include in your code a plot of the
    true function f(x) compared to the linear, quadratic and cubic approximations.
    Attach a copy of the code and output.
    (b) Find the degree n that will guarantee an accuracy of 10−3 when ln(1.5) is approximated by Pn(1.5) using the result from part(a).
  5. (25 points) Consider the sequence {xk} defined by xk+1 =
    x
    2
    k + 9
    2xk
    , k = 0, 1, 2, . . . ,.
    (a) Show that for the initial guess x0 = 4, the sequence has a limit x
    ∗ = 3.
    (b) Show that the convergence of the sequence to the limit x
    ∗ = 3 is quadratic.
    (c) Write a computer program (in MATLAB or PYTHON) that will implement the
    recursive relation to compute the first 10 terms of the sequence and print them.
    Attach a copy of the code and output.
  6. (25 points) Consider finding the integral: I(x) = Z x
    0
    sin(t
    2
    ) dt. While this integral
    cannot be evaluated in terms of elementary functions, the following approximating
    technique may however be used.
    (a) Derive a Taylor Series expansion about x = 0 for I(x).
    (b) Write a computer program (in MATLAB or PYTHON) to approximate I(x) by
    the approximation in part (a) for n terms. Use the program to plot the approximation of I(x) for 2 terms, for 5 terms and for 10 terms. Plot the three approximate
    functions respectively by plotting over the domain [0, 1]. Attach a copy of the
    code and output.

Sample Solution

ACED ESSAYS