Show at least two different ways to prove that the equation x = 2−x has
exactly one real solution.
- (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). - (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
. - (10 points) As h → 0, find the rate of convergence of the function
F(h) =
sin h − h +
h
3
6
h
5
. - (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). - (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. - (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
