## Calculating Equations

Show at least two different ways to prove that the equation x = 2−x has

exactly one real solution.

2. (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).

3. (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

.

4. (10 points) As h → 0, find the rate of convergence of the function

F(h) =

sin h − h +

h

3

6

h

5

.

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).

6. (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.

7. (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.