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Makeup Material for Midterm Exam 1

Due: Thursday, March 15 (in class)

This assignment gives the opportunity to earn at most 60 extra points toward the first mid-term exam. You will not be able to earn more than the total 150 points available on the exam.


  1. [30 pts] Consider the following function:

    f(x)=1(1x)x.

    Obviously, if you cancel the 1's in the numerator f(x)=1 for all values of x. However if you compute f(x) as written above in MATLAB, it does not return 1 when x is small. Using MATLAB we can plot f(x) for x[2ϵm,2ϵm] using the following code

    f = @(x) (1- (1-x))./x;
    x = -2:.01:2;
    x = eps*x;
    plot(x,f(x))
    axis([-2*eps 2*eps 0 2])
    title('Plot of $$f(x) = \frac{1-(1-x)}{x}$$','interpreter','latex');
    xlabel('$x$','interpreter','latex');
    ylabel('$f(x)$','interpreter','latex');

    which produces the following image:

    Please explain why the graph looks like it does. (I suggest testing out f in MATLAB at certain values of x).

    In particular please answer the following questions: (5pts each)

    1. Why is f(x)=0 for 0x<ϵm/4?

    2. Why is f(x)=0 for ϵm/2<x0? Specifically, why is the f zero for twice as long on the left than on the right?

    3. Why does f(x) jump from zero to the value 2 at around x=.250001ϵm?

    4. Why does f(ϵm)=1?

    5. Why does f oscillate around 1 for .25ϵm<x<2ϵm?

    6. Why are there twice as many oscillations for positive x as there are negative x?

  2. [15 pts] Suppose that f(x) has infinitely many continuous derivatives on [a,b] and that ˉx[a,b] is a nth order root for f, namely

    f(ˉx)=0,f(ˉx)=0,,f(n1)(ˉx)=0,f(n)(ˉx)0.

    We want to apply Newton's method to find the root ˉx. Derive a formula relating the i+1th error ei+1=xi+1ˉx for Newton's method to the ith error ei, to leading order in ei. What is the corresponding order of convergence?

  3. [15 pts] Let

    A=[1223]

    Answer the following questions.

    1. Find a vector x such that

    2. Find a vector such that

    3. What are and respectively?