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UCB Math 128A, Spring 2020: Midterm Exam
Prof. Persson, April 9, 2020
Instructions:
❼ This exam is an open book, unproctored online exam.
❼ Please write your name and your SID on your first page.
❼ Write your answers on a separate page for each problem.
❼ You have 80 minutes to work on the exam and 15 minutes to upload your work to Gradescope.
❼ You must justify your answers for full credit.
❼ You may access all the course material, including notes/textbooks, homework/project solutions, and your own notes.
❼ But you may not:
– Use any other material that was not part of the course, including any internet resources.
– Use any computing environment such as MATLAB, calculators, or any other programming language.
– Have any type of communication with anyone about the exam. This includes direct calls, texts, or emails, but also indirect communication e.g. through shared online documents or by uploading the problems to any websites.
❼ At the conclusion of the exam, please copy and sign the honesty statement in the last problem. If we or the university discover any violation of these rules, it will be reported to the UC Berkeley Office of Student Conduct.
1. Consider the equation 
a) (1 point) Show that the equation has at least one solution.
b) (2 points) Show that the equation has exactly one solution p.
2. Consider the fixed point iteration pn+1 = g(pn), where 
a) (3 points) Show that 
converges to a unique fixed point p for any initial guess p0 ∈ R.
b) (2 points) Find another fixed point iteration pn+1 = g2(pn) which converges to the same fixed point p but quadratically, provided the initial guess is close enough (including a justification for the quadratic convergence).
3. Consider the function f(x) = |x|.
a) (3 points) Find a polynomial p(x) which interpolates the function f(x) at the points x = −2, −1, 0, 1, 2.
b) (2 points) Suppose you use p(x) to approximate the original function f(x). Using the remainder term for interpolating polynomials, what bound can you obtain for the error if x ∈ [−2, 2]?
4. (3 points) Find an O(h2) formula to approximate f‘’ (x0) that uses f(x0 − h), f(x0), and f(x0 + 2h). Include the details of the error term.