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Learning Goal: I’m working on a linear algebra multi-part question and need an explanation and answer to help me learn.The problems that follow illustrate the methods covered in class. They are typical of
the types of problems that will be on the tests.
1. Solving Equations
Problem 1. Suppose that f : R → R is continuous and suppose that for a < b ∈ R,
f(a) · f(b) < 0. Show that there is a c with a < c < b such that f(c) = 0.
Problem 2. Solve the equation x
5 − 3x
4 + 2x
3 − x
2 + x = 3. Solve using the Bisection
method. Solve using the Newton-Raphson method. How many solutions are there?
Problem 3. Solve the equation x = cos x by the Bisection method and by the NewtonRaphson method. How many solutions are there? Solve the equation sin(x) = cos x by the
Bisection method and by the Newton-Raphson method. How many solutions are there?
Problem 4. Let h be a continuous function h : Rn → Rn
. Let x0 ∈ Rn
. Suppose that
h
n
(x0) → z as n → ∞. Show that h(z) = z.
Problem 5. Solve the equation x
4 = 2 by the Newton-Raphson method. How many real
solutions are there? For which starting values x0 will the method converge?
Requirements: Not Applicable