Math 2710 Second Exam Study Guide

Note: This guide is offered without warranty.

The second exam will cover the following topics.

Key topics from Chapter Three

Key Topics from Chapter 4

Sample problems on induction from Gilbert and Vanstone – see for example: 11-18, 29, 31, 33, 37, 43, 45, 47, 53, 56, 65-67, 74, 76.

Prove that \(u(x)\) is a solution to the linear second order differential equation \[ \frac{d^2u}{dx^2}-a\frac{du}{dx}-bu. \]

Let \(t^2-at-b=0\) be the characteristic polynomial of this differential equation and let \(r_0\) and \(r_1\) be its roots. Assume for simplicity that these roots are distinct (so that \(a^2-4b\not=0\).) By expressing \(u(x)\) as a linear combination of \(\exp(r_0x)\) and \(\exp(r_1x)\), derive a formula for \(f(n)\) involving \(r_0\) and \(r_1\).

Sequences and series

Decimal expansions

Prove that an eventually repeating decimal expansion (or base r expansion) converges to a rational number. Prove that a rational number has an eventually repeating decimal (or base r) expansion.

See: Gilbert and Vanstone, Chapter 5, Problems 23-32, 37, 41, 42, 44,