Chapter 5.7

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. For each rational function \(f\) that is given below, classify all zeros of \(f\) by finding the largest natural number \(n\) and a real number \(x_0\) so that \(f\) is \(O((x-x_0)^n).\)
    1. \(f(x) = (x-2)(x+6)^2(x-10)^4\)
    2. \(f(x) = \frac{(x-4)^6(x+5)^3}{x+7}\)
  2. For each function \(f\) and \(x_0\) that is given below, find the largest natural numbers \(m\) and \(n\) so that \(f\) is \(O((x-x_0)^m)\) and \(o((x-x_0)^n).\)
    1. \(f(x) = (x-2)(x+6)^2(x-10)^4\) and \(x_0=2\)
    2. \(f(x) = (x+5)\sin((x+6)^2)\) and \(x_0=-6\)
    3. \(f(x) = (x+6)\sin(x+6)\) and \(x_0=-6\)
  3. For each function \(f\) and each real number \(x_0\), write a formula for \(\Delta_{x_0}f(h)\) and simplify the expression as much as possible.
    1. \(f(x) = -x^2+2x+1\) and \(x_0 = -2\)
    2. \(f(x) = \cos(x)\) and \(x_0 = \tfrac{\pi}{4}\)
    3. \(f(x) = \sqrt{x-1}\) and \(x_0 = 5\)
  4. Take \(f\) and \(g\) to be the functions that are given by \[g(x) = 4+\sin(x) \quad \text{and}\quad f(x) = 1-\cos(x-4).\] Perform the appropriate calculations in order to justify the following statements.
    1. \(g-g(0)\) is \(\displaystyle O(x)\)
    2. \(f\) is \(\displaystyle o(x-g(0))\)
    3. \(f\circ g\) is \(\displaystyle o(x)\)

Remember, do the knowledge checks before checking the solutions.

Practice Problems

Return

© Copyright 2025 by the POC Writing Team: Bryan Carrillo, Yat Sun Poon, and David Weisbart. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the POC Writing Team.