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. \(x_0 = 2\), \(x_0=10\) and \(f(x) = (x-2)(x+6)^2(x-10)^4\)
    2. \(x_0 = -5\), \(x_0=-6\) and \(f(x) = (x+5)\sin((x+6)^2)\)
    3. \(x_0=-6\) and \(f(x) = (x+6)\sin(x+6)\)
  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. \(x_0 = -2\) and \(f(x) = -x^2+2x+1\)
    2. \(x_0 = \frac{\pi}{4}\) and \(f(x) = \cos(x)\)
    3. \(x_0 = 5\) and \(f(x) = \sqrt{x-1}\)
  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 \(\displaystyle o(x)\)

Practice Problems

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