Chapter 5.6

Notes

Linguistic Mapping Exercises

Knowledge Checks

  1. Take \(f\) to be a function with the property that \[\lim_{x\to -1^-} f(x) = 5 = \lim_{x\to -1^+} f(x).\] Determine the value of \(f(-1)\) so that \(f\) is continuous at \(-1\).
  2. Sketch a \(f\) that is continuous at \(x=3\) but not continuous at \(x=4\).
  3. Show that the function \(f\) is continuous at \(4\), where \(f\) is given by: \[f(x) = \begin{cases}\frac{1-\cos(x-4)}{x-4} &\text{if } x\ne4\\2x-8&\text{if } x= 4.\end{cases}\]
  4. Show that the function \(f\) is continuous at \(1\), where \(f\) is given by \[f(x) = \begin{cases}x^2+5 &\text{if } x<1\\6&\text{if } x=1\\\log_2(x+63)&\text{if } x> 1.\end{cases}\]
  5. Take \(f\) to be the function that is given by \[f(x) = \frac{1-\cos(x-4)}{x-4}.\] Determine the maximal domain of \(f\) and find a continuous extension of \(f\) to all of \(\mathbb R\).
  6. Carefully show that the function \(f\) is not continuous at \(2\), where \(f\) is given by: \[f(x) = \begin{cases} \text{exp}_5(x) & \text{if } x \leq 2\\ 2x+5 & \text{if } x> 2\end{cases}\quad\text{where}\quad\text{exp}_5(x)=5^x.\]
  7. Take \(f\) to be a function with a domain of \(\mathcal{D}(f)=(-\infty,\infty)\) and that is continuous on \((-\infty,3)\cup(3,6)\cup(6,\infty)\). Take \(g\) to be a function with a domain of \(\mathcal{D}(g)=[-6,\infty)\) and that is continuous on \([-6,-2)\cup(-2,0)\cup(0,2)\cup(2,\infty).\)
    1. Determine whether \(f\) is defined at \(x=3\).
    2. Determine whether \(f\) is continuous at \(x=3\) and \(x=9.\)
    3. Determine whether \(g\) is defined at \(x=-7\).
    4. Determine whether \(g\) continuous at \(x=-2\) and \(x=2.\)
    5. Determine the maximal set on which \(f+g\) is continuous on.
    6. Determine the maximal set on which \(fg\) is continuous on.
  8. Decompose the function \(f\) into sums, products, and quotients of continuous functions to find a set \(S\) on which \(f\) is continuous: \[f(x) = \frac{\sqrt{-x+10}}{x^2 -6}+\ln(x)+\sin(x).\]
  9. Write the function \(f\) that is given below as a compound function to determine \(\displaystyle \lim_{x\to 4}f(x)\), where \(f\) is given by \[f(x) = \cos\left(\frac{1-\cos(x-4)}{x-4}\right).\]
  10. Take \(f\) to be the polynomial that is given by \[f(x) = x^5 - 4x^3 +2x - 1.\] Show that \(f\) has at least one real root by using the intermediate value theorem.
  11. Use the bisection method to approximate a solution to the equation \[x^2 = 7\] to within an error of no greater than \(\frac{1}{10}\).

Practice Problems

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