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 function \(f\) 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 \[f(x) = \begin{cases}\dfrac{1-\cos(x-4)}{x-4} &\text{if } x\ne4\\2x-8&\text{if } x= 4.\end{cases}\]

  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 \(f\) and \(g\) to be functions that are defined on \[\mathcal{D}(f)=(-\infty,\infty)\quad\text{and}\quad\mathcal{D}(g)=[-6,\infty)\] and that are continuous on \[S_f=(-\infty,3)\cup(3,6)\cup(6,\infty)\quad\text{and}\quad S_g=[-6,-2)\cup(-2,0)\cup(0,2)\cup(2,\infty),\] respectively.

    1. Determine whether \(f\) is defined at \(x=3\).
    2. Determine whether \(f\) is continuous at \(x=3\) and at \(x=9.\)
    3. Determine whether \(g\) is defined at \(x=-7\).
    4. Determine whether \(g\) continuous at \(x=-2\) and at \(x=2.\)
    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, where \(f\) is given by \[f(x) = \frac{\sqrt{10-x}}{x^2 -6}+\ln(x)+\sin(x).\]

  9. Write the function \(f\) as a composite function to determine \(\lim\limits_{x\to 4}f(x)\), where \(f\) is given by \[f(x) = \cos\left(\frac{1-\cos(x-4)}{x-4}\right).\]

  10. Construct a function that is continuous everywhere except at \(1\), that is strictly increasing to the left of \(1\), is asymptotically equal to \(-2x^2\) to the left, is strictly decreasing to the right of \(1\), is right continuous at \(1\), is asymptotically equal to \(-3x^3\), and has the property that \[\lim_{x\to 1^-}f(x)=2\quad\text{and}\quad \lim_{x\to 1^+}f(x)=3.\]

  11. Construct a continuous path \(c\) with domain \([0,\infty)\) that describes the position of a particle that moves to the right on the line segment from \((1,3)\) to \((4,5)\), is at \((1,3)\) at time \(0\), is never at the same point at different time points, that never reaches \((4,5),\) but that has the property that \[\lim_{t\to\infty}\|(4,5)-c(t)\|=0.\]

  12. 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.

  13. Use the bisection method to approximate a solution to the following equation \[x^2 = 7\] to within an error of no greater than \(\frac{1}{10}\).

Remember, do the knowledge checks before checking the solutions.

Practice Problems

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