2.1: Continuity
- Page ID
- 143115
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)- Sketch the graph of a continuous function that passes through the points \((-3, -1)\), \((-1, 2)\), \((0, -1)\), and \((2, 3)\).
- Sketch the graph of a function that has a removable discontinuity at \(x = 1\).
- Sketch the graph of a function that has a jump discontinuity at \(x = 2\).
- Sketch the graph of a function that has an infinite discontinuity at \(x = -1\).
- Sketch the graph of the function \(f(x) = x^2 - 1\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(f(x) = \dfrac{1}{x}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(g(x) = \dfrac{1}{x^2 - 1}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(h(x) = \dfrac{1}{x^2 + 1}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(f(x) = \dfrac{x^2}{x}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(g(x) = \dfrac{x+2}{x^2-x-6}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(h(x) = \tan(x)\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(f(x) = \dfrac{|x|}{x}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(g(x) = \dfrac{1}{\sqrt{x}}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(h(x) = \sqrt{x^2-1}\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function
\[f(x) = \left\{\begin{align*}
& x^2, & x \neq -1, \\
& 2, & x = -1.
\end{align*}\right.\]
For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function
\[g(x) = \left\{\begin{align*}
& -x, & x < 0, \\
& \sqrt{x}, & x \ge 0.
\end{align*}\right.\]
For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function
\[h(x) = \left\{\begin{align*}
& 2^x, & x \le 0, \\
& \frac{1}{x}, & x > 0.
\end{align*}\right.\]
For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function
\[h(x) = \left\{\begin{align*}
& \frac{1}{x^2}, & x < 2, \\
& \pi, & x = 2, \\
& 2x-5, & x > 2.
\end{align*}\right.\]
For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Sketch the graph of the function \(f(x) = \sin\left(\dfrac{1}{x}\right)\). For what values of \(x\) is the function continuous? For what values of \(x\) is the function discontinuous? Classify any discontinuity as removable, jump, infinite, or other.
- Consider the function \(g(x) = x^2\), and recall that \(\sqrt{2} = 1.414213562\ldots\). The \(x\)-values in the table below approximate \(x = \sqrt{2}\). The \(x\)-values on the left side of the table approach \(\sqrt{2}\) from the left; the \(x\)-values on the right right side of the table approach \(\sqrt{2}\) from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of \(g(\sqrt{2})\)? Why or why not?
\(x\) \(g(x)\) \(x\) \(g(x)\) 1.41 1.42 1.414 1.415 1.4142 1.4143 - Consider the function \(h(x) = \dfrac{x-1}{x^2-3x+2}\), and recall that \(\sqrt{2} = 1.414213562\ldots\). The \(x\)-values in the table below approximate \(x = \sqrt{2}\). The \(x\)-values on the left side of the table approach \(\sqrt{2}\) from the left; the \(x\)-values on the right right side of the table approach \(\sqrt{2}\) from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of \(h(\sqrt{2})\)? Why or why not?
\(x\) \(h(x)\) \(x\) \(h(x)\) 1.41 1.42 1.414 1.415 1.4142 1.4143 - Consider the function \(f(x) = \dfrac{|x^2 - 2|}{x^2 - 2}\), and recall that \(\sqrt{2} = 1.414213562\ldots\). The \(x\)-values in the table below approximate \(x = \sqrt{2}\). The \(x\)-values on the left side of the table approach \(\sqrt{2}\) from the left; the \(x\)-values on the right right side of the table approach \(\sqrt{2}\) from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of \(f(\sqrt{2})\)? Why or why not?
\(x\) \(f(x)\) \(x\) \(f(x)\) 1.41 1.42 1.414 1.415 1.4142 1.4143 - Consider the function \(g(x) = \dfrac{1}{x^2 - 2}\), and recall that \(\sqrt{2} = 1.414213562\ldots\). The \(x\)-values in the table below approximate \(x = \sqrt{2}\). The \(x\)-values on the left side of the table approach \(\sqrt{2}\) from the left; the \(x\)-values on the right right side of the table approach \(\sqrt{2}\) from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of \(g(\sqrt{2})\)? Why or why not?
\(x\) \(g(x)\) \(x\) \(g(x)\) 1.41 1.42 1.414 1.415 1.4142 1.4143 - Consider the function \(h(x) = \dfrac{x^2 - 2}{x^4 - 4}\), and recall that \(\sqrt{2} = 1.414213562\ldots\). The \(x\)-values in the table below approximate \(x = \sqrt{2}\). The \(x\)-values on the left side of the table approach \(\sqrt{2}\) from the left; the \(x\)-values on the right right side of the table approach \(\sqrt{2}\) from the right. Use a calculator to complete the table by evaluating the function at these values. Do the function values give a good approximation of \(h(\sqrt{2})\)? Why or why not?
\(x\) \(h(x)\) \(x\) \(h(x)\) 1.41 1.42 1.414 1.415 1.4142 1.4143 -
Consider the function \(f(x) = \dfrac{1}{x^2} - 2\). Evaluate \(f(1)\) and \(f(2)\). Does the Intermediate Value Theorem guarantee that the function has a zero on the interval \((1, 2)\)? Why or why not? Confirm your answer using a graphing calculator.
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Consider the function \(g(x) = \dfrac{1}{x^2 - 2}\). Evaluate \(g(1)\) and \(g(2)\). Does the Intermediate Value Theorem guarantee that the function has a zero on the interval \((1, 2)\)? Why or why not? Confirm your answer using a graphing calculator.