2.1: Continuity
- Page ID
- 143115
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- 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.