4.3: The Mean Value Theorem
- Page ID
- 144283
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
- Verify that \(f(x) = x^2\) satisfies the hypotheses of the mean value theorem on the interval \([-1, 2]\), then find all values \(c\) that satisfy the conclusion of the theorem.
- Verify that \(f(x) = \dfrac{x}{x+2}\) satisfies the hypotheses of the mean value theorem on the interval \([1, 4]\), then find all values \(c\) that satisfy the conclusion of the theorem.
- Verify that \(f(x) = \sin(\pi x)\) satisfies the hypotheses of the mean value theorem on the interval \([0, 2]\), then find all values \(c\) that satisfy the conclusion of the theorem.
- Let \(f(x) = \dfrac{1}{x}\). Show that there is no value \(c \in [-1, 1]\) that satisfies the conclusion of the mean value theorem. Explain why the mean value theorem does not apply to the function on the interval \([-1, 1]\).
- Let \(f(x) = \left|x - 3\right|\). Show that there is no value \(c \in [2, 5]\) that satisfies the conclusion of the mean value theorem. Explain why the mean value theorem does not apply to the function on the interval \([2, 5]\).
- Let \(f(x) = x^{1/3}\). Show that there is no value \(c \in [-1, 1]\) that satisfies the conclusion of the mean value theorem. Explain why the mean value theorem does not apply to the function on the interval \([-1, 1]\).