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diff --git a/macros/latex/contrib/studenthandouts/samplecode/worksheets/handout-3-2.tex b/macros/latex/contrib/studenthandouts/samplecode/worksheets/handout-3-2.tex new file mode 100644 index 0000000000..6d5cefe0a8 --- /dev/null +++ b/macros/latex/contrib/studenthandouts/samplecode/worksheets/handout-3-2.tex @@ -0,0 +1,48 @@ +\sethandouttitle{The Mean Value Theorem} + + +\begin{enumerate} + + + \item Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. + Then find all numbers $c$ that satisfy the conclusion of Rolle's Theorem. + \begin{enumerate} + \item $f(x) = 5 - 12x + 3x^2$ on $[1,3]$. + \item $f(x) = \sqrt{x} - \frac{1}{3}x$ on $[0,9]$. + \end{enumerate} + + \item Let $f(x) = 1 - x^{2/3}$. Show that $f(-1)=f(1)$ but there is no number $c$ in $(-1,1)$ such that $f'(c)=0$. + Why does this not contradict Rolle's Theorem? + + \item Verify that the function satisfies the hypotheses of the Mean Value Theorem on the given interval. + Then find all numbers $c$ that satisfy the conclusion of the Mean Value Theorem. + \begin{enumerate} + \item $f(x) = 2x^2 - 3x +1$ on $[0,2]$. + \item $f(x) = \ln(x)$ on $[1,4]$. + \end{enumerate} + + \item Let $f(x) = 2 - |2x-1|$. Show that there is no value of $c$ such that + \[ + f(3)-f(0) = f'(c)(3-0). + \] + Why does this not contradict the Mean Value Theorem? + + \item Show that the equation $2x + \cos(x)$ has exactly one real root. + + \item If $f(1)=10$ and $f'(2) \geq 2$ for $1 \leq x \leq 4$, how small can $f(4)$ possibly be? + + \item Does there exist a function $f$ such that $f(0) = -1$, $f(2) = 4$ and $f'(x) \leq 2$? + + + \item Prove that + \[ + \arcsin \left( \frac{x-1}{x+1} \right) = 2 \arctan \sqrt{x} - \frac{\pi}{2}. + \] + + + +\end{enumerate} + + + + |