Calculus1 Theorems - Maple Help
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Calculus 1:  Theorems about Differentiation

The Student[Calculus1] package contains three routines that can be used to both work with and visualize Rolle's theorem and the mean value theorem.  This worksheet demonstrates this functionality.

For further information about any command in the Calculus1 package, see the corresponding help page.  For a general overview, see Calculus1.

Getting Started

While any command in the package can be referred to using the long form, for example, Student[Calculus1][MeanValueTheorem],  it is easier, and often clearer, to load the package, and then use the short form command names.

 > $\mathrm{restart}$
 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$

The following sections show how the various routines work.

Rolle's Theorem

Rolle's theorem states that if $f$ is a function that satisfies:

1.  f is continuous on the closed interval $\left[a,b\right]$,

2.  f is differentiable on the open interval ($a,b$), and

3.  $f\left(a\right)=f\left(b\right)$

then there exists a point $c$ in the open interval ($a,b$) such that f'($c$) = 0.

The routine RollesTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative is zero.

 > $\mathrm{RollesTheorem}\left(x\left(x-4\right),x=1..3,\mathrm{output}=\mathrm{points}\right)$
 $\left[{2}\right]$ (1.1)
 > $\mathrm{RollesTheorem}\left(x\left(x-4\right),x=1..3\right)$
 > $\mathrm{RollesTheorem}\left(\mathrm{sin}\left(x\right),1..2\mathrm{π}+1\right)$

The Mean Value Theorem

The mean value theorem is a generalization of Rolle's theorem which states that if $f$ is a function that satisfies:

1.  f is continuous on the closed interval $\left[a,b\right]$, and

2.  f is differentiable on the open interval ($a,b$),

then there exists a point $c$ in the open interval ($a,b$) such that f'($c$) = $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ where the right-hand side is the slope of the line connecting the points ($a,f\left(a\right)$) and ($b,f\left(b\right)$).  The Mean Value Theorem can be derived from Rolle's Theorem by considering the function $g\left(x\right)=f\left(x\right)-\frac{\left(f\left(b\right)-f\left(a\right)\right)\left(x-a\right)}{b-a}$.

The routine MeanValueTheorem takes an expression representing the function, checks that the requirements of the theorem hold, and then plots the expression and all points where the derivative equals the slope of the secant line connecting the end points of the graph of $f$ on $\left[a,b\right]$.

 > $\mathrm{MeanValueTheorem}\left({x}^{3}-5{x}^{2}+8x-1,x=1..3,\mathrm{output}=\mathrm{points}\right)$
 $\left[\frac{{7}}{{3}}\right]$ (2.1)
 > $\mathrm{MeanValueTheorem}\left({x}^{3}-5{x}^{2}+8x-1,x=1..3\right)$
 > $\mathrm{MeanValueTheorem}\left(\mathrm{sin}\left(x\right),x=-4..2\mathrm{π}\right)$

You can also learn about the Mean Value Theorem using the MeanValueTheoremTutor command.

 > $\mathrm{MeanValueTheoremTutor}\left(\right)$

Main: Visualization

Previous: Derivatives