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Student[Calculus1]

 FunctionAverage
 find the average value of a function or expression

 Calling Sequence FunctionAverage(f(x), x = a..b, opts) FunctionAverage(f(x), a..b, opts)

Parameters

 f(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify the end points of the curve opts - equation(s) of the form option=value where option is one of averageoptions, functionoptions, output, showaverage, showfunction, or Student plot options; specify output options

Description

 • The FunctionAverage(f(x), x=a..b) command returns the average value of the expression $f\left(x\right)$ on the interval from a to b. By using options, you can specify that the command returns a plot or inert integral instead.
 • If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.
 • The average value of an expression f(x) from x = a to x = b is the value of the expression:

$\frac{{\int }_{a}^{b}f\left(x\right)\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}ⅆx}{b-a}$

 • The opts argument can contain any of the Student plot options or any of the following equations that (excluding output) set plot options.
 averageoptions = list
 A list of options for the plot of the average value of $f\left(x\right)$. By default, the average is plotted as a solid blue line. For more information on plot options, see plot/options.
 functionoptions = list
 A list of options for the plot of the expression $f\left(x\right)$.  By default, the expression is plotted as a solid red line. For more information on plot options, see plot/options.
 output = value, integral, or plot
 This option controls the return value of the function.
 – output = value specifies that the average value is returned. Other options are ignored if output = value.  This is the default.
 – output = integral specifies that the expression $\frac{1}{b-a}$ times the inert integral with the integrand f(x) is returned. Other options are ignored if output = integral.
 – output = plot specifies that a plot, which shows the expression and its average value from a to b, is displayed.
 showaverage = true or false
 Whether the average of the expression $f\left(x\right)$ over the range $a..b$ is plotted. By default, the value is true.
 showfunction = true or false
 Whether the expression $f\left(x\right)$ is plotted.  By default, the value is true.
 caption = anything
 A caption for the plot.
 The default caption is constructed from the parameters and the command options. caption = "" disables the default caption. For more information about specifying a caption, see plot/typesetting.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{FunctionAverage}\left(x-3,x=-1..1\right)$
 ${-3}$ (1)
 > $\mathrm{FunctionAverage}\left(\mathrm{ln}\left(x\right),x=1..4\right)$
 ${-}{1}{+}\frac{{8}{}{\mathrm{ln}}{}\left({2}\right)}{{3}}$ (2)
 > $\mathrm{FunctionAverage}\left({x}^{\frac{3}{2}},1..4,\mathrm{output}=\mathrm{integral}\right)$
 $\frac{\left({{\int }}_{{1}}^{{4}}{{x}}^{{3}}{{2}}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{x}\right)}{{3}}$ (3)
 > $\mathrm{FunctionAverage}\left(\frac{1}{x},x=1..4,\mathrm{output}=\mathrm{plot}\right)$
 > $\mathrm{FunctionAverage}\left(\mathrm{sin}\left(x\right)+x,3..10,\mathrm{output}=\mathrm{plot},\mathrm{functionoptions}=\left[\mathrm{thickness}=2\right],\mathrm{averageoptions}=\left[\mathrm{linestyle}=\mathrm{dash}\right]\right)$