Newton Quotient - Maple Help

Student[Calculus1]

 NewtonQuotient
 return the Newton quotient of an expression

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

Parameters

 f(x) - algebraic expression in variable 'x' x - name; specify the independent variable a, b - algebraic expressions; specify the plot range c - algebraic expression; specify a point in a..b opts - equation(s) of the form option=value where option is one of central, derivativeoptions, functionoptions, h, iterations, output, pointoptions, quotientoptions, showderivative, showfunction, showpoints, showquotient, view, or Student plot options; specify output options

Description

 • The NewtonQuotient(f(x), x) command returns the Newton quotient of the expression $f\left(x\right)$, that is, the slope of the secant line connecting $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$.
 • The NewtonQuotient(f(x), x=a..b) command returns a plot of $f\left(x\right)$ from a to b and the function defined by the Newton quotient for step size h at each point in the interval $\left(a,b\right)$.
 • The NewtonQuotient(f(x), x=c) command returns a plot of $f\left(x\right)$ around c and shows the secant line connecting the points $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$, that is, the secant line whose slope is the corresponding Newton quotient.
 • The optional argument a..b gives the range of the plot.  If no range is given, the interval $\left[c-2\left|h\right|,c+2\left|h\right|\right]$ is used.
 • If the independent variable can be uniquely determined from the expression, the parameter x need not be included in the calling sequence.
 • The opts argument can contain any of the Student plot options or any of the following equations that (excluding central, h, iterations, and output) set plot options.
 central = true or false
 Whether the secant line that approximates the tangent is drawn between  $x=c+\frac{h}{2}$ and $x=c-\frac{h}{2}$ instead of $x=c$ and $x=c+h$. By default, the value is false, meaning the secant line is drawn using $x=c$ and $x=c+h$.
 derivativeoptions = list
 A list of options for the plot of the derivative of the expression $f\left(x\right)$.  By default, the expression is plotted as a solid green 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.
 h = positive, negative, list of positive or negative, or name
 The value of the parameter h.  The default value is $0.1$. If the option output is plot or animation, this option must be of type positive, negative, or a list of these types. (See the description of the central option for more information.)
 iterations = posint
 In an animation, by default $5$ iterations are shown with the value of h divided by $2$ each step.  This option is ignored if the option h is a list.
 output = value, line, plot, or animation
 This option controls the return value of the function.
 – output = value specifies that the value of the Newton quotient at the point (and corresponding value of h) are returned. Plotting options are ignored if output = value.  This is the default.
 – output = line specifies that the expression of the secant line passing through the points $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$ is returned.  Plotting options are ignored if output = line.
 – output = plot specifies that a plot, which shows the expression and the secant through the points $\left(c,f\left(c\right)\right)$ and $\left(c+h,f\left(c+h\right)\right)$, is displayed.
 – output = animation specifies that an animation, which shows the expression and a sequence of secants through the points $\left(c,f\left(c\right)\right)$ and $\left(c+\frac{h}{{2}^{n}},f\left(c+\frac{h}{{2}^{n}}\right)\right)$,  $n=1,2,3,\dots$ , is displayed.
 pointoptions = list
 A list of options for any points in the plot. By default, all points are plotted as blue circles.  If the second argument is a range, this option is ignored. For more information on plot options, see plot/options.
 quotientoptions = list
 A list of options for the plot of the Newton quotient or secant lines. By default, the Newton quotients on an interval or secant lines on a point are plotted in blue. For more information on plot options, see plot/options.
 showderivative = true or false
 Whether the tangent to the graph of f(x) at the point $\left(c,f\left(c\right)\right)$ is plotted.  By default, the value is false.
 showfunction = true or false
 Whether the expression $f\left(x\right)$ is plotted.  By default, the value is true.
 showpoints = true or false
 Whether the expression f(x) evaluated at the points $c$, $c+\frac{h}{2}$, or $c+h$, as appropriate, is marked.  (See the description of the central option for more information.) By default, the value is true.
 showquotient = true or false
 Whether the quotient of 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.
 title = anything
 A title for the plot.
 The default title is constructed from the parameters and the command options. title = "" disables the default title. For more information about specifying a title, see plot/typesetting.

Examples

 > $\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Calculus1}\right]\right):$
 > $\mathrm{NewtonQuotient}\left(\mathrm{sin}\left(x\right),1.0,'h'=0.1\right)$
 ${0.4973637530}$ (1)
 > $\mathrm{NewtonQuotient}\left(\mathrm{sin}\left(x\right),1.0,\mathrm{output}=\mathrm{line},'h'=0.1\right)$
 ${0.4973637530}{}{x}{+}{0.3441072318}$ (2)
 > $\mathrm{NewtonQuotient}\left({x}^{3}-2{x}^{2}-x+1,x=-2..2,\mathrm{output}=\mathrm{plot}\right)$

To play the following animation in this help page, right-click (Control-click, on Macintosh) the plot to display the context menu.  Select Animation > Play.

 > $\mathrm{NewtonQuotient}\left({x}^{3}-2{x}^{2}-x+1,x=\frac{1}{2},-1..2,\mathrm{output}=\mathrm{animation},'h'=0.4\right)$

The command to create the plot from the Plotting Guide is

 > $\mathrm{NewtonQuotient}\left({x}^{3}-2{x}^{2}-x+1,x=\frac{1}{2},\mathrm{output}=\mathrm{plot},'h'=1.0,\mathrm{view}=\left[-1..2,-2..1.5\right]\right)$