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Student[Calculus1]

  

NewtonQuotient

  

return the Newton quotient of an expression

 

Calling Sequence

Parameters

Description

Examples

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 fx, that is, the slope of the secant line connecting c,fc and c+h,fc+h.

• 

The NewtonQuotient(f(x), x=a..b) command returns a plot of fx from a to b and the function defined by the Newton quotient for step size h at each point in the interval a,b.

• 

The NewtonQuotient(f(x), x=c) command returns a plot of fx around c and shows the secant line connecting the points c,fc and c+h,fc+h, 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 c2h,c+2h 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+12h and x=c12h 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 fx.  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 fx.  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 c,fc and c+h,fc+h 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 c,fc and c+h,fc+h, is displayed.

– 

output = animation specifies that an animation, which shows the expression and a sequence of secants through the points c,fc and c+h2n,fc+h2n,  n=1,2,3, , 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 c,fc is plotted.  By default, the value is false.

  

 

  

showfunction = true or false

  

Whether the expression fx is plotted.  By default, the value is true.

  

 

  

showpoints = true or false

  

Whether the expression f(x) evaluated at the points c, c+12h, 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 fx 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

withStudent[Calculus1]:

NewtonQuotientsinx,1.0,'h'=0.1

0.4973637530

(1)

NewtonQuotientsinx,1.0,output=line,'h'=0.1

0.4973637530x+0.3441072318

(2)

NewtonQuotientx32x2x+1,x=2..2,output=plot

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.

NewtonQuotientx32x2x+1,x=12,1..2,output=animation,'h'=0.4

The command to create the plot from the Plotting Guide is

NewtonQuotientx32x2x+1,x=12,output=plot,'h'=1.0,view=1..2,2..1.5

See Also

plot/options

plot/typesetting

Student

Student plot options

Student[Calculus1]

Student[Calculus1][NewtonsMethod]

Student[Calculus1][TangentSecantTutor]

Student[Calculus1][VisualizationOverview]

 


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