Calculus 1: Applications of Differentiation - Maple Help

Online Help

All Products    Maple    MapleSim


Home : Support : Online Help : Education : Student Package : Calculus 1 : Example Worksheets : examples/Calculus1DiffApps

Calculus 1:  Applications of Differentiation

The Student[Calculus1] package contains three routines that can be used to both work with and visualize the concepts of Taylor approximations, charting a function, and using Newton's method.  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][FunctionChart],  it is easier, and often clearer, to load the package, and then use the short form command names.

restart

withStudentCalculus1:

The following sections show how the routines work.

Taylor Approximations

If at a point c, a function f has a power series expansion
                    fx=n=0anxcn
the coefficients an are given by:
                   an=fncn!

where fnc is the nth derivative of f evaluated at the point c.  Named after the English mathematician Brook Taylor, this infinite series is called the Taylor expansion of the function f at c.

 

The Taylor expansion of the exponential function ⅇx is:
                  ⅇx=n=0xnn!

from which it follows that:

                    ⅇ=n=01n!

 

Taylor approximations require both an expression and a point around which to expand.

TaylorApproximationx75x5+4x47x2+3,x=1,order=3

1+11x15x2+x3

(1.1)

Thus, around the point x=1 the polynomial x75x5+4x47x2+3 behaves like x315x2+11x1.

TaylorApproximationx75x5+4x47x2+3,x=1,order=3,output=plot

TaylorApproximationsinx,x=1,output=animation,order=1..16

The derivative of the arctan function has singularities at I and I. Therefore, the radius of convergence of the Taylor approximation around the origin is 1.

TaylorApproximationarctanx,x=0,output=animation,order=1..20

You can compute and visualize Taylor approximations using the TaylorApproximationTutor command.

TaylorApproximationTutor

Function Chart

The FunctionChart routine plots a function and shows regions of positive and negative sign, increasing and decreasing, and positive and negative concavity.  By default:

1.  Roots are marked by circles.
2.  Extreme points are marked by diamonds.
3.  Inflection points are marked by crosses.
4.  Regions of increase and decrease are marked by red and blue lines, respectively.

5.  Regions of positive and negative concavity are marked by azure and purple fill, respectively, with arrows pointing in the direction of the concavity.

FunctionChartx4+2x39x23x+6,x=5..4

FunctionChartsinx,x=0..2π

FunctionChartx32x24x+2x4,x=3..3

You can also perform curve analysis using the CurveAnalysisTutor command.

CurveAnalysisTutor

Newton's Method

Given a point a and an expression fx, the x-intercept of the tangent line through (a, fa) can be used as an approximation to a root of the expression fx.  The equation of the tangent line is:

Tangentfx,x=a,output=line

ⅆⅆafax+faⅆⅆafaa

(3.1)

collectconvert,D,Dfa

xaDfa+fa

(3.2)

Solving for zero:

 

solve=0,x

fa+DfaaDfa

(3.3)

expand

faDfa+a

(3.4)

As an example, consider the function Fx=x21 and an initial point x=2.0.

F:=x→x21

F:=x→x21

(3.5)

aroot:=2.0F2.0DF2.0

aroot:=1.250000000

(3.6)

Repeating this another 9 times:

forito5doaroot:=arootFarootDFarootend do

aroot:=1.025000000

aroot:=1.000304878

aroot:=1.000000046

aroot:=1.000000000

aroot:=1.000000000

(3.7)

The routine NewtonsMethod performs the same process.

NewtonsMethodFx,x=2,output=sequence

2,1.250000000,1.025000000,1.000304878,1.000000046,1.000000000

(3.8)

NewtonsMethodFx,x=2,output=plot

NewtonsMethodsinxx,x=1,output=plot

The root to which a sequence of Newton iterations converges to depends on the initial point.  For example,

NewtonsMethodsinxx,x=2,output=plot

In general, when the root is not a double root, Newton's method is very efficient.  In the following example with Digits set to 30, Newton's method converges to the root after only 7 iterations.

Digits:=30

NewtonsMethodx44x3+4x23x+3,x=1,output=sequence,iterations=10

Digits:=10

Digits:=30

1,1.33333333333333333333333333333,1.28318584070796460176991150443,1.28231623816647766759714731909,1.28231595363411166690275078928,1.28231595363408116582940754743,1.28231595363408116582940754709,1.28231595363408116582940754707,1.28231595363408116582940754707,1.28231595363408116582940754707,1.28231595363408116582940754707

Digits:=10

(3.9)

You can also learn about Newton's method using the NewtonsMethodTutor command.

NewtonsMethodTutor

Main: Visualization

Previous: Various Theorems about Derivatives

Next: Integration

 


Download Help Document

Was this information helpful?



Please add your Comment (Optional)
E-mail Address (Optional)
What is ? This question helps us to combat spam