Classroom Tips and Techniques: Stepwise Solutions in Maple - Part 1
Robert J. Lopez
Emeritus Professor of Mathematics and Maple Fellow
Maplesoft
Introduction
Maple's powerful mathematical engine is primarily designed to provide the results of
mathematical operations. But there are commands, Assistants, Tutors, and Task Templates that
show stepwise calculations in algebra, calculus (single-variable, multivariable, vector), and linear
algebra. In this article we discuss Maple's functionality for providing these stepwise solutions to
mathematical problems in algebra and calculus (both of one and several variables). In our next
article, we will continue with stepwise tools in linear algebra and vector calculus.
However, we hasten to point out that often, the underlying algorithms Maple uses are not the
ones students see in their textbooks. For example, the standard calculus text contains a detailed
section on methods of integration, a collection of manipulations designed to produce the
antiderivatives of most of the elementary functions. Maple, on the other hand, will use a
number of other devices, including the Risch algorithm, to obtain these antiderivatives.
Because Maple "does" symbolic math, it is always possible to guide Maple through nearly any
segment of mathematical calculations. Thus, if Maple does not have a built-in tool for
displaying a calculation stepwise, the calculation can always be reduced to its rudiments by
simply directing Maple to take the required steps.
Stepwise Algebra
Solving Equations
Maple's solve and fsolve commands solve equations analytically and numerically, respectively.
Stepwise solutions are provided by the Equation Manipulator, an Assistant that can be accessed
either from the Tools menu, or from the Context Menu by choosing the option "Manipulate
Equation."
Demonstrations of stepwise equation-solving can be viewed in the recorded webinar "Clickable
Calculus: Precalculus, and Calculus of One and Several Variables."
Partial Fraction Decomposition
The indefinite integral of the function
requires the partial fraction decomposition
which can also be obtained from the Context Menu under the Conversions option. A stepwise
decomposition is available via the Task Template in Table 1.
Tools‫ظ‬Tasks‫ظ‬Browse: Algebra‫ظ‬Partial Fractions‫ظ‬Stepwise
Stepwise Partial Fraction Decomposition
Initialize
Write rational function here
3
2
x C x C 4 xC 1
2
2
x C 4 x C 1
Factor Denominator
Clear
Write the partial-fraction decomposition template in this box
3
2
x C x C 4xC 1
2
2
x C 4 x C 1
c xC d
a xC b
C
2
2
x C 1
x C 4
≡
*
Check Template Fractions
Clear
Correct!
To determine the constants, multiply both sides of the identity (*) by the denominator
of the fraction on the left.
Expand (Clear Parentheses)
3
Clear
2
C x C 4xC 1
3
2
3
2
a x C a xC b x C bC cx C 4c xC d x C 4d
Collect Like Terms
Clear
0
3
2
a C c K 1 x C K 1C b C d x C a C 4 c K 4 x K 1C b C 4 d
Form Equations
Clear
-1 C b C 4 d
aC 4c-4
-1 C b C d
aC c-1
0
=
0
0
0
Solve Equations
Clear
↓
a = 0, b = 1, c = 1, d = 0
Partial Fractions:
Clear
3
2
x C x C 4xC 1
2
2
x C 4 x C 1
1
2
x C 4
Table 1
C
=
x
2
x C 1
Stepwise partial fraction Task Template
The algebra for obtaining the equations that determine the coefficients
is not unique.
This Task Template adopts one particular strategy for this, but there are other methods.
These algebraic steps can also be implemented directly in Maple, either with the appropriate
commands, or even via the Context Menu, as we show in Table 2. The left-hand column in this
table states the action to perform, and the right-hand column shows the effect of carrying out that
instruction. The initial identity
1. Enter identity.
Press Enter key.
2. Context Menu:
Move to Left
Left-hand Side
Simplify
Numerator
Collect‫ظ‬
Coefficients‫ظ‬
Solve
3. Using equation
labels and the
evaluation template
from the
Expression palette,
transfer the values
of the coefficients
to the identity.
Table 2
Stepwise partial-fractions by first principles via the Context Menu
Stepwise Calculus of a Single Variable
Limits
The Limit Methods tutor, shown in Table 3 as a screen-shot, will guide the evaluation of a limit.
This tutor is available from the Tools menu, or from the Context Menu after the Student Calculus
1 package has been loaded.
Table 3
The Limit Methods tutor applied to
The sequence of steps can be copied and pasted from the tutor, but that does not salvage the
per-step annotations. Closing the tutor only provides the final answer, not the intermediate
steps. The greatest value in the tutor seems to be its use as an experimental platform for
investigating how a limit might be calculated.
The annotated stepwise solution shown in Table 4 is available via the ShowSteps command, or
better yet, from the Context Menu under the Solve‫ظ‬Show Solution Steps option (after loading
the Student Calculus 1 package with the Tools menu option: Load Package).
Loading Student:-Calculus1
Table 4 Stepwise limit via the Solve‫ظ‬Show Solution Steps option
in the Context Menu
Derivatives
The Differentiation Methods tutor, shown in Table 5 as a screen-shot, will guide the
differentiation process. This tutor is available from the Tools menu, or from the Context Menu
after the Student Calculus 1 package has been loaded.
Table 5
The Differentiation Methods tutor applied to
The sequence of steps can be copied and pasted from the tutor, but that does not salvage the
per-step annotations. Closing the tutor only provides the final answer, not the intermediate
steps. The greatest value in the tutor seems to be its use as an experimental platform for
investigating how a derivative might be evaluated.
The annotated stepwise solution shown in Table 6 is available via the ShowSteps command, or
better yet, from the Context Menu under the Solve‫ظ‬Show Solution Steps option (after loading
the Student Calculus 1 package with the Tools menu option: Load Package).
Loading Student:-Calculus1
Table 6 Stepwise differentiation via the Solve‫ظ‬Show Solution Steps
option in the Context Menu
Notice that the differentiation operator "d" is gray, not black. This indicates the inert form of the
operator, obtained by applying the Context Menu: 2-D Math‫ظ‬Convert To‫ظ‬Inert Form to the
operator
in the Expression palette.
Integrals
The Integration Methods tutor, shown in Table 7 as a screen-shot, will guide the integration
process. This tutor is available from the Tools menu, or from the Context Menu after the Student
Calculus 1 package has been loaded.
Table 7
The Integration Methods tutor applied to
The sequence of steps can be copied and pasted from the tutor, but that does not salvage the
per-step annotations. Closing the tutor only provides the final answer, not the intermediate
steps. The greatest value in the tutor seems to be its use as an experimental platform for
investigating how an integral might be evaluated.
The annotated stepwise solution shown in Table 8 is available via the ShowSteps command, or
better yet, from the Context Menu under the Solve‫ظ‬Show Solution Steps option (after loading
the Student Calculus 1 package with the Tools menu option: Load Package).
Loading Student:-Calculus1
Table 8 Stepwise integration via the Solve‫ظ‬Show Solution Steps option
in the Context Menu
The integral operator is not black, but gray, the inert form of the operator, obtained by applying
in the Expression
the Context Menu: 2-D Math‫ظ‬Convert To‫ظ‬Inert Form to the operator
palette.
The change annotation in Table 8 includes the required change of variable, something that the
tutor does not provide. Note too, that the procedure followed by Maple is not the only method of
solution. It is also possible to "factor out the 4" and set
to obtain
Tangent Line
It is a staple of the calculus course to find the equation of the line tangent to a curve at a given
point. Since this requires computing a derivative to obtain the slope of the tangent line, and a
bit of algebra to simplify the point-slope form of the equation of the line, this calculation can
certainly be implemented "from first principles" directly in Maple. However, as shown in Table
9, there is the Tangent Line task template, which we have used to find, at
, the line tangent
to
. Both the solution and the details of the calculation are provided by this task
template.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Derivatives‫ظ‬Applications‫ظ‬Tangent Line
Tangent Line
2
C xC 1
2
(Default value:
)
Find Tangent Line
=5xK 3
Clear
Compute Details
2xC 1
5
7
Graph
5 x- 2 C 7
Clear Details
Table 9
Clear Graph
Equation of a tangent line by the Tangent Line task template
Normal Line
It is also a staple of the calculus course to find the equation of the line normal to a curve at a
given point. Since this requires computing a derivative to obtain the slope of the tangent line,
and a bit of algebra to simplify the point-slope form of the equation of the line, this calculation
can certainly be implemented "from first principles" directly in Maple. However, as shown in
, the line
Table 10, there is the Normal Line task template, which we have used to find, at
normal to
. Both the solution and the details of the calculation are provided by
this task template.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Derivatives‫ظ‬Applications‫ظ‬Normal Line
Normal Line
2
C xC 1
2
(Default value:
)
Find Normal Line
= K
1
37
xC
5
5
Clear
Compute Details
Graph
2xC 1
5
7
= -
Normal Line:
1
x- 2 C 7
5
Clear Details
Table 10
Clear Graph
Equation of a normal line by the Normal Line task template
Derivative by Definition
Table 11 contains the "Derivatives by Definition" task template.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Derivatives‫ظ‬Derivatives by Definition
Derivatives by Definition
Enter the function
and the value of
for which
is to be obtained.
2
C 2 x
a
(Default value: )
Difference Quotient
2aC hC 2
Simplify
Derivative
2aC 2
Clear All
Table 11 The derivative of
task template
by definition, using the "Derivatives by Definition"
Difference (or Newton) Quotient
The difference (or Newton) quotient is the slope of the secant line, which, in the limit, becomes
the slope of the tangent line. In essence, this is the expression whose limit yields the derivative.
This calculation is captured by the Difference (or Newton) Quotient task template, as shown in
Table 12.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Derivatives‫ظ‬Difference (or Newton) Quotient
The Difference (or Newton) Quotient
Enter the function
to be evaluated, the -coordinate of the point of tangency, , and , where
is the
-coordinate of the point at which the secant line will be found.
3
1
1
Clear All
Slope of Secant Line
7
Equation of Secant Line
=7xK 6
Equation of Tangent Line
=3xK 2
Launch Tutor
Graph
Table 12
Animation
The difference quotient for
Clicking the "Launch Tutor" button in the task template will launch the Tangent (Newton
Quotient) tutor that is shown in Table 13. This tutor could be accessed independently from the
Tools‫ظ‬Tutors menu.
Table 13
The Tangent (Newton Quotient) tutor for
Implicit Differentiation
defined by the equation
can be
The implicit derivative of
obtained with the Context Menu option "Differentiate Implicitly." It can be obtained stepwise
with the task template in Table 14.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Derivatives‫ظ‬Implicit Differentiation‫ظ‬
Implicit Differentiation
Enter an equation in two
variables:
2
2
3 x C 2 x y K 5 y C 12 = 0
Clear All
Dependent variable:
Implicit Derivative:
Independent variable:
3xC y
dy
=
K xC 5y
dx
Execute
Stepwise Calculation
Make dependent variable
explicit:
2
2
3 x C 2 x y x K 5 y x C 12 = 0
Execute
Differentiate with respect
d
6xC 2yx C 2x
y x K 10 y x
to independent variable:
dx
Execute
Stepwise
Isolate Derivative:
d
K 6xK 2yx
yx =
dx
2 x K 10 y x
Execute
Make independent variable dy
3xC y
= K
implicit:
dx
xK 5y
Execute
d
yx =0
dx
Table 14
Stepwise implicit differentiation via task template
Clicking the "Stepwise" button will launch the Differentiation Methods tutor in which the
derivative can be computed step-by-step.
Riemann Sums
The Riemann sum for finding the area bounded by
and the -axis can be
explored graphically and numerically by tutor; and analytically, by task template.
Table 15 shows the Riemann Sums tutor applied to this function.
Table 15
Application of the Riemann Sums tutor to the function
By default, a midpoint sum is chosen, but we have elected to demonstrate the left sum. The graph
divided into
equal subintervals, each one supporting a rectangle
shows the interval
whose height is determined at the left edge of the subinterval. The area under curve is displayed,
along with the approximate area, namely, the sum of the areas in the left-rectangles.
Table 16 shows via task template the analytic evaluation of the corresponding Riemann sum for
, arbitrary, rectangles.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Integration‫ظ‬Riemann Sums‫ظ‬Left
The Left Riemann Sum
Enter
:
>
Enter the interval
:
>
Enter the value of : >
The left Riemann
sum:
>
Value of the Riemann >
sum:
>
Table 16
Analytic approach to left Riemann sum for
by task template
Of course, the analytic expression obtained for this left Riemann sum approaches
as
Mean Value Theorem
for
The Mean Value theorem states that under suitable conditions,
some in the interval
. In this form, the theorem relates to the linear (or tangent line)
.
approximation. If rearranged to
the theorem has a geometric interpretation: in the interval
, there is a point
where the
tangent line is parallel to the secant line connecting
with
. This is well
illustrated by the Mean Value Theorem tutor shown in Table 17, where the tutor is applied to the
function
on the interval
.
Table 17
Mean Value Theorem tutor applied to
on
The graph in the tutor shows the geometry - the tangent line is parallel to the secant line. The
value of is also determined to be
, and the linear "approximation"
is exact at this value because
.
Table 18 contains a task template that might be a more convenient implementation of the Mean
Value theorem calculations.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Mean Value Theorem
Mean Value Theorem
Enter and an interval
6C x K x
2
Clear
K 2
Clear a
1
Clear b
Computational Mode:
Analytic
Numeric
Mean Value Theorem
Clear
K
1
2
Table 18
Mean Value theorem via task template
The task template has two advantages: the value of can be obtained exactly, when possible;
and the display of the linear approximation is easier to read.
Rolle's Theorem
Rolle's theorem states that under suitable conditions, when
, there is in the interval
where the tangent line is horizontal, that is, where
. This theorem, used to prove
the Mean Value theorem, is illustrated by the graph in Table 19, constructed with the
RollesTheorem command in the Student Calculus 1 package.
Loading Student:-Calculus1
Table 19
Rolle's theorem illustrated by the RollesTheorem command
The usage
returns the value of
at which the horizontal tangent is found.
Table 20 contains a task template that might be a more convenient implementation of the Rolle's
theorem calculations.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Rolle's Theorem
Rolle's Theorem
Enter
6C x K x
and an interval
for which
2
Clear
K 2
3
Clear a
Clear b
Computational Mode:
Analytic
Points where
Numeric
:
1
2
Table 20
Rolle's Theorem
Clear
Rolle's theorem via task template
Curve Analysis
In the era before the widespread availability of graphing hardware and software, a significant
portion of a first calculus course was devoted to curve sketching. Surprisingly, few modern
calculus texts deviate from this historic practice, in spite of the reasonable cost of graphing
technology.
Maple has a Curve Analysis tutor that implements its FunctionChart (equivalently,
FunctionPlot) command. In addition to drawing an annotated graph, the tutor provides much of
the data upon which the traditional approach to curve sketching is based. Unfortunately, when
the tutor is closed, only the graph is preserved. Hence, the task template "Find Special Points on
a Function" is a useful addition to the tutor.
Table 21 shows the tutor applied to the function
Table 21
on the interval
.
The Curve Analysis tutor applied to
Clicking on the eight radio-buttons provides the raw data with which a graph could be sketched
in the historic approach to this task.
Table 22 shows, for the function
some of this information being captured with a task template.
Tools‫ظ‬Tasks‫ظ‬Browse: Calculus‫ظ‬Find Special Points on a Function
>
>
>
>
>
>
>
Table 22
The task template "Find Special Points on a Function" applied to
The graph in Table 21 shows that
has three -intercepts in the interval
, yet the Roots
command did not find any zeros. The following modification of the Roots command
yields the three -intercepts as floating-point numbers. These values are the same as those
computed via
Maple's solve command returns the exact solutions on the left in Table 23. Although these
, they are actually real, as can be seen from their equivalents shown on
solutions contain
the right.
Table 23
Exact zeros of the cubic function
Surface Area of a Surface of Revolution
The surface area of the surface of revolution formed when
, is rotated about the
-axis can be computed by means of the Surface of Revolution tutor, as shown in Table 24.
Table 24 Surface of Revolution tutor used to obtain the surface area of a
surface of revolution
In addition to the graph, this tutor displays the integral whose value is the required surface area,
the exact value of the integral, and its floating-point equivalent. Clicking the "Frustums" radio
button and then the "Display" button will show the surface approximated by segments
(frustrums) of cones. After these choices have been made, the display will include a
Riemann-sum approximation corresponding to the discretization.
Volume of a Solid of Revolution
, is rotated about the -axis
The volume of the solid of revolution formed when
can be computed by means of the Volume of Revolution tutor, as shown in Table 25.
Table 25 Volume of Revolution tutor used to obtain the volume of a
solid of revolution.
In addition to the graph, this tutor displays the integral whose value is the required volume, the
exact value of the integral, and its floating-point equivalent. For a horizontal axis of rotation, the
"Disks" radio button is available; for a vertical axis, the "Shells" radio button is available. If
"Disks" are selected, the solid is shown segmented into the chosen number of disks, and the
display will include the corresponding Riemann sum. A similar statement can be made for shells,
mutatis mutandis. In either event, the corresponding Riemann-sum approximation is provided.
Stepwise Calculus of Several Variables
The MultiInt Command
The MultiInt command of the Student Multivariable Calculus package will formulate and
evaluate an iterated multiple integral. One of its output options is a display of the steps involved
in executing the calculation. Table 26 shows the use of this command to evaluate the volume of
the region
Loading Student:-MultivariateCalculus
>
Table 26
Volume of the region
computed stepwise by the MultiInt command
The first line of the output is the unevaluated integral; and the last, the value of the integral. The
second line shows the outer integral after the inner integral has been evaluated as far as the
antiderivative with respect to . For this antiderivative, has been held fixed. The
antiderivative must be evaluated at the limits in the inner integral. The third line shows the outer
integral completely in . The fourth line is the antiderivative with respect to that must be
evaluated at the limits in the outer integral. The final value is in the last line.
This integration tool is available as the task template in Table 27.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Multiple Integration‫ظ‬Cartesian 2-D
Iterated Double Integral in Cartesian Coordinates
Integrand:
Region:
>
>
>
>
>
Inert integral:
>
Value:
>
Stepwise
Evaluation:
>
>
Table 27
Access to the MultiInt command through a task template
Critical Points and the Second-Derivative Test
A common task in the first multivariate calculus course is the determination and classification of
critical points of a multivariate function. Table 28 addresses this with a task template.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Critical Points & Second Derivative Test
Critical Points and the Second Derivative Test
Objective
Function
>
List of
Independent
Variables
>
Equations
>
Critical Points
>
Second Derivative >
Test
Hessians and their >
Eigenvalues
>
Table 28
Finding and classifying critical points for a multivariate function
The given function has two critical points, both found with the Solve command. However, the
format of the solution is not "points" so the output has to put into the form of a list of lists. The
second-derivative test is applied to each point. The origin cannot be classified by this test, so
nothing is said about it by the test. The other point is found to be a saddle point. In the final
"row" of the template, the Hessian matrix (the matrix of second derivatives) and its eigenvalues
is given for each point. Since the Hessian is symmetric, the signs of its eigenvalues suffice to
determine if the matrix is positive or negative definite, or even indefinite. At the origin, the
Hessian has a zero eigenvalue, and is singular. That is why the origin cannot be classified by
the second-derivative test. The eigenvalues at the other point are of opposite sign, so the
Hessian there is indefinite. That's why the second point is a saddle.
Center of Mass
The Student Precalculus package contains a CenterOfMass command that will determine the
center of mass of a discrete distribution of masses in . The Student Multivariate Calculus
package contains a CenterOfMass command that will determine the center of mass of a
or , using Cartesian, polar, spherical, or cylindrical
continuous distribution of mass in
coordinates. In each case, this command writes the expressions for the coordinates of the center
of mass, then evaluates the integrals expressing the appropriate moments and total mass. In
(Cartesian and polar), the CenterOfMass command can draw a graph of the density function
over the planar region on which it is defined. All of the continuous cases are implemented in task
templates.
Cartesian 2-D
To find the center of mass of the planar region
whose density is
, use the task template in Table 29.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Center of Mass‫ظ‬Cartesian 2-D
Center of Mass for Planar Region in Cartesian Coordinates
Density:
>
Region:
>
>
>
>
Moments Mass: >
Inert Integral -
Explicit values for >
and
Plot:
>
>
Table 29
Center of mass of a planar region in Cartesian coordinates
The red region in the graph is the planar region whose center of mass is located at the green dot,
whereas the blue surface is a graph of the density function
.
Polar
To find the center of mass of the planar region
whose density is
, use the task template in Table 30.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Center of Mass‫ظ‬Polar
Center of Mass for Planar Region in Polar Coordinates
Density:
>
Region:
>
>
>
>
Moments Mass:
Inert Integral -
>
Explicit values for
and
>
Plot:
>
>
Table 30
Center of mass of a planar region in polar coordinates
The red region in the graph is the planar region whose center of mass is located at the green dot,
whereas the blue surface is a graph of the density function
.
Cartesian 3-D
To find the center of mass of the region
whose density is
, use the task template in Table 31.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Center of Mass‫ظ‬Cartesian 3-D
Center of Mass for 3D Region in Cartesian Coordinates
Density:
>
Region:
>
>
>
>
>
>
Moments Mass >
:
Inert Integral -
Explicit values
for , , and
>
>
Table 31
Center of mass of a spatial region in Cartesian coordinates
The task template fixes the order of integration, but the CenterOfMass command will accept
any of the other five possible orders for integration over a region in .
Cylindrical
To find the center of mass of the region
, use the task template in Table 32.
whose density is
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Center of Mass‫ظ‬Cylindrical
Center of Mass for 3D Region in Cylindrical Coordinates
Density:
>
Region:
>
>
>
>
>
>
Moments ÷
Mass:Inert
Integral -
>
Explicit values >
for , , and ,
the center of mass
given in
cylindrical
coordinates:
>
Table 32
Center of mass of a spatial region in cylindrical coordinates
Spherical
To find the center of mass of the region
whose density is , use the task template in Table 33.
Tools‫ظ‬Tasks‫ظ‬Browse: Multivariate Calculus‫ظ‬Center of Mass‫ظ‬Spherical
Center of Mass for 3D Region in Spherical Coordinates
( is the colatitude, measured down from the -axis)
Density:
>
Region:
>
>
>
>
>
>
Moments ÷
Mass:Inert
Integral -
>
Explicit values >
for , and , the
center of mass
given in spherical
coordinates:
>
Table 33
Center of mass of a spatial region in spherical coordinates
Unfortunately, the value of is being computed incorrectly by the CenterOfMass command.
, not
. Corrected code will be available in versions of Maple
In the present example,
after Maple 13.02.
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