Differential Calculus
Limits
The Limit Methods tutor, shown in Table 3 as a screenshot, 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 perstep 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 Solve≻Show Solution Steps option (after loading the Student Calculus 1 package with the Tools menu option: Load Package).
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Table 4 Stepwise limit via the Solve≻Show Solution Steps option in the Context Menu

Derivatives
The Differentiation Methods tutor, shown in Table 5 as a screenshot, 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 perstep 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 Solve≻Show Solution Steps option (after loading the Student Calculus 1 package with the Tools menu option: Load Package).
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Table 6 Stepwise differentiation via the Solve≻Show 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: 2D Math≻Convert To≻Inert Form to the operator in the Expression palette.
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 pointslope form of the equation of the line, this calculation can certainly be implemented "from first principles" directly in Maple. However, as shown in Table 7, 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.
Tools≻Tasks≻Browse:
Calculus  Differential≻Applications≻Tangent Line


Table 7 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 pointslope form of the equation of the line, this calculation can certainly be implemented "from first principles" directly in Maple. However, as shown in Table 8, there is the Normal Line task template, which we have used to find, at , the line normal to . Both the solution and the details of the calculation are provided by this task template.
Tools≻Tasks≻Browse:
Calculus  Differential≻Applications≻Normal Line


Table 8 Equation of a normal line by the Normal Line task template

Derivative by Definition
Table 9 contains the "Derivatives by Definition" task template.
Tools≻Tasks≻Browse:
Calculus  Differential≻Derivatives≻Derivatives by Definition


Table 9 The derivative of by definition, using the "Derivatives by Definition" task template

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 10.
Tools≻Tasks≻Browse:
Calculus  Differential≻Derivatives≻Difference (or Newton) Quotient


Table 10 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 11. This tutor could be accessed independently from the Tools≻Tutors menu.

Table 11 The Tangent (Newton Quotient) tutor for

Implicit Differentiation
The implicit derivative of defined by the equation can be obtained with the Context Menu option "Differentiate Implicitly." It can be obtained stepwise with the task template in Table 12.
Tools≻Tasks≻Browse:
Calculus  Differential≻Derivatives≻Implicit Differentiation≻

Implicit Differentiation

Enter an equation in two variables:


Dependent variable: Independent variable:

Implicit Derivative:


Stepwise Calculation

Make dependent variable explicit:


Differentiate with respect to independent variable:


Isolate Derivative:


Make independent variable implicit:



Table 12 Stepwise implicit differentiation via task template

Clicking the "Stepwise" button will launch the Differentiation Methods tutor in which the derivative can be computed stepbystep.
Mean Value Theorem
The Mean Value theorem states that under suitable conditions, for 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 13, where the tutor is applied to the function on the interval .

Table 13 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 14 contains a task template that might be a more convenient implementation of the Mean Value theorem calculations.
Tools≻Tasks≻Browse:
Calculus  Differential≻Theorems≻Mean Value Theorem


Table 14 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 15, constructed with the RollesTheorem command in the Student Calculus 1 package.
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Table 15 Rolle's theorem illustrated by the RollesTheorem command

The usage
returns the value of at which the horizontal tangent is found.
Table 16 contains a task template that might be a more convenient implementation of the Rolle's theorem calculations.
Tools≻Tasks≻Browse:
Calculus  Differential≻Theorems≻Rolle's Theorem


Table 16 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 17 shows the tutor applied to the function on the interval .

Table 17 The Curve Analysis tutor applied to

Clicking on the eight radiobuttons provides the raw data with which a graph could be sketched in the historic approach to this task.
Table 18 shows, for the function
some of this information being captured with a task template.
Tools≻Tasks≻Browse:
Calculus  Differential≻Graphical Analysis≻Find Special Points on a Function

> 


(11) 
> 

> 


(13) 
> 


(14) 
> 


(15) 
> 

> 


Table 18 The task template "Find Special Points on a Function" applied to

The graph in Table 17 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 floatingpoint numbers. These values are the same as those computed via
Maple's solve command returns the exact solutions on the left in Table 19. Although these solutions contain , they are actually real, as can be seen from their equivalents shown on the right.


Table 19 Exact zeros of the cubic function

Integral Calculus
Methods of Integration
The Integration Methods tutor, shown in Table 20 as a screenshot, 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 20 The Integration Methods tutor applied to

The sequence of steps can be copied and pasted from the tutor, but that does not salvage the perstep 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 21 is available via the ShowSteps command, or better yet, from the Context Menu under the Solve≻Show Solution Steps option (after loading the Student Calculus 1 package with the Tools menu option: Load Package).
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Table 21 Stepwise integration via the Solve≻Show Solution Steps option in the Context Menu

The integral operator is not black, but gray, the inert form of the operator, obtained by applying the Context Menu: 2D Math≻Convert To≻Inert Form to the operator in the Expression 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
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 22 shows the Riemann Sums tutor applied to this function.

Table 22 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 shows the interval divided into equal subintervals, each one supporting a rectangle 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 leftrectangles.
Table 23 shows via task template the analytic evaluation of the corresponding Riemann sum for , arbitrary, rectangles.
Tools≻Tasks≻Browse:
Calculus  Integral≻Integration≻Riemann Sums≻Left

The Left Riemann Sum

Enter :

> 


(17) 

Enter the interval :

> 


(18) 

Enter the value of :

> 


(19) 

The left Riemann sum:

> 


(20) 

Value of the Riemann sum:

> 


(21) 


Table 23 Analytic approach to left Riemann sum for by task template

Of course, the analytic expression obtained for this left Riemann sum approaches as .
Numeric Integration
The Riemann Sums tutor is actually the Approximate Integration tutor. This one tutor can be used to explore Riemann sums, or different methods of numeric integration. Underlying this tutor is the ApproximateInt command from the Student Calculus1 package.
Table 24 shows the ApproximateInt command applied to the function , integrated by the trapezoid rule. The command can output a graph, a sum, the value of the sum, or an animation.



Table 24 The ApproximateInt command

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 25.

Table 25 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 floatingpoint equivalent. Clicking the "Frustums" radio button and then the "Display" button will show the surface approximated by segments (frustums) of cones. After these choices have been made, the display will include a Riemannsum approximation corresponding to the discretization.
Volume of a Solid of Revolution
The volume of the solid of revolution formed when , is rotated about the axis can be computed by means of the Volume of Revolution tutor, as shown in Table 26.

Table 26 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 floatingpoint 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 Riemannsum approximation is provided.