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Maple 2023 includes a number of improvements to support teaching and learning of mathematics and science.


Step-by-Step Solutions

New Physics Courseware Support: Mechanics

Step-by-Step Solutions

Maple 2023 improves the existing suite of commands for showing step-by-step solutions to standard math problems.  It also adds some new methods as follows:

Implicit Differentiation Steps

The function f whose rule is given by fx=x2+x+1, is said to be defined explicitly. The function yx whose rule must be extracted from an equation of the form Fx,y=0 is said to be defined implicitly.

A simple example is the circle, defined by x2+y2=9, where y±x= ±9x2 are two different explicit functions that can be extracted from the equation of the circle. The semicircle above the x-axis is defined by y+x=9x2; and below, by yx= 9x2.

Implicit differentiation is a technique by which yx can be obtained without necessarily having to solve for yx explicitly. It is merely the Chain rule applied to the identity Fx,yx=0.

Maple can show you the steps required to implicitly differentiate with the new command ImplicitDiffSolution.


Student:-Calculus1:-ImplicitDiffSolutionx2+y2=9, y, x

Implicit Differentiation Stepsx2+y2=9Rewriteyas a functionyx:x2+yx2=9Differentiate the left sideⅆⅆxx2+yx21. Apply thesumruleRecall the definition of thesumruleⅆⅆxfx+gx=ⅆⅆxfx+ⅆⅆxgxfx=x2gx=yx2This gives:ⅆⅆxx2+ⅆⅆxyx22. Apply thepowerrule to the termⅆⅆxx2Recall the definition of thepowerrulexx=x1This means:ⅆⅆxx2=So,ⅆⅆxx2=We can rewrite the derivative as:3. Apply thechainrule to the termyx2Recall the definition of thechainruleⅆⅆxfgx=f'gxⅆⅆxgxOutside functionfv=v2Inside functiongx=yxDerivative of outside functionⅆⅆvfv=2vApply compositionf'gx=2yxDerivative of inside functionⅆⅆxgx=ⅆⅆxyxPut it all togetherⅆⅆxfgxⅆⅆxgx=This gives:The final result isDifferentiate the right sideⅆⅆx91. Apply theconstantrule to the termⅆⅆx9Recall the definition of theconstantruleⅆCⅆx=0This means:ⅆⅆx9=0We can now rewrite the derivative as:0Rewriteⅆⅆxyxasy'and solve fory'=0Subtract2xfrom both sides=Simplify=Divide both sides by2y=Simplifyy'=Reduce fraction by gcdy'=xy


Complete the Square Steps

Completing the square is a standard approach that takes a trinomial of degree 2 and rewrites it as a binomial made up of a perfect square plus a remainder.  This is a useful method for getting a quadratic into a form that is easier to work with, and is often used as a first step in solving a quadratic equation.

There is a new command CompleteSquareSteps that shows the algebraic steps required to complete the square:

Student:-Basics:-CompleteSquareSteps3x2+2x + 1,x

Add and subtract13222Simplify termsThe first3terms can be regrouped as a perfect squareSimplify the remaining term


Long Division Result

Maple 2023 adds a new option to the LongDivision command that makes it clear how the inputs relate back to the computed result, especially when the remainder is not zero.

In the examples below, the division is carried out, and then, below the long division, it shows an answer derived from the long division that is equal to the dividend/divisor.  

Student:-Basics:-LongDivision 2, 3,digits=3,  'appendresult'=true 

 — — —.—6—6—6— 3 ) 2.000      1—8—       20       1—8—        20        1—8—         2=%+0.666,11500


 Student:-Basics:-LongDivision 3x2+2x+1, x+3, 'appendresult'=true 




New Physics Courseware Support: Mechanics

Maple 2023 now has Physics Courseware support for Mechanics.  This new set of content is a helpful complement for a physics mechanics course.  It contains typical symbolic problems and shows how they can be solved in a Maple worksheet, demonstrating how computer algebra can support the learning activity.
The material covers several key topics such as equations of motion, curvilinear coordinates, conservation laws, integration of the equations of motion, Kepler's problem, oscillations, rigid-body motion and non-inertial coordinate systems.  It utilizes the Physics:-Vectors package to handle abstract vectors as well as projections using unit vectors.