Maple 2023 includes a number of improvements to support teaching and learning of mathematics and science.
New Physics Courseware Support: Mechanics
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= ±9−x2 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=9−x2; and below, by y−x= −9−x2.
Implicit differentiation is a technique by which y′x 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=9•Rewriteyas a functiony⁡x:x2+y⁡x2=9•Differentiate the left sideⅆⅆxx2+y⁡x2▫1. Apply thesumrule◦Recall the definition of thesumruleⅆⅆxf⁡x+g⁡x=ⅆⅆxf⁡x+ⅆⅆxg⁡xf⁡x=x2g⁡x=y⁡x2This gives:ⅆⅆxx2+ⅆⅆxy⁡x2▫2. Apply thepowerrule to the termⅆⅆxx2◦Recall the definition of thepowerrule∂∂xx=⁢x−1◦This means:ⅆⅆxx2=◦So,ⅆⅆxx2=We can rewrite the derivative as:▫3. Apply thechainrule to the termy⁡x2◦Recall the definition of thechainruleⅆⅆxf⁡g⁡x=f'⁡g⁡x⁢ⅆⅆxg⁡x◦Outside functionf⁡v=v2◦Inside functiong⁡x=y⁡x◦Derivative of outside functionⅆⅆvf⁡v=2⁢v◦Apply compositionf'⁡g⁡x=2⁢y⁡x◦Derivative of inside functionⅆⅆxg⁡x=ⅆⅆxy⁡x◦Put it all togetherⅆⅆxf⁡g⁡x⁢ⅆⅆxg⁡x=This gives:•The final result is•Differentiate the right sideⅆⅆx9▫1. Apply theconstantrule to the termⅆⅆx9◦Recall the definition of theconstantruleⅆCⅆx=0◦This means:ⅆⅆx9=0We can now rewrite the derivative as:0•Rewriteⅆⅆxy⁡xasy'and solve fory'=0•Subtract2⋅xfrom both sides=•Simplify=•Divide both sides by2⋅y=•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:-CompleteSquareSteps3⁢x2+2⁢x + 1,x
•Add and subtract13⋅222•Simplify terms•The first3terms can be regrouped as a perfect square•Simplify 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 3⋅x2+2⋅x+1, x+3, 'appendresult'=true
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.
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