Steps for Sketching a Curve
Maple 2022 includes new commands for showing the steps needed to manipulate algebraic expressions in order to reduce them to their simplest form.
In general the generated steps try to find that hard balance between being too verbose and too cryptic. The SimplifySteps command errs on the side of adding more steps, and is aimed to help someone who wants to learn what the steps are, even for fundamentals like adding fractions. Depending on the problem it will adjust somewhat; recognizing higher level problems for which it will decide to skip more easy-level steps.
Here are some examples of different categories of problems that SimplifySteps as well as related command FractionSteps can handle:
There is a dedicated FractionSteps command that goes into slightly more detail than SimplifySteps
FractionSteps 1/2 + 1/6
Let's Simplify Fractions•Find fractions to get lowest common denominator of6•Multiply•Add numerators•Multiply3⋅1•Multiply1⋅1•Add3+1•Cancel out factor of2
SimplifySteps 1/2 + 1/6
Let's simplify•Find fractions to get lowest common denominator of6•Multiply•Add fractions23
Let's simplify•Pull out a factor of4=2from12•Multiply in order to rationalize the denominator•Multiply the denominator•Factor roots•Combine2⋅2,3⋅3•Multiply3⋅2⁢5•Cancel out factor of65
Let's simplify•Multiply in order to rationalize the denominator•Multiply the denominator•Factor roots•Combine323⋅3,213⋅2⋅2•Multiply3⁢316⋅2⁢213⁢5•Multiply2⋅6⁢5⁢213⁢316•Cancel out factor of125⁢213⁢316
Let's simplify•Evaluate exponent3⁢x⁢y22•Multiply2⋅x3⋅y3⋅9⁢x2⁢y418⁢x5⁢y7
Let's simplify•Apply the product rulean⁢am=an+mto add exponents with common base•Add exponents•Solution
Let's simplify•Apply the integer power of a power rule,anm=an⁢m
Let's simplify•Cancel out factor ofy4providedy4≠0y
Let's simplify•Divide assumingy4≠ 01y9
Let's simplify•Apply the quotient rule:anam=an−m•Evaluate exponent1231123
Let's simplify•Use the log rule,loga⁡x=logb⁡xlogb⁡ato express as a single logarithm•Solution
Note: This is different than how Maple's simplify command treats expressions like this, always converting to ln:
Let's simplify•Apply the log ruleloga⁡mn=n⁢loga⁡m•Apply the log ruleloga⁡a=1•Multiply5⋅420
You can also use the new PowerSteps command to get step by step results for problems with radicals, exponents, and logarithms.
Let's simplify•ApplyPythagorastrig identity,cot⁡x2=csc⁡x2−11+csc⁡x2−1•ApplyReciprocal Functiontrig identity,csc⁡x=1sin⁡x1sin⁡x2•Evaluate1sin⁡x2
SimplifySteps"sinPi + Pi + x"
Let's simplify•ApplyReciprocal Functiontrig identity,sec⁡x=1cos⁡xsec⁡x2−11cos⁡x2•ApplyPythagorastrig identity,sec⁡x2=1+tan⁡x21+tan⁡x2−1⁢cos⁡x2•ApplyQuotienttrig identity,tan⁡x=sin⁡xcos⁡xsin⁡xcos⁡x2⁢cos⁡x2•Evaluatesin⁡x2
Let's simplify•ApplyPythagorastrig identity,cot⁡x2=csc⁡x2−1−cos⁡x2+1⁢1+csc⁡x2−1•ApplyReciprocal Functiontrig identity,csc⁡x=1sin⁡x−cos⁡x2+1⁢1sin⁡x2•ApplyPythagorastrig identity,cos⁡x2=1−sin⁡x2−1−sin⁡x2+1sin⁡x2•Evaluate1
You can also use the new TrigSteps command to get these step by step results.
Showing the steps to solving an integral, limit, or derivative has been available in past versions of Maple via the Student:-Calculus1:-ShowSolution command. You can now also access those step by step solutions through SimplifySteps, further unifying the ability to do step by step solutions using a single command.
Let's simplify•Integralto evaluate∫xⅆx▫1. Apply thepowerrule to the term∫xⅆx◦Recall the definition of thepowerrule, for n≠-1∫xⅆx=◦This means:∫xⅆx=◦So,∫xⅆx=x22We can rewrite the integral as:x22•Sub evaluatedintegralback in expression3⁢x22
Maple 2022 includes a new command for showing the steps needed to sketch the graph of an expression by identifying the basic function and the transformations done to the function. Various kinds of expressions are handled, including trig, logs, and polynomials to pick just a few. Here are some examples:
Let's plot2⁢sin⁡3⁢x+π3+1•Compared to the plot ofsin⁡x, we have a vertical stretch by a factor of2•Then, we have a horizontal compression by a factor of13•Then, we have a vertical shift of1•Then, we have a horizontal shift of−π9•Apply the horizontal shift and stretch to the range,x=−2⁢π..2⁢π+π9..+π9=−2.443460953..1.745329252•We can now plot using the information extractedPLOT⁡...
Let's plot2⁢x2+4⁢x+10•Complete the square2⁢x+12+8•With the expression in vertex form we can extract valuable information•The coefficient2of thex+12term indicates a parabola that opensupand has a verticalstretchof2•We have a horizontal shift of−1and a vertical shift of8which gives a vertex of (−1,8)•We can now plot using the information extractedPLOT⁡...
Let's plot4⁢x+10•This is a line; find two points and draw a line through themy=4⁢x+10•Setx= 0 to solve for y intercepty=10•This gives a y intercept of (0,10)y=10•Set expresson to 0 to solve forxintercept0=•Subtract4⋅xfrom both sides=•Simplify=10•Divide both sides by−4=•Simplifyx=−52•This gives anxintercept of (−52,0)x=−52•By connecting through the two points we can plot the linePLOT⁡...
Let's plot23⁢x+2•Rewrite the equation in the following form•Compared to the plot of1x, we have a vertical stretch by a factor of2PLOT⁡...•Then, we have a horizontal compression by a factor of13PLOT⁡...•Then, we have a horizontal shift of23PLOT⁡...•The final plot with asymptotes in cyan aty=0andx=−23isPLOT⁡...
For more information, see the help page Student:-Basics:-CurveSketchSteps.
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