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6.1 Introductory Calculus • 223You can now plot f, its Taylor approximation, and a pair of curvesindicating the error band.> plot( [ f(x), poly, f(x)+max_error, f(x)-max_error ],> x=2..4,> color=[ red, blue, brown, brown ] );2.521.510.52 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 4xThe plot shows that the actual error stays well inside the error estimate.The IntegralThe integral of a function can be considered as a measure of the areabetween the x-axis and the graph of the function. The definition of theRiemann integral relies on this graphical interpretation of the integral.> f := x -> 1/2 + sin(x);f := x → 1 2 + sin(x)For example, the ApproximateInt command with method = left,partition = 6, and output = plot specified, from the Student[Calculus1]package draws the graph of f along with 6 boxes. The height of each boxis the value of f evaluated at the left-hand side of the box.> with(Student[Calculus1]):> ApproximateInt( f(x), x=0..10, method=left, partition=6,> output=plot);
224 • Chapter 6: Examples from CalculusChange output = plot to output = sum to calculate the area of theboxes.> ApproximateInt( f(x), x=0..10, method=left, partition=6,> output=sum);( 5∑)5( 1 3 2 + sin(5 3 i)) i=0Approximately, this number is> evalf( % );6.845601766The approximation of the area improves as you increase the numberof boxes. Increase the number of boxes to 12 and calculate the value ofApproximateInt for each of these boxes.> seq( evalf(ApproximateInt( f(x), x=0..10, method=left,> partition=n^2)), n=3..14);6.948089404, 6.948819107, 6.923289158, 6.902789479,6.888196447, 6.877830054, 6.870316620, 6.864739771,6.860504866, 6.857222010, 6.854630209, 6.852550665Use the option output = animation to create an animation for theleft Reimann sum.
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Learning Guidecommand the brillianc
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ii•Waterloo Maple Inc.57 Erb Stre
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iv • ContentsThe numer and denom
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vi • ContentsThe Parts of an Expr
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viii • Contents
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2 • Chapter 1: Introduction to Ma
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98 • Chapter 4: Graphics> f := x
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100 • Chapter 4: Graphics> plot(
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146 • Chapter 5: Evaluation and S
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294 • Indexexact, 8finite rings a
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296 • Indexspherical, 114viewing,
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298 • IndexXML, 283ExportMatrix,
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300 • IndexHP LaserJet, 284HTML,
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