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6 Examples from CalculusThis chapter provides examples of how Maple can help you present andsolve problems from calculus. The first section describes elementary conceptssuch as the derivative and the integral, the second section discussesordinary differential equations in some depth, and the third section concernspartial differential equations.6.1 Introductory CalculusThis section contains a number of examples of how to illustrate ideas andsolve problems from calculus. The Student[Calculus1] package containsmany commands that are especially useful in this area.The DerivativeThis section illustrates the graphical meaning of the derivative: the slopeof the tangent line. Then it shows you how to find the set of inflectionpoints for a function.Define the function f: x ↦→ exp(sin(x)) in the following manner.> f := x -> exp( sin(x) );f := x → e sin(x)Find the derivative of f evaluated at x 0 = 1.> x0 := 1;x0 := 1p 0 and p 1 are two points on the graph of f.205
206 • Chapter 6: Examples from Calculus> p0 := [ x0, f(x0) ];p0 := [1, e sin(1) ]> p1 := [ x0+h, f(x0+h) ];p1 := [1 + h, e sin(1+h) ]The NewtonQuotient command from the Student package finds theslope of the secant line through p 0 and p 1 .> with(Student[Calculus1]):Use NewtonQuotient command to find the expression for the slope.> m := NewtonQuotient(f(x), x=x0, h=h);m := esin(1+h) − e sin(1)hIf h = 1, the slope is> eval(%, h=1);e sin(2) − e sin(1)The evalf command gives a floating-point approximation.> evalf( % );0.162800903As h tends to zero, the secant slope values seem to converge.> h_values := seq( NewtonQuotient( f(x), x=1.0, h=1.0/i^2 ),> i=1..20);h_values := 0.1628009030, 1.053234748, 1.174305789,1.210917616, 1.226806975, 1.235154852, 1.240091510,1.243256512, 1.245408615, 1.246939100, 1.248066842,1.248921648, 1.249585493, 1.250111128, 1.250534250,1.250880512, 1.251167122, 1.251406908, 1.251609743,1.251782800
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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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4 • Chapter 1: Introduction to Ma
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6 • Chapter 2: Mathematics with M
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8 • Chapter 2: Mathematics with M
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98 • Chapter 4: Graphics> f := x
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100 • Chapter 4: Graphics> plot(
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104 • Chapter 4: Graphics0.40.2-1
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146 • Chapter 5: Evaluation and S
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270 • Chapter 7: Input and Output
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288 • Chapter 8: Maplets8.2 Termi
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292 • Chapter 8: Maplets8.10 Conc
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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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310 • Indexunordered lists (sets)
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