By C F Koval; University of Wisconsin--Extension. Cooperative Extension Programs
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Gewöhnliche Differentialgleichungen: Theorie und Praxis - vertieft und visualisiert mit Maple®
Die Theorie der Gewöhnlichen Differentialgleichungen ist ein grundlegendes und unverändert aktuelles Gebiet der Mathematik. Das vorliegende Buch führt nicht nur äußerst sorgfältig und umfassend in die Theorie ein, sondern vermittelt auch aufgrund der zahlreichen vollständig durchgerechneten Beispiele einen Einblick in deren Anwendungspraxis.
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Example text
F (x) and g(x) are not returned because a colon is included at the end of each command. SOLUTION: After defining f (x) and g(x), > f:=x->(-1-2*x)/(-4+x): > g:=x->(4*x-1)/(x+2): we compute and simplify the compositions f (g(x)) and g(f (x)). Because both results are x, f (x) and g(x) are inverse functions. 3 Graphing Functions, Expressions, and Equations 47 10 5 -10 0 -5 0 x 5 10 -5 -10 Figure 2-7 f (x) in black, g(x) in gray, and y = x dashed > simplify((g@f)(x)); x To see that the graphs of f (x) and g(x) are symmetric about the line y = x, we use plot to graph f (x), g(x), and y = x together in Figure 2-7, illustrating the use of the color and linestyle options.
With evalf(expr,n), Maple yields a numerical approximation of expr to n digits of precision, if possible. At other times, simplify can be used to produce the expected results. Remark. If the expression b in ab contains more than one symbol, be sure that the exponent is included in parentheses. Entering aˆn/m computes an /m = m1 an while entering aˆ(n/m) computes an/m . 2: Compute (a) √ 27 and (b) 21 √ 3 2 8 = 82/3 . SOLUTION: (a) Maple automatically simplifies √ √ 27 = 3 3. > sqrt(27); 3 √ 3 We use evalf to obtain an approximation of √ 27.
7182818284590452353602874713526624977572470937000 returns a 50-digit approximation of e. 141592653589793238462643 returns a 25-digit approximation of π. Entering > (3+I)/(4-I); 7 11 + i 17 17 performs the division (3 + i)/(4 − i) and writes the result in standard form. 3 Built-In Functions Maple contains numerous mathematical functions. Functions frequently encountered by beginning users include the exponential function, exp(x); the natural logarithm, ln(x); the absolute value function, abs(x); the square root function, sqrt(x); the trigonometric functions sin(x), cos(x), tan(x), sec(x), csc(x), and cot(x); the inverse trigonometric functions arcsin(x), arccos(x), arctan(x), arcsec(x), arccsc(x), and arccot(x); the hyperbolic trigonometric functions sinh(x), cosh(x), and tanh(x); and their inverses arcsinh(x), arccosh(x), and arctanh(x).