rlopez

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17 years, 212 days

Dr. Robert J. Lopez, Emeritus Professor of Mathematics at the Rose-Hulman Institute of Technology in Terre Haute, Indiana, USA, is an award winning educator in mathematics and is the author of several books including Advanced Engineering Mathematics (Addison-Wesley 2001). For over two decades, Dr. Lopez has also been a visionary figure in the introduction of Maplesoft technology into undergraduate education. Dr. Lopez earned his Ph.D. in mathematics from Purdue University, his MS from the University of Missouri - Rolla, and his BA from Marist College. He has held academic appointments at Rose-Hulman (1985-2003), Memorial University of Newfoundland (1973-1985), and the University of Nebraska - Lincoln (1970-1973). His publication and research history includes manuscripts and papers in a variety of pure and applied mathematics topics. He has received numerous awards for outstanding scholarship and teaching.

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These are answers submitted by rlopez

RJL Maplesoft The following commands will result in a graph of the surface, its tangent plane at (1,1,1), and the normal line through (1,1,1). However, this isn't a "directional derivative" issue per se. It might arise while studying the meaning of the directional derivative, but in itself, the construction does not require the concept of "directional derivative." with(VectorCalculus): BasisFormat(false): with(plots): q := x*y+y*z+z*x-3; V := Gradient(q,cartesian[x,y,z]); N := evalVF(V,<1,1,1>); L := <1,1,1>+t*N; Z := solve(q,z); TPZ := mtaylor(Z,[x=1,y=1],2); p1 := implicitplot3d(q=0,x=-1..3,y=-1..3,z=-1..3): p2 := plot3d(TPZ, x=0..2,y=0..2): p3 := spacecurve(L,t=-1/2..1/2, color=black, thickness=2): display([p||(1..3)], scaling=constrained, axes=box);
For example, to apply the ratio test to the series Sum(1/2^n, n=0..infinity), define > a := n -> 1/2^n: > 'a'[n] = a(n); then compute > r = limit(a(n+1)/a(n), n=infinity); We suggest two lines to define a[n]. The first defines it as a functin of n, the second displays it more naturally. The left side of the output of the second line appears as a[n], whereas the right side appears as 1/2^n. Similar computations apply to the nth-root test, etc. There is no single command for determining the convergence or divergence of a series. If the whole series is entered as > S := Sum(1/2^n, n=0..infinity); then the command > value(S); will produce a finite result, in which case it's clear the series converged. If Maple can determine the series diverges, it will return the symbol "infinity" or -infinity. If Maple returns unevaluated, it means that Maple could not determine a limit for the partial sums, but that does not mean such a limit does not exist. It only means that Maple couldn't determine that limit. There is no single command in Maple for writing the general term in a series. The taylor (or series) commands produce a partial sum, with the terms written out (not in in sigma-notation). However, the nth derivative of an expression can be computed. Hence, it is possible to build the general form of a series expansion. For example, to obtain the general expansion of sin(x) at x = 0, implement the following commands. > q := eval(diff(sin(x),x$n)/n!, x=0); > q1 := simplify(eval(q, n=2*k+1)) assuming k::posint; The first command produces sin(n*Pi/2)/n! whereas the second produces (-1)^k/(2*k+1)!. User insight and intervention is definitely needed here! The requisite Maclaurin expansion is then obtained with > Sum(q1*x^(2*k+1), k=0..infinity); The Share Library, a repository for code contributed by users, contained the FPS package with its FormalPowerSeries command. This command would return the formal power series in sigma notation. However, the Share Library is no longer available.
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