diff --git a/changelog b/changelog index 0aeec85..19aff4b 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20070812 tpd re-merge input branch 20070811 tpd src/input/Makefile add classtalk, calcprob 20070811 tpd src/input/limit.input.pamphlet updated with new tests 20070811 tpd src/input/intbypart.input.pamphlet updated with new tests diff --git a/src/input/Makefile.pamphlet b/src/input/Makefile.pamphlet index 621d8ed..34850f2 100644 --- a/src/input/Makefile.pamphlet +++ b/src/input/Makefile.pamphlet @@ -292,9 +292,10 @@ REGRES= algaggr.regress algbrbf.regress algfacob.regress alist.regress \ binary.regress bop.regress bstree.regress bouquet.regress \ bug10069.regress \ bugs.regress bug10312.regress bug6357.regress bug9057.regress \ + calcprob.regress \ calculus2.regress calculus.regress cardinal.regress card.regress \ carten.regress cclass.regress char.regress ch.regress \ - chtheorem.regress \ + chtheorem.regress classtalk.regress \ clifford.regress clif.regress coercels.regress collect.regress \ complex.regress conformal.regress \ constant.regress contfrac.regress contfrc.regress \ @@ -486,10 +487,10 @@ FILES= ${OUT}/algaggr.input ${OUT}/algbrbf.input ${OUT}/algfacob.input \ ${OUT}/bernpoly.input ${OUT}/binary.input ${OUT}/bop.input \ ${OUT}/bouquet.input ${OUT}/bstree.input ${OUT}/bug6357.input \ ${OUT}/bug9057.input ${OUT}/bug10069.input ${OUT}/bug10312.input \ - ${OUT}/calculus.input \ + ${OUT}/calcprob.input ${OUT}/calculus.input \ ${OUT}/cardinal.input ${OUT}/card.input ${OUT}/carten.input \ ${OUT}/cclass.input ${OUT}/cdraw.input ${OUT}/char.input \ - ${OUT}/ch.input ${OUT}/chtheorem.input \ + ${OUT}/ch.input ${OUT}/chtheorem.input ${OUT}/classtalk.input \ ${OUT}/clifford.input ${OUT}/clif.input \ ${OUT}/coercels.input ${OUT}/collect.input ${OUT}/color.input \ ${OUT}/complex.input ${OUT}/cone.input ${OUT}/conformal.input \ @@ -669,12 +670,13 @@ DOCFILES= \ ${DOC}/c06fqf.input.dvi ${DOC}/c06frf.input.dvi \ ${DOC}/c06fuf.input.dvi ${DOC}/c06gbf.input.dvi \ ${DOC}/c06gcf.input.dvi ${DOC}/c06gqf.input.dvi \ - ${DOC}/c06gsf.input.dvi ${DOC}/calculus2.input.dvi \ + ${DOC}/c06gsf.input.dvi ${DOC}/calcprob.input.dvi \ + ${DOC}/calculus2.input.dvi \ ${DOC}/calculus.input.dvi ${DOC}/cardinal.input.dvi \ ${DOC}/card.input.dvi ${DOC}/carten.input.dvi \ ${DOC}/cclass.input.dvi ${DOC}/cdraw.input.dvi \ ${DOC}/char.input.dvi ${DOC}/ch.input.dvi \ - ${DOC}/chtheorem.input.dvi \ + ${DOC}/chtheorem.input.dvi ${DOC}/classtalk.input.dvi \ ${DOC}/clifford.input.dvi ${DOC}/clif.input.dvi \ ${DOC}/coercels.input.dvi ${DOC}/collect.input.dvi \ ${DOC}/color.input.dvi ${DOC}/complex.input.dvi \ diff --git a/src/input/calcprob.input.pamphlet b/src/input/calcprob.input.pamphlet new file mode 100644 index 0000000..f41808d --- /dev/null +++ b/src/input/calcprob.input.pamphlet @@ -0,0 +1,124 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input calcprob.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +Cover a range of calculus problems +\end{abstract} +\eject +\tableofcontents +\eject +<<*>>= +)spool calcprob.output +)set message test on +)set message auto off +)clear all + +--S 1 +solve(3*x-(x-7)=4*x-5,x) +--R +--R (1) [x= 6] +--R Type: List Equation Fraction Polynomial Integer +--E 1 + +--S 2 +solve(4*x-3*y=9,y)::List Equation Polynomial Fraction Integer +--R +--R 4 +--R (2) [y= - x - 3] +--R 3 +--R Type: List Equation Polynomial Fraction Integer +--E 2 + +--S 3 +solve(A*x+B*y=C,y) +--R +--R - A x + C +--R (3) [y= ---------] +--R B +--R Type: List Equation Fraction Polynomial Integer +--E 3 + +--S 4 +m:=3*x-4*(x-(2/3)*y)=(4/5)*x-(7*y+3) +--R +--R 8 4 +--R (4) - y - x= - 7y + - x - 3 +--R 3 5 +--R Type: Equation Polynomial Fraction Integer +--E 4 + +--S 5 +n:=solve(m*15,y) +--R +--R 27x - 45 +--R (5) [y= --------] +--R 145 +--R Type: List Equation Fraction Polynomial Integer +--E 5 + +--S 6 +p:=n.1*145-27*x +--R +--R (6) 145y - 27x= - 45 +--R Type: Equation Fraction Polynomial Integer +--E 6 + +--S 7 +(x1,y1):=(-3,-8) +--R +--R (7) - 8 +--R Type: Integer +--E 7 + +--S 8 +(x2,y2):=(-6,2) +--R +--R (8) 2 +--R Type: PositiveInteger +--E 8 + +--S 9 +m:=(y2-y1)/(x2-x1) +--R +--R 10 +--R (9) - -- +--R 3 +--R Type: Fraction Integer +--E 9 + +--S 10 +solve(y1=m*x1+b,b) +--R +--R (10) [b= - 18] +--R Type: List Equation Fraction Polynomial Integer +--E 10 + +--S 11 +b:=-18 +--R +--R (11) - 18 +--R Type: Integer +--E 11 + +--S 12 +y=m*x+b +--R +--R 10 +--R (12) y= - -- x - 18 +--R 3 +--R Type: Equation Polynomial Fraction Integer +--E 12 +)spool +)lisp (bye) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} + + diff --git a/src/input/classtalk.input.pamphlet b/src/input/classtalk.input.pamphlet new file mode 100644 index 0000000..7f89df1 --- /dev/null +++ b/src/input/classtalk.input.pamphlet @@ -0,0 +1,733 @@ +\documentclass{article} +\usepackage{axiom} +\begin{document} +\title{\$SPAD/src/input classtalk.input} +\author{Timothy Daly} +\maketitle +\begin{abstract} +These are examples from the talk ``Axiom in an Educational Setting''. +\end{abstract} +\eject +\tableofcontents +\eject +<<*>>= +)spool classtalk.output +)set message test on +)set message auto off +)set break resume +)clear all + +@ +\section{Numbers} +<<*>>= +--S 1 +1 +--R +--R (1) 1 +--R Type: PositiveInteger +--E 1 + +--S 2 +1/2 +--R +--R 1 +--R (2) - +--R 2 +--R Type: Fraction Integer +--E 2 + +--S 3 +3+4*%i +--R +--R (3) 3 + 4%i +--R Type: Complex Integer +--E 3 + +--S 4 +3.4 +--R +--R (4) 3.4 +--R Type: Float +--E 4 + +--S 5 +X::ROMAN +--R +--R (5) X +--R Type: RomanNumeral +--E 5 + +--S 6 +binary(5) +--R +--R (6) 101 +--R Type: BinaryExpansion +--E 6 + +--S 7 +factor(60) +--R +--R 2 +--R (7) 2 3 5 +--R Type: Factored Integer +--E 7 + +--S 8 +q:=(y-1)*x*(z+5) +--R +--R (8) (x y - x)z + 5x y - 5x +--R Type: Polynomial Integer +--E 8 + +--S 9 +factor q +--R +--R (9) x(y - 1)(z + 5) +--R Type: Factored Polynomial Integer +--E 9 + +--S 10 +eval(q,[x=5,y=6,z=7]) +--R +--R (10) 300 +--R Type: Polynomial Integer +--E 10 + +--S 11 +eval(q,[x=5,y=6]) +--R +--R (11) 25z + 125 +--R Type: Polynomial Integer +--E 11 + +@ +\section{Trigonometry} +<<*>>= +--S 12 +b:=[log a, exp a, asin a, acos a, atan a, acot a, sinh a] +--R +--R a +--R (12) [log(a),%e ,asin(a),acos(a),atan(a),acot(a),sinh(a)] +--R Type: List Expression Integer +--E 12 + +--S 13 +[exp b.1, log b.2, sin b.3, cos b.4, tan b.5, cot b.6, asinh b.7] +--R +--R (13) [a,a,a,a,a,a,a] +--R Type: List Expression Integer +--E 13 + +--S 14 +a:=.7 +--R +--R (14) 0.7 +--R Type: Float +--E 14 + +--S 15 +b:=[log a, exp a, asin a, acos a, atan a, acot a, sinh a] +--R +--R (15) +--R [- 0.3566749439 3873237891, 2.0137527074 704765216, 0.7753974966 1075306374, +--R 0.7953988301 8414355549, 0.6107259643 8920861654, 0.9600703624 0568800269, +--R 0.7585837018 3953350346] +--R Type: List Float +--E 15 + +--S 16 +[exp b.1, log b.2, sin b.3, cos b.4, tan b.5, cot b.6, asinh b.7] +--R +--R (16) [0.7,0.7,0.7,0.7,0.7,0.7,0.7] +--R Type: List Float +--E 16 + +--S 17 +simplify(sin(x)**2+cos(x)**2) +--R +--R (17) 1 +--R Type: Expression Integer +--E 17 + +@ +\section{Polynomial Manipulations} +<<*>>= +)clear all +--S 18 +eq1:=A*x^2 + B*x*y + C*y^2 + D*x + E*y + F +--R +--R 2 2 +--R (1) C y + (B x + E)y + A x + D x + F +--R Type: Polynomial Integer +--E 18 + +--S 19 +rotatex:=x'*cos(t)-y'*sin(t) +--R +--R (2) - y' sin(t) + x' cos(t) +--R Type: Expression Integer +--E 19 + +--S 20 +rotatey:=x'*sin(t)+y'*cos(t) +--R +--R (3) x' sin(t) + y' cos(t) +--R Type: Expression Integer +--E 20 + +--S 21 +eval(eq1,[x=rotatex, y=rotatey]) +--R +--R (4) +--R 2 2 2 +--R (A y' - B x' y' + C x' )sin(t) +--R + +--R 2 2 +--R ((- B y' + (2C - 2A)x' y' + B x' )cos(t) - D y' + E x')sin(t) +--R + +--R 2 2 2 +--R (C y' + B x' y' + A x' )cos(t) + (E y' + D x')cos(t) + F +--R Type: Expression Integer +--E 21 + +@ +\section{Polynomials over Simple Algebraic Extension Fields} +<<*>>= +)clear all +--S 22 +a:=rootOf(a^2+a+1) +--R +--R (1) a +--R Type: AlgebraicNumber +--E 22 + +--S 23 +factor(x^2+3) +--R +--R 2 +--R (2) x + 3 +--R Type: Factored Polynomial Integer +--E 23 + +--S 24 +factor(x^2+3,[a]) +--R +--R (3) (x - 2a - 1)(x + 2a + 1) +--R Type: Factored Polynomial AlgebraicNumber +--E 24 + +--S 25 +definingPolynomial(a) +--R +--R 2 +--R (4) a + a + 1 +--R Type: AlgebraicNumber +--E 25 + +--S 26 +zerosOf(b^2+b+1,b) +--R +--R +---+ +---+ +--R \|- 3 - 1 - \|- 3 - 1 +--R (5) [----------,------------] +--R 2 2 +--R Type: List Expression Integer +--E 26 + +@ +\section{Derivatives} +<<*>>= +--S 27 +differentiate(sin(x),x) +--R +--R (6) cos(x) +--R Type: Expression Integer +--E 27 + +--S 28 +differentiate(sin(x),x,2) +--R +--R (7) - sin(x) +--R Type: Expression Integer +--E 28 + +--S 29 +differentiate(cos(z)/(x^2+y^3),[x,y,z],[1,2,3]) +--R +--R 4 3 +--R (- 84x y + 24x y)sin(z) +--R (8) -------------------------------- +--R 12 2 9 4 6 6 3 8 +--R y + 4x y + 6x y + 4x y + x +--R Type: Expression Integer +--E 29 + +--S 30 +y:=operator y +--R +--R (9) y +--R Type: BasicOperator +--E 30 + +--S 31 +deqx:=D(y(x),x,2)+D(y(x),x)+y(x) +--R +--R +--R ,, , +--R (10) y (x) + y (x) + y(x) +--R +--R Type: Expression Integer +--E 31 + +--S 32 +solve(deqx,y,x) +--R +--R x x +--R +-+ - - - - +-+ +--R x\|3 2 2 x\|3 +--R (11) [particular= 0,basis= [cos(-----)%e ,%e sin(-----)]] +--R 2 2 +--RType: Union(Record(particular: Expression Integer,basis: List Expression Integer),...) +--E 32 + +@ +\section{Limits} +<<*>>= +)clear all +--S 33 +limit((x^2-3*x+2)/(x^2-1),x=1) +--R +--R 1 +--R (1) - - +--R 2 +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 33 + +--S 34 +limit(x*log(x),x=0) +--R +--R (2) [leftHandLimit= "failed",rightHandLimit= 0] +--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +--E 34 + +--S 35 +limit(sinh(a*x)/tan(b*x),x=0) +--R +--R a +--R (3) - +--R b +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 35 + +--S 36 +limit(sqrt(3*x^2+1)/(5*x),x=%plusInfinity) +--R +--R +-+ +--R \|3 +--R (4) ---- +--R 5 +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 36 + +--S 37 +complexLimit((2+z)/(1-z),z=%infinity) +--R +--R (5) - 1 +--R Type: OnePointCompletion Fraction Polynomial Integer +--E 37 + +@ +\section{Indefinite Integration} +<<*>>= +)clear all +--S 38 +integrate(1+sqrt(x)/x,x) +--R +--R +-+ +--R (1) 2\|x + x +--R Type: Union(Expression Integer,...) +--E 38 + +--S 39 +integrate(sin(x)/x,x) +--R +--R (2) Si(x) +--R Type: Union(Expression Integer,...) +--E 39 + +@ +This used to give the answer: +$$\frac{\sqrt{x}\sqrt{\pi} erf(x\sqrt{a})}{2a}$$ +<<*>>= +--S 40 +integrate(exp(-a*x^2),x) +--R +--R x 2 +--R ++ - %Q a +--R (3) | %e d%Q +--R ++ +--R Type: Union(Expression Integer,...) +--E 40 + +--S 41 +integrate(sin(x)/x^2,x) +--R +--R x +--R ++ sin(%Q) +--R (4) | ------- d%Q +--R ++ 2 +--R %Q +--R Type: Union(Expression Integer,...) +--E 41 + +@ +\section{Definite Integration} +<<*>>= +)clear all +--S 42 +integrate(exp(-x)/sqrt(x),x=0..%plusInfinity) +--R +--R _ 1 +--R (1) | (-) +--R 2 +--R Type: Union(f1: OrderedCompletion Expression Integer,...) +--E 42 + +--S 43 +integrate(1/x^2,x=-1..1) +--R +--R +--RDaly Bug +--R >> Error detected within library code: +--R integrate: pole in path of integration +--R +--R Continuing to read the file... +--R +--E 43 + +)clear all + +@ +This used to return +$$\frac{4\log{(4)}-8\log{(2)}+3\pi}{12}$$ +<<*>>= +--S 44 +integrate(sin(x)^3/(sin(x)^3+cos(x)^3),x=0..%pi/2,"noPole") +--R +--R 2log(16) - 4log(4) + 3%pi +--R (1) ------------------------- +--R 12 +--R Type: Union(f1: OrderedCompletion Expression Integer,...) +--E 44 + +--S 45 +integrate(exp(-x^2)*log(x)^2,x=0..%plusInfinity) +--R +--R _ 1 1 _ 1 1 2 +--R | (-)polygamma(1,-) + | (-)digamma(-) +--R 2 2 2 2 +--R (2) -------------------------------------- +--R 8 +--R Type: Union(f1: OrderedCompletion Expression Integer,...) +--E 45 + +@ +\section{Laplace Transformations} +<<*>>= +)clear all + +--S 46 +laplace(sin(a*t)*cosh(a*t)-cos(a*t)*sinh(a*t),t,s) +--R +--R 3 +--R 4a +--R (1) -------- +--R 4 4 +--R s + 4a +--R Type: Expression Integer +--E 46 + +--S 47 +laplace(2/t * (1-cos(a*t)),t,s) +--R +--R 2 2 +--R (2) log(s + a ) - 2log(s) +--R Type: Expression Integer +--E 47 + +--S 48 +laplace((exp(a*t)-exp(b*t))/t,t,s) +--R +--R (3) - log(s - a) + log(s - b) +--R Type: Expression Integer +--E 48 + +--S 49 +laplace(exp(a*t+b)*Ei(c*t),t,s) +--R +--R b s + c - a +--R %e log(---------) +--R c +--R (4) ----------------- +--R s - a +--R Type: Expression Integer +--E 49 + +@ +\section{Clifford Algebras} +{\tt CliffordAlgebra(n,K,Q)} defines a vector space of dimension $2^n$ +over $K$, given a quadratic form $Q$ on $K^n$ (e.q. quaternions). +<<*>>= +)clear all +--S 50 +K:=Fraction Polynomial Integer +--R +--R (1) Fraction Polynomial Integer +--R Type: Domain +--E 50 + +--S 51 +qf:QFORM(2,K):=quadraticForm matrix([[-1,0],[0,-1]])$(SQMATRIX(2,K)) +--R +--R +- 1 0 + +--R (2) | | +--R + 0 - 1+ +--R Type: QuadraticForm(2,Fraction Polynomial Integer) +--E 51 + +--S 52 +i:=e(1)$CLIF(2,K,qf) +--R +--R (3) e +--R 1 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 52 + +--S 53 +j:=e(2)$CLIF(2,K,qf) +--R +--R (4) e +--R 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 53 + +--S 54 +k:=i*j +--R +--R (5) e e +--R 1 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 54 + +--S 55 +x:=a+b*i+c*j+d*k +--R +--R (6) a + b e + c e + d e e +--R 1 2 1 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 55 + +--S 56 +y:=m+f*i+g*j+h*k +--R +--R (7) m + f e + g e + h e e +--R 1 2 1 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 56 + +--S 57 +x+y +--R +--R (8) m + a + (f + b)e + (g + c)e + (h + d)e e +--R 1 2 1 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 57 + +--S 58 +x*y +--R +--R (9) +--R a m - d h - c g - b f + (b m + c h - d g + a f)e +--R 1 +--R + +--R (c m - b h + a g + d f)e + (d m + a h + b g - c f)e e +--R 2 1 2 +--R Type: CliffordAlgebra(2,Fraction Polynomial Integer,MATRIX) +--E 58 + +@ +\section{Taylor Series} +<<*>>= +)clear all +--S 59 +taylor(sin(x),x=0) +--R +--R 1 3 1 5 1 7 1 9 11 +--R (1) x - - x + --- x - ---- x + ------ x + O(x ) +--R 6 120 5040 362880 +--R Type: UnivariateTaylorSeries(Expression Integer,x,0) +--E 59 + +@ +\section{Laurent Series} +<<*>>= +--S 60 +laurent(x/log(x),x=1) +--R +--R (2) +--R - 1 3 5 1 2 11 3 11 4 +--R (x - 1) + - + -- (x - 1) - -- (x - 1) + --- (x - 1) - ---- (x - 1) +--R 2 12 24 720 1440 +--R + +--R 271 5 13 6 7297 7 425 8 +--R ----- (x - 1) - ---- (x - 1) + ------- (x - 1) - ------ (x - 1) +--R 60480 4480 3628800 290304 +--R + +--R 530113 9 10 +--R --------- (x - 1) + O((x - 1) ) +--R 479001600 +--R Type: UnivariateLaurentSeries(Expression Integer,x,1) +--E 60 + +@ +\section{Puiseux Series} +<<*>>= +--S 61 +puiseux(sqrt(sec(x)),x=3*%pi/2) +--R +--R +--R 1 3 7 +--R - - - - +--R 3%pi 2 1 3%pi 2 1 3%pi 2 3%pi 5 +--R (3) (x - ----) + -- (x - ----) + --- (x - ----) + O((x - ----) ) +--R 2 12 2 160 2 2 +--R Type: UnivariatePuiseuxSeries(Expression Integer,x,(3*pi)/2) +--E 61 + +@ +\section{General Series} +<<*>>= +--S 62 +series(x^x,x=0) +--R +--R (4) +--R 2 3 4 5 +--R log(x) 2 log(x) 3 log(x) 4 log(x) 5 +--R 1 + log(x)x + ------- x + ------- x + ------- x + ------- x +--R 2 6 24 120 +--R + +--R 6 7 8 9 10 +--R log(x) 6 log(x) 7 log(x) 8 log(x) 9 log(x) 10 11 +--R ------- x + ------- x + ------- x + ------- x + -------- x + O(x ) +--R 720 5040 40320 362880 3628800 +--R Type: GeneralUnivariatePowerSeries(Expression Integer,x,0) +--E 62 + +@ +\section{Matrices} +<<*>>= +)clear all +--S 63 +m:=matrix [[1,2],[3,4]] +--R +--R +1 2+ +--R (1) | | +--R +3 4+ +--R Type: Matrix Integer +--E 63 + +--S 64 +4*m*(-5) +--R +--R +- 20 - 40+ +--R (2) | | +--R +- 60 - 80+ +--R Type: Matrix Integer +--E 64 + +--S 65 +n:=matrix [[1,0,-2],[-3,5,1]] +--R +--R + 1 0 - 2+ +--R (3) | | +--R +- 3 5 1 + +--R Type: Matrix Integer +--E 65 + +--S 66 +m*n +--R +--R +- 5 10 0 + +--R (4) | | +--R +- 9 20 - 2+ +--R Type: Matrix Integer +--E 66 + +--S 67 +hilb:=matrix([[1/(i+j) for i in 1..3] for j in 1..3]) +--R +--R +1 1 1+ +--R |- - -| +--R |2 3 4| +--R | | +--R |1 1 1| +--R (5) |- - -| +--R |3 4 5| +--R | | +--R |1 1 1| +--R |- - -| +--R +4 5 6+ +--R Type: Matrix Fraction Integer +--E 67 + +--S 68 +inverse(hilb) +--R +--R + 72 - 240 180 + +--R | | +--R (6) |- 240 900 - 720| +--R | | +--R + 180 - 720 600 + +--R Type: Union(Matrix Fraction Integer,...) +--E 68 + +@ +\section{Systems of Equations} +<<*>>= +)clear all +--S 69 +solve([x+y+z=8,3*x-2*y+z=0,x+2*y+2*z=17],[x,y,z]) +--R +--R (1) [[x= - 1,y= 2,z= 7]] +--R Type: List List Equation Fraction Polynomial Integer +--E 69 + +--S 70 +solve([x+2*y+3*z=2,2*x+3*y+4*z=2,3*x+4*y+5*z=2],[x,y,z]) +--R +--R (2) [[x= %W - 2,y= - 2%W + 2,z= %W]] +--R Type: List List Equation Fraction Polynomial Integer +--E 70 + +--S 71 +solve([[1,1,1],[3,-2,1],[1,2,2]],[8,0,17]) +--R +--R (3) [particular= [- 1,2,7],basis= [[0,0,0]]] +--RType: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer) +--E 71 + +--S 72 +solve([[1,2,3],[2,3,4],[3,4,5]],[2,2,2]) +--R +--R (4) [particular= [- 2,2,0],basis= [[1,- 2,1]]] +--RType: Record(particular: Union(Vector Fraction Integer,"failed"),basis: List Vector Fraction Integer) +--E 72 +)spool +)lisp (bye) + +@ +\eject +\begin{thebibliography}{99} +\bibitem{1} nothing +\end{thebibliography} +\end{document} diff --git a/src/input/intbypart.input.pamphlet b/src/input/intbypart.input.pamphlet index 72a04e5..0b26c61 100644 --- a/src/input/intbypart.input.pamphlet +++ b/src/input/intbypart.input.pamphlet @@ -91,6 +91,7 @@ integrate(x*exp(x),x) --R Type: Union(Expression Integer,...) --E 2 @ + \section{integrate $e^x cos(x) dx$} This integral will require the substituion by parts rule to be applied twice. @@ -165,8 +166,8 @@ integrate(x^3*exp(x^2),x) --R 2 --R Type: Union(Expression Integer,...) --E 4 -@ +@ \section{integrate $ln(x^2+2)dx$} To integrate $$\int{ln(x^2+2)dx}$$ @@ -194,49 +195,123 @@ integrate(log(x^2+2),x) --E 5 @ -\section{integrate $ln(x) dx$} +\section{integrate $x\ sin(x) dx$} To integrate -$$\int{ln(x) dx}$$ +$$\int{x\ sin(x)\ dx}$$ Let -$$u=ln(x)$$ -$$dv=dx$$ -$$du=\frac{1}{x}dx$$ -$$v=x$$ +$$u=x$$ +$$dv=sin(x)\ dx$$ +$$du = dx$$ +$$v = -cos(x)$$ so -$$\int{ln(x)\ dx}$$ -$$=x\ ln(x)=\int{1\ dx}$$ -$$=x\ ln(x) - x + C$$ -$$=x(ln(x)-1)+C$$ +$$\int{x\ sin(x)\ dx}$$ +$$= -x\ cos(x) - \int{-cos(x)\ dx}$$ +$$= -x\ cos(x)+sin(x)+C$$ <<*>>= --S 6 -integrate(log(x),x) +integrate(x*sin(x),x) --R ---R (6) x log(x) - x +--R (6) sin(x) - x cos(x) --R Type: Union(Expression Integer,...) --E 6 @ -\section{integrate $x\ sin(x) dx$} +\section{integrate $x\ cos(x) dx$} To integrate -$$\int{x\ sin(x)\ dx}$$ +$$\int{x\ cos(x)\ dx}$$ Let $$u=x$$ -$$dv=sin(x)\ dx$$ +$$dv=cos(x)\ dx$$ $$du = dx$$ -$$v = -cos(x)$$ +$$v = sin(x)$$ so -$$\int{x\ sin(x)\ dx}$$ -$$= -x\ cos(x) - \int{-cos(x)\ dx}$$ -$$= -x\ cos(x)+sin(x)+C$$ +$$\int{x\ cos(x)\ dx}$$ +$$= x\ sin(x) - \int{sin(x)\ dx}$$ +$$= x\ sin(x)+cos(x)+C$$ <<*>>= --S 7 -integrate(x*sin(x),x) +integrate(x*cos(x),x) +--R --R ---R (7) sin(x) - x cos(x) +--R (7) x sin(x) + cos(x) --R Type: Union(Expression Integer,...) --E 7 @ +\section{integrate $x^2\ sin(x) dx$} +To integrate +$$\int{x^2\ sin(x)\ dx}$$ +Let +$$u=x^2$$ +$$dv=sin(x)\ dx$$ +$$du = 2x\ dx$$ +$$v = -cos(x)$$ +so +$$\int{x^2\ sin(x)\ dx}$$ +$$= -x^2\ cos(x) - \int{-2x\ cos(x)\ dx}$$ +$$= -x^2\ cos(x)+2\int{x\ cos(x)\ dx}$$ +$$=-x^2\ cos(x)+2(x\ sin(x)+cos(x))+C$$ +<<*>>= +--S 8 +integrate(x^2*cos(x),x) +--R +--R +--R 2 +--R (8) (x - 2)sin(x) + 2x cos(x) +--R Type: Union(Expression Integer,...) +--E 8 +@ + +\section{integrate $sin(x)\ cos(x)\ dx$} +To integrate +$$\int{sin(x)\ cos(x)\ dx}$$ +Let +$$u=sin(x)$$ +$$dv=cos(x)\ dx$$ +$$du=cos(x)\ dx$$ +$$v=sin(x)$$ +so +$$\int{sin(x)cos(x)dx} +=sin(x)sin(x) - \int{sin(x)cos(x)dx}$$ +but the integral appears on both sides of the equation so +$$2\int{sin(x)cos(x)dx}=sin^2(x)$$ +so +$$\int{sin(x)cos(x)dx}=\frac{1}{2}sin^2(x)+C$$ +<<*>>= +--S 9 +integrate(sin(x)*cos(x),x) +--R +--R +--R 2 +--R cos(x) +--R (9) - ------- +--R 2 +--R Type: Union(Expression Integer,...) +--E 9 +@ + +\section{integrate $ln(x) dx$} +To integrate +$$\int{ln(x) dx}$$ +Let +$$u=ln(x)$$ +$$dv=dx$$ +$$du=\frac{1}{x}dx$$ +$$v=x$$ +so +$$\int{ln(x)\ dx}$$ +$$=x\ ln(x)=\int{1\ dx}$$ +$$=x\ ln(x) - x + C$$ +$$=x(ln(x)-1)+C$$ +<<*>>= +--S 10 +integrate(log(x),x) +--R +--R (10) x log(x) - x +--R Type: Union(Expression Integer,...) +--E 10 +@ + \section{integrate $x^2\ ln(x)\ dx$} To integrate $$\int{x^2\ ln(x)\ dx}$$ @@ -251,15 +326,38 @@ $$\frac{x^3}{3} ln(x) - \int{\frac{x^3}{3}\frac{dx}{x}}$$ $$\frac{x^3}{3} ln(x)-\frac{1}{3}\int{x^2\ dx}$$ $$\frac{x^3}{3}ln(x)-\frac{1}{9}x^3 + C$$ <<*>>= ---S 8 +--S 11 integrate(x^2*log(x),x) --R ---R 3 3 ---R 3x log(x) - x ---R (8) -------------- ---R 9 +--R 3 3 +--R 3x log(x) - x +--R (11) -------------- +--R 9 --R Type: Union(Expression Integer,...) ---E 8 +--E 11 +@ + +\section{integrate $x^2\ e^x\ dx$} +To integrate +$$\int{x^2\ e^x\ dx}$$ +Let +$$u=x^2$$ +$$dv=e^x\ dx$$ +$$du = 2x\ dx$$ +$$v = e^x$$ +so +$$\int{x^2\ e^x\ dx}$$ +$$x^2\ e^x - 2x\ e^x - \int{e^x\ 2dx}$$ +$$x^2\ e^x - 2x\ e^x+2\ e^x+C$$ +<<*>>= +--S 12 +integrate(x^2*exp(x),x) +--R +--R +--R 2 x +--R (12) (x - 2x + 2)%e +--R Type: Union(Expression Integer,...) +--E 12 @ \section{integrate $sin^{-1}(x)\ dx$} @@ -278,15 +376,128 @@ $$=x\ sin^{-1}(x) + \frac{1}{2}(2(1-x^2)^{1/2})+C$$ $$=x\ sin^{-1}(x)+(1-x^2)^{1/2}+C$$ $$=x\ sin^{-1}(x)+\sqrt{1-x^2}+C$$ <<*>>= ---S 9 -integrate(1/sin(x),x) +--S 13 +integrate(asin(x),x) +--R --R ---R sin(x) ---R (9) log(----------) ---R cos(x) + 1 +--R +--------+ +--R | 2 +--------+ +--R 2x\|- x + 1 | 2 +--R - x atan(-------------) + 2\|- x + 1 +--R 2 +--R 2x - 1 +--R (13) -------------------------------------- +--R 2 --R Type: Union(Expression Integer,...) ---E 9 -)spool +--E 13 +@ + +\section{integrate $\tan^{-1}(x)\ dx$} +$$\int{\tan^{-1}(x)\ dx}$$ +Let +$$u=tan^{-1}$$ +$$dv=dx$$ +$$du=\frac{1}{1+x^2}\ dx$$ +$$v=x$$ +so +$$\int{\tan^{-1}(x)\ dx}$$ +$$=x\ \tan^{-1}-\int{\frac{x}{1+x^2}\ dx}$$ +$$=x\ tan^{-1}(x)-\frac{1}{2}\int{\frac{2x}{1+x^2}\ dx}$$ +$$=x\ tan^{-1}(x)-\frac{1}{2}ln(1+x^2)+C$$ +<<*>>= +--S 14 +integrate(atan(x),x) +--R +--R +--R 2 2x +--R - log(x + 1) - x atan(------) +--R 2 +--R x - 1 +--R (14) ------------------------------ +--R 2 +--R Type: Union(Expression Integer,...) +--E 14 +@ + +\section{integrate $\sec^3(x)\ dx$} +$$\int{\sec^3(x)\ dx}$$ +Let +$$u=sec(x)$$ +$$dv=sec^2(x)\ dx$$ +$$du=sec(x)tan(x)\ dx$$ +$$v=tan(x)$$ +so +$$\int{\sec^3(x)\ dx}$$ +$$=sec(x)tan(x)-\int{sec(x)tan^2(x)\ dx}$$ +$$=sec(x)tan(x)-\int{sec(x)(sec^2(x)-1)\ dx}$$ +$$=sec(x)tan(x)-\int{sec^3(x)\ dx}+\int{sec(x)\ dx}$$ +$$=sec(x)tan(x)-\int{sec^3(x)\ dx}+ln(\vert sec(x)+\tan(x)\vert )$$ +but +$$=2\int{sec^3(x)\ dx}=sec(x)tan(x)+ln(\vert sec(x)+\tan(x)\vert )$$ +so +$$\int{sec^3(x)\ dx}= +\frac{1}{2}(sec(x)tan(x)+ln(\vert sec(x)+\tan(x)\vert ))+C$$ +<<*>>= +--S 15 +integrate(sec(x)^3,x) +--R +--R +--R (15) +--R 2 sin(x) + cos(x) + 1 2 sin(x) - cos(x) - 1 +--R cos(x) log(-------------------) - cos(x) log(-------------------) + sin(x) +--R cos(x) + 1 cos(x) + 1 +--R -------------------------------------------------------------------------- +--R 2 +--R 2cos(x) +--R Type: Union(Expression Integer,...) +--E 15 +@ + +\section{integrate $x^3\ e^{2x}$} +$$\int{x^3\ e^{2x}\ dx}$$ +Let +$$u=x^3$$ +$$dv=e^{2x}\ dx$$ +$$du=3x^2\ dx$$ +$$v=\frac{1}{2}e^{2x}$$ +$$\int{x^3\ e^{2x}\ dx}$$ +$$=\frac{1}{2}x^3\ e^{2x} - \frac{3}{2}\int{x^2\ e^{2x}\ dx}$$ + +To solve +$$\frac{3}{2}\int{x^2\ e^{2x}\ dx}$$ +Let +$$u=x^2$$ +$$dv=e^{2x}\ dx$$ +$$du=2x\ dx$$ +$$v=\frac{1}{2}e^{2x}$$ +so after substitution the new result is +$$=\frac{1}{2}x^3\ e^{2x} - +\frac{3}{2}\left(\frac{1}{2}x^2\ e^{2x}-\int{xe^{2x}\ dx}\right)$$ +$$=\frac{1}{2}x^3\ e^{2x} - \frac{3}{4}x^2e^{2x}+ +\frac{3}{2}\int{xe^{2x}\ dx}$$ +Let +$$u=x$$ +$$dv=e^{2x}\ dx$$ +$$du=dx$$ +$$v=\frac{1}{2}e^{2x}$$ +so +$$\int{x^3\ e^{2x}\ dx}$$ +$$=\frac{1}{2}x^3\ e^{2x}-\frac{3}{4}x^2\ e^{2x}+ +\frac{3}{2}\left(\frac{1}{2}xe^{2x}-\frac{1}{2}\int{e^{2x}\ dx}\right)$$ +$$=\frac{1}{2}x^3\ e^{2x}-\frac{3}{4}x^2\ e^{2x}+ +\frac{3}{4}xe^{2x}-\frac{3}{8}e^{2x}+C$$ +<<*>>= +--S 16 +integrate(x^3*exp(2*x),x) +--R +--R +--R 3 2 2x +--R (4x - 6x + 6x - 3)%e +--R (16) ------------------------ +--R 8 +--R Type: Union(Expression Integer,...) +--E 16 +)spool )lisp (bye) @ diff --git a/src/input/limit.input.pamphlet b/src/input/limit.input.pamphlet index c210c87..ec80f68 100644 --- a/src/input/limit.input.pamphlet +++ b/src/input/limit.input.pamphlet @@ -5,24 +5,276 @@ \author{Timothy Daly} \maketitle \begin{abstract} +Exercise the limit function. \end{abstract} \eject \tableofcontents \eject -\section{License} -<>= ---Copyright The Numerical Algorithms Group Limited 1991. -@ +\section{Limit of a Function} +If $f$ is a function, then +$$\lim_{x -> a}{f(x) = A}$$ +if the value of $f(x)$ gets arbitrarily close to $A$ as $x$ gets +arbitrarily close to $a$. For example, +$$\lim_{x->3}{x^2}=9$$ +since $x^2$ gets arbitrarily close to 9 as $x$ approaches 3. + +By definition, the limit +$$\lim_{x -> a}{f(x) = A}$$ if and only if, for any chosen positive +number $\epsilon$, however small, there exists a positive number +$\delta$ such that +$$0 < \vert x-a \vert < \delta {\tt\ implies\ } +\vert f(x)-A \vert < \epsilon$$ +Note that $f(x)$ does not need to be defined at $a$. + <<*>>= )spool limit.output )set message test on )set message auto off )clear all +--S 1 of 15 +limit((x^2-4)/(x-2),x=2) +--R +--R +--R (1) 4 +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 1 +@ +\section{Right and Left Limits} +The limit +$$\lim_{x -> a^-}{f(x) = A}$$ +given that $f$ is defined in the interval $(c,a)$ and $f(x)$ +approaches $A$ as $x$ approaches $a$ through the values less than +$a$, that is, as $x$ approaches $a$ from the left. + +Similarly +$$\lim_{x -> a^+}{f(x) = A}$$ +means that $f$ is defined in some interval $(a,d)$ and $f(x)$ +approaches $A$ as $x$ approaches $a$ from the right. + +If $f$ is defined in an interval to the left of $a$ and in an interval +to the right of $a$ then the statement +$$\lim_{x -> a}{f(x) = A}$$ +is equivalent to the conjunction of the two statements +$$\lim_{x -> a^-}{f(x) = A} {\rm\ and\ } \lim_{x -> a^+}{f(x) = A}$$. + +The existence of the limit from the left does not imply the existence +of the limit from the right. The existence of the limit from the +right does not imply the existence of the limit from the left. + +When a function is defined only on one side of a point $a$, then +$$\lim_{x -> a}{f(x)}$$ +is identical with the one-sided limit, if it exists. A two-sided +limit might not exist because it is not defined outside a certain +bound. For example, if +$$f(x)=\sqrt{x}$$ +then $f$ is defined only at and to the right of 0. Hence, +$$\lim_{x->0}{\sqrt{x}}=\lim_{x->0^+}{\sqrt{x}}=0$$ +But the limit +$$\lim_{x->0^-}{\sqrt{x}}$$ +does not exist since $\sqrt{x}$ is not defined for real values of $x$. + +A limit might not exist because the function grows without bound +as it approaches the limit point. For example, the function +$\sqrt{\frac{1}{x}}$ is only defined when $x > 0$. So +$$\lim_{x->0^+}{\sqrt{\frac{1}{x}}}$$ +does not exist since $\frac{1}{x}$ gets larger as $x -> 0$ from +the right. So +$$\lim_{x->0}{\sqrt{\frac{1}{x}}}$$ +does not exist. + +<<*>>= +--S 2 of 15 +limit(sqrt(9-x^2),x=-4) +--R +--R +--R (2) "failed" +--R Type: Union("failed",...) +--E 2 + +--S 3 of 15 +limit(sqrt(9-x^2),x=-3) +--R +--R +--R (3) [leftHandLimit= "failed",rightHandLimit= 0] +--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +--E 3 + +--S 4 of 15 +limit(sqrt(9-x^2),x=-2) +--R +--R +--R +-+ +--R (4) \|5 +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 4 + +--S 5 of 15 +limit(sqrt(9-x^2),x=0) +--R +--R +--R (5) 3 +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 5 + +--S 6 of 15 +limit(sqrt(9-x^2),x=2) +--R +--R +--R +-+ +--R (6) \|5 +--R Type: Union(OrderedCompletion Expression Integer,...) +--E 6 + +--S 7 of 15 +limit(sqrt(9-x^2),x=3) +--R +--R +--R (7) [leftHandLimit= 0,rightHandLimit= "failed"] +--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Expression Integer,"failed"),rightHandLimit: Union(OrderedCompletion Expression Integer,"failed")),...) +--E 7 + +--S 8 of 15 +limit(sqrt(9-x^2),x=4) +--R +--R +--R (8) "failed" +--R Type: Union("failed",...) +--E 8 + @ +\section{Theorems on Limits} +If $$f(x)=c$$ where $c$ is a constant, then +$$\lim_{x->a}{f(x)=c}$$ + +Assume that +$$\lim_{x->a}{f(x)=A}{\rm\ and\ }\lim_{x->a}{g(x)=B}$$ +then the limit of a constant times a function is +the constant times limit of a function is +the constant times the limit. +$$\lim_{x->a}{cf(x)}= +c*\lim_{x->a}{f(x)= +cA}$$ + +The limit of a function plus (or minus) another function is +the limit of the first function plus (or minus) +the limit of the second function is +the sum (or difference) of their limits. +$$\lim_{x->a}{[f(x)\pm{}g(x)]}= +\lim_{x->a}{f(x)}\pm\lim_{x->a}{g(x)}= +A\pm{}B$$ + +The limit of the product of functions is +the limit of the first function times the limit of the second function is +the product of their limits. +$$\lim_{x->a}{[f(x)g(x)]}= +\lim_{x->a}{f(x)}*\lim_{x->a}{g(x)}= +A*B$$ + +The limit of the ratio of two functions is +the the limit of the first function divided by the limit of the +second function provided the limit of the second function is non-zero. +$$\lim_{x->a}{\left(\frac{f(x)}{g(x)}\right)}= +\frac{\lim_{x->a}{f(x)}}{\lim_{x->a}{g(x)}}= +\frac{A}{B}, +{\rm\ if\ }B\ne{}0$$ + +The limit of the ${\rm{}n}^{th}$ root of a function is +the ${\rm{}n}^{th}$ root of the limit of the function provided +the ${\rm{}n}^{th}$ root exists. +$$\lim_{x->a}{\sqrt[n]{f(x)}}=\sqrt[n]{\lim_{x->a}{f(x)}}= +\sqrt[n]{A},{\rm\ if\ }\sqrt[n]{A}{\rm\ is\ defined}$$ + The returned limit seems to be wrong: + +\section{Infinity} +Let +$$\lim_{x->a}{f(x)}=+\infty$$ +mean that, as $x$ approaches $a$, $f(x)$ eventually becomes and +therefore remains greater than any preassigned positive number, +however large. In such a case, we say that $f(x)$ approaches +$+\infty$ as $x$ approaches $a$. The +$$\lim_{x->a}{f(x)}=+\infty$$ +if and only if, for any positive number $M$, there exists a +positive number $\delta$ such that, whenever +$$0 < \vert x-a \vert < \delta {\rm\ then\ }f(x) > M$$ + +Similarly, let +$$\lim_{x->a}{f(x)}=-\infty$$ +mean that, as $x$ approaches $a$, $f(x)$ eventually becomes and +thereafter remains less than any preassigned negative number. In +that case, we say that $f(x)$ approaches +$-\infty$ as $x$ approaches $a$. + +Let +$$\lim_{x->a}{f(x)}=\infty$$ +mean that, as $x$ approaches $a$, $\vert f(x)\vert$ eventually becomes and +thereafter remains greater than any preassigned positive number. Hence, +$$\lim_{x->a}{f(x)}=\infty$$ if and only if +$$\lim_{x->a}{\vert f(x)\vert}=+\infty$$ + +These definitions can be extended to one-sided limits. + +<<*>>= +--S 9 of 15 +limit(1/x^2,x=0) +--R +--R +--R (9) + infinity +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 9 + +--S 10 of 15 +limit(-1/(x-1)^2,x=1) +--R +--R +--R (10) - infinity +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 10 + +--S 11 of 15 +limit(1/x,x=0) +--R +--R +--R (11) [leftHandLimit= - infinity,rightHandLimit= + infinity] +--RType: Union(Record(leftHandLimit: Union(OrderedCompletion Fraction Polynomial Integer,"failed"),rightHandLimit: Union(OrderedCompletion Fraction Polynomial Integer,"failed")),...) +--E 11 + +@ +The limit concepts already introduced can be extended to the case +where the variable approaches $+\infty$ or $-\infty$. For example, +$$\lim_{x->+\infty}{f(x)}=A$$ +means that $f(x)$ approaches $A$ as $x -> +\infty$ or, in more +precise terms, given any positive $\epsilon$, there exists a +number $N$ such that, whenever $x>N$, then +$$\vert f(x)-A\vert < \epsilon$$ + +Similar statements can be made for: +$$\lim_{x->-\infty}{f(x)}=A$$ +$$\lim_{x->+\infty}{f(x)}=+\infty$$ +$$\lim_{x->-\infty}{f(x)}=-\infty$$ +$$\lim_{x->+\infty}{f(x)}=-\infty$$ +$$\lim_{x->-\infty}{f(x)}=+\infty$$ <<*>>= ---S 1 of 2 +--S 12 of 15 +limit(1/x,x=%plusInfinity) +--R +--R +--R (12) 0 +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 12 + +--S 13 of 15 +limit(2+(1/x^2),x=%plusInfinity) +--R +--R +--R (13) 2 +--R Type: Union(OrderedCompletion Fraction Polynomial Integer,...) +--E 13 + +)clear all + +--S 14 of 15 f := exp(n) * (sin(1/n + exp(-n)) - sin(1/n)) --R --R @@ -31,22 +283,24 @@ f := exp(n) * (sin(1/n + exp(-n)) - sin(1/n)) --R (1) %e sin(-----------) - %e sin(-) --R n n --R Type: Expression Integer ---E 1 +--E 14 ---S 2 of 2 +--S 15 of 15 limit(f,n=%plusInfinity) --R --R --R (2) "failed" --R Type: Union("failed",...) ---E 2 +--E 15 )spool )lisp (bye) @ \eject \begin{thebibliography}{99} -\bibitem{1} nothing +\bibitem{1} Ayres, Frank Jr. and Mendelson, Elliott +``Calculus'' Schaum's Outlines 4th edition 1999 +ISBN 0-07-041973-6 pp61-63 \end{thebibliography} \end{document}