Sindbad~EG File Manager
/* Initialize menu text for MATHPERT */
/* Translator: translate text enclosed in quotation marks,
but do NOT translate text (usually formulas)
enclosed in dollar signs. Use the ISO-Latin1
character set.
8.6.98 last modified
Sent to translator 8.12.98
8.13 improper_integrals choice 3 modified
improper_integrals choice 5 added
8.14 text containing Erf modified to be parseable
8.17 added menu title Logarithmic Limits, and text for logarithmic_limits menu
8.20 more ops on series3 menu
8.28 corrected complex_functions choices 8 and 9
9.29 altered the last entry in trig_ineq
11.24.98 last modified
12.29.98 added four more factorial operations in binomial_theorem menu
1.6.99 added operations to series2, changed the wording on first line of series3
1.7.99 changed menu titles for series operator menus
1.13.99 more operations in series_geom5
1.14.99 added last operation in advanced_sigma_notation
1.19.99 added last operation in numerical_calculation menu
1.22.99 four more operations in series_appearances
3.9.99 modified
3.30.99 changed if b >= 0 to if b > 0 in roots_and_fractions menu where
b occurs in a denom.
12.30.99 added closing brackets in logarithmic_limits
1.4.00 added four more complex_hyperbolic operations
1.9.00 added new operation in advanced_factoring and moved one from
there to numerical_calculation
1.12.00 added missing commas in complex_hyperbolic
2.27.00 added text for series_convergence2
4.10.00 corrected improper_integrals text
7.10.00 deleted a line under special_limits
Added a missing parenthesis under advanced_limits
6.16.04 added "$$integral(u,t,a,a) = 0$$" in definite integration
6.21.04 modified text for complexroot and complexsqrt under complex_functions menu
6.24.04 modified text for complex_numbers menu for four new operations.
1.25.06 added another operation to minima_and_maxima
1.27.06 add 4 new operations to sg_function2
*/
#define ENGLISH_DLL
#include "export.h" /* do not translate this or the next 3 lines */
#include "mtext.h"
#include "operator.h"
#include "english1.h"
const char arithstr[] = "arithmetic"; /* save space with ONE copy of this */
const char *menutext[MAXMENUS][MAXLENGTH] =
{
{ /* numerical_calculation */
arithstr,
"decimal calculation",
"calculate decimal $\\sqrt $ or $^n\\sqrt $",
"decimal value of $x^n$",
"decimal value of function",
"factor integer",
"evaluate numerically at a point",
"decimal value of $\\pi $",
"decimal value of e",
"compute function value",
"factor polynomial numerically"
},
{ /* numerical_calculation2 */
"decimal to fraction",
"express as square",
"express as cube",
"express as ?-th power",
"express as power of ?",
"write integer as a^n",
"x = ? + (x-?)"
},
{ /* complex_arithmetic */
"$i^2 = -1$",
"i^(4n) = 1",
"i^(4n+1) = i",
"i^(4n+2) = -1",
"i^(4n+3) = -i",
"complex arithmetic",
"power of complex number",
"complex arithmetic and powers",
"complex decimal calculation",
"integer factors of integer",
"complex factors of integer",
"factor n+mi (n not zero)",
"calculate decimal $\\sqrt $ or $^n\\sqrt $",
"decimal value of $x^n$",
"decimal value of function",
"evaluate numerically at a point"
},
{ /* simplify_sums */
"cancel double minus -(-a)=a",
"push minus in -(a+b) = -a-b",
"-a-b = -(a+b)",
arithstr,
"regroup terms",
"put terms in order",
"drop zero terms x+0 = x",
"cancel $\\pm $ terms",
"collect $\\pm $ terms (once)",
"collect all $\\pm $ terms in a sum",
"a+b = b+a",
"a(b-c) = -a(c-b)",
"-ab = a(-b)",
"-abc = ab(-c)",
"a(-b)c = ab(-c)"
},
{ /*simplify_products */
"$x\\times 0 = 0\\times x = 0$",
"$x\\times 1 = 1\\times x = x$",
"a(-b) = -ab",
"a(-b-c) = -a(b+c)",
"(-a-b)c = -(a+b)c",
"regroup factors",
"collect numbers",
"order factors",
"collect powers",
"a(b+c)=ab+ac",
"$(a-b)(a+b) = a^2-b^2$",
"$(a + b)^2 = a^2 + 2ab + b^2$",
"$(a - b)^2 = a^2 - 2ab + b^2$",
"$(a-b)(a^2+ab+b^2)=a^3-b^3$",
"$(a+b)(a^2-ab+b^2)=a^3+b^3$",
"ab = ba"
},
{ /* expand_menu */
"multiply out product of sums",
"multiply out numerator",
"multiply out denominator",
"$na = a +...+ a$"
},
{ /* fractions */
"0/a = 0",
"a/1 = a",
"a(1/a) = 1",
"multiply fractions (a/c)(b/d)=ab/cd",
"a(b/c) = ab/c",
"cancel ab/ac = b/c",
"add fractions $a/c \\pm b/c=(a\\pm b)/c$",
"apart $(a \\pm b)/c = a/c \\pm b/c$",
"apart and cancel $(ac\\pm b)/c = a\\pm b/c$",
"polynomial division",
"cancel by polynomial division",
"au/bv=(a/b)(u/v) (integers a,b)",
"a/b = (1/b) a",
"au/b=(a/b)u (real numbers a,b)",
"ab/cd = (a/c)(b/d)",
"ab/c = (a/c) b"
},
{ /* signed_fractions */
"cancel minus (-a)/(-b) = a/b",
"-(a/b) = (-a)/b",
"-(a/b) = a/(-b)",
"(-a)/b = -(a/b)",
"a/(-b)= -a/b",
"(-a-b)/c = -(a+b)/c",
"a/(-b-c) = -a/(b+c)",
"a/(b-c) = -a/(c-b)",
"-a/(-b-c) = a/(b+c)",
"-a/(b-c) = a/(c-b)",
"-(-a-b)/c = (a+b)/c",
"(a-b)/(c-d) = (b-a)/(d-c)",
"ab/c = a(b/c)"
},
{ /* compound_fractions */
"(a/c)/(b/c) = a/b",
"a/(b/c)=ac/b (invert and multiply)",
"1/(a/b) = b/a",
"(a/b)/c = a/(bc)",
"(a/b)/c = (a/b)(1/c)",
"(a/b)c/d = ac/bd",
"factor denominator",
"common denom in fraction",
},
{ /* common_denominators */
"factor denominator",
"find common denominator",
"find common denom (fracts only)",
"multiply fractions (a/b)(c/d)=ac/bd",
"multiply fractions a(c/d)= ac/d",
"order factors",
"add fractions $a/c \\pm b/c=(a \\pm b)/c$",
"common denominator",
"common denom (fractions only)",
"common denom and simplify numerator",
"common denom and simp (fracts only)",
"multiply num and denom by ?"
},
{ /* exponents */
"a^0 = 1 (a not zero)",
"a^1 = a",
"0^b = 0 if b > 0",
"1^b = 1",
"$(-1)^n = \\pm 1$ (n even or odd)",
"(a^b)^c = a^(bc) if a>0 or $c\\in Z$",
"$(-a)^n = (-1)^na^n$",
"$(a/b)^n = a^n/b^n$",
"$(ab)^n = a^nb^n$",
"$(a+b)^2 = a^2+2ab+b^2$",
"expand by binomial theorem",
"collect powers",
"a^(b+c) = a^b a^c", /* reversecollectpowers */
"$a^n/b^n = (a/b)^n$",
"b^n/b^m = b^(n-m)",
"ab^n/b^m = a/b^(m-n)"
},
{ /* expand_powers */
"a^2 = aa",
"a^3 = aaa",
"a^n = aaa...(n times)",
"a^n = a^?a^(n-?)",
"$(a \\pm b)^2 = a^2 \\pm 2ab + b^2$",
"(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3",
"(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3",
"a^(bc) = (a^b)^c if $a>0$ or $c\\in Z$",
"a^(bc) = (a^c)^b if $a>0$ or $c\\in Z$",
"a^(b?) = (a^b)^?",
"1/a^n = (1/a)^n"
},
{ /* negative_exponents */
"a^(-n) = $1/a^n$ (n constant)",
"$a^(-n)/b = 1/(a^nb)$ (n constant)",
"a^(-1) = 1/a",
"$a^(-n) = 1/a^n$",
"$a^(-n)/b = 1/(a^nb)$",
"a/b^(-n) = ab^n",
"$a/b^n = ab^(-n)$",
"a/b = ab^(-1)",
"$(a/b)^(-n) = (b/a)^n$",
"b^n/b^m = b^(n-m)",
"ab^n/b^m = a/b^(m-n)",
"a^(b-c) = a^b/a^c"
},
{ /* square_roots */
"$\\sqrt x\\sqrt y = \\sqrt (xy)$",
"$\\sqrt (xy) = \\sqrt x\\sqrt y$",
"$\\sqrt (x^2y) = x\\sqrt y$ or $|x|\\sqrt y$",
"$\\sqrt (x^2)=x$ if $x\\ge 0$",
"$\\sqrt (x^2)=|x|$",
"factor integer x in $\\sqrt x$",
"$\\sqrt (x/y) = \\sqrt x/\\sqrt y$",
"$\\sqrt (x/y) = \\sqrt |x|/\\sqrt |y|$",
"$\\sqrt x/\\sqrt y = \\sqrt (x/y)$",
"$x/\\sqrt x = \\sqrt x$",
"$\\sqrt x/x = 1/\\sqrt x$",
"$(\\sqrt x)^2^n = x^n$ if $x\\ge 0$",
"$(\\sqrt x)^(2n+1) = x^n\\sqrt x$",
"evaluate $\\sqrt $ to rational",
"evaluate $\\sqrt $ to decimal",
"simple arithmetic" /* that is, doesn't compute roots */
},
{ /* advanced_square_roots */
"show common factor in $\\sqrt u/\\sqrt v$",
"factor polynomial under $\\sqrt $",
"rationalize denominator",
"rationalize numerator",
"$\\sqrt (x^2)=|x|$ or $\\sqrt (x^2^n)=|x|^n$",
"cancel $\\sqrt $: $\\sqrt (xy)/\\sqrt y = \\sqrt x$",
"multiply out under $\\sqrt $",
"$a^2-b = (a-\\sqrt b)(a+\\sqrt b)$",
"$^2\\sqrt u = \\sqrt u$",
"$\\sqrt u = ^2^n\\sqrt u^n$",
"$\\sqrt u = (^2^n\\sqrt u)^n$",
"$\\sqrt (u^2^n) = u^n$ if $u^n\\ge 0$",
"$\\sqrt (u^(2n+1)) = u^n\\sqrt u$ if $u^n\\ge 0$",
"$a\\sqrt b = \\sqrt (a^2b)$ if $a\\ge 0$",
"rationalize denom and simplify"
},
{ /* fractional_exponents */
"$a ^ \\onehalf = \\sqrt a$",
"$a^(n/2) = \\sqrt (a^n)$",
"$a^(b/n) = ^n\\sqrt (a^b)$",
"$\\sqrt a = a ^ \\onehalf $",
"$^n\\sqrt a = a^(1/n)$",
"$^n\\sqrt (a^m) = a^(m/n)$",
"$(^n\\sqrt a)^m = a^(m/n)$",
"$(\\sqrt a)^m = a^(m/2)$",
"$1/\\sqrt a = a^(-\\onehalf )$",
"$1/^n\\sqrt a = a^(-1/n)$",
"evaluate (-1)^(p/q)",
"factor integer a in a^(p/q)",
"a/b^(p/q) = (a^q/b^p)^(1/q)",
"a^(p/q)/b = (a^p/b^q)^1/q)",
"arithmetic"
},
{ /*nth_roots */
"$^n\\sqrt x^n\\sqrt y = ^n\\sqrt (xy)$",
"$^n\\sqrt (xy) = ^n\\sqrt x ^n\\sqrt y$",
"$^n\\sqrt x^m = (^n\\sqrt x)^m$ if $x\\ge 0$ or n odd", /* rootofpower5 */
"$^n\\sqrt (x^ny) = x ^n\\sqrt y$ or $|x|^n\\sqrt y$",
"$^n\\sqrt (x^n) = x$ if $x\\ge 0$ or n odd", /* rootofpower */
"$^n\\sqrt (x^(nm))=x^m$ if $x\\ge 0$ or n odd", /* rootofpower3 */
"$^2^n\\sqrt (x^n) = \\sqrt x$", /* rootofpower2 */
"$^m^n\\sqrt x^m) = ^n\\sqrt x$", /* rootofpower4 */
"$(^n\\sqrt x)^n = x$", /* powerofroot */
"$(^n\\sqrt a)^m = ^n\\sqrt (a^m)$", /* powerofroot2 */
"$(^n\\sqrt a)^(qn+r) = a^q ^n\\sqrt (a^r)$", /* powerofroot3 */
"factor integer x in $^n\\sqrt x$", /* factorunderroot */
"$^n\\sqrt (-a) = -^n\\sqrt a$, n odd",
"evaluate to rational",
"factor polynomial under $^n\\sqrt $",
"multiply out under $^n\\sqrt $"
},
{ /* roots_of_roots */
"$\\sqrt (\\sqrt x) = ^4\\sqrt x$", /* sqrtofsqrt */
"$\\sqrt (^n\\sqrt x) = ^2^n\\sqrt x$", /* sqrtofroot */
"$^n\\sqrt (\\sqrt x) = ^2^n\\sqrt x$", /* rootofsqrt */
"$^n\\sqrt (^m\\sqrt x) = ^n^m\\sqrt x$", /* rootofsqrt */
},
{ /* roots_and_fractions */
"$^n\\sqrt (x/y) = ^n\\sqrt x/^n\\sqrt y$",
"$^n\\sqrt x/^n\\sqrt y = ^n\\sqrt (x/y)$",
"$x/^n\\sqrt x = (^n\\sqrt x)^(n-1)$",
"$^n\\sqrt x/x = 1/(^n\\sqrt x)^(n-1)$",
"cancel under $^n\\sqrt : ^n\\sqrt (ab)/^n\\sqrt (bc)=^n\\sqrt a/^n\\sqrt b$",
"cancel $^n\\sqrt $: $^n\\sqrt (xy)/^n\\sqrt y = ^n\\sqrt x$",
"show common factor in $^n\\sqrt u/^n\\sqrt v$",
"$a(^n\\sqrt b) = ^n\\sqrt (a^nb)$ if n odd",
"$a(^n\\sqrt b) = ^n\\sqrt (a^nb)$ if $a\\ge 0$",
"$-^n\\sqrt a = ^n\\sqrt (-a)$ if n odd",
"$a/^n\\sqrt b = ^n\\sqrt (a^n/b)$ (n odd or $a\\ge 0$)",
"$^n\\sqrt a/b = ^n\\sqrt (a/b^n)$ (n odd or $b>0$)",
"$\\sqrt a/b = \\sqrt (a/b^2)$ if $b>0$",
"$a/\\sqrt b = \\sqrt (a^2/b)$ if $a\\ge 0$",
"$(^m^n\\sqrt a)^n = ^m\\sqrt a$",
"$(^2^n\\sqrt a)^n = \\sqrt a$"
},
{ /* complex_numbers */
"1/i = -i",
"a/i = -ai",
"a/(bi) = -ai/b",
"$\\sqrt (-1) = i$",
"$\\sqrt (-a) = i\\sqrt a$ if $a\\ge 0$",
"clear denominator of i",
"$(a-bi)(a+bi) = a^2+b^2$",
"$a^2+b^2 = (a-bi)(a+bi)$",
"$|u + vi|^2 = u^2 + v^2$",
"$|u + vi| = \\sqrt (u^2+v^2)$",
"(u+vi)/w = u/w + (v/w)i",
"write in form u+vi",
"$\\sqrt(bi)= \\sqrt(b/2)+\\sqrt(b/2)i$, if b >= 0",
"$\\sqrt(-bi)= \\sqrt(b/2)-\\sqrt(b/2)i$, if b >= 0",
"$\\sqrt(a+bi)= \\sqrt((a+c)/2)+\\sqrt((a-c)/2)i$, if b \\ge 0 and $c^2=a^2+b^2$",
"$\\sqrt(a-bi)= \\sqrt((a+c)/2)-\\sqrt((a-c)/2)i$, if b \\ge 0 and $c^2=a^2+b^2$"
},
{ /* factoring */
"factor out number",
"clear numerical denominators",
"ab + ac = a(b+c)",
"factor out highest power",
"$a^2+2ab+b^2 = (a+b)^2$",
"$a^2-2ab+b^2 = (a-b)^2$",
"$a^2-b^2 = (a-b)(a+b)$",
"factor quadratic trinomial",
"use quadratic formula",
"$a^2^n = (a^n)^2$",
"$a^nb^n = (ab)^n$",
"factor integer coefficients",
"factor integer",
"make a substitution, u = ?",
"eliminate defined variable",
"regard a variable as constant"
},
{ /* advanced_factoring */
"write it as a function of ?",
"write it as a function of ? and ?",
"a^(3n) = (a^n)^3",
"a^(?n) = (a^n)^?",
"a^3 - b^3 = (a-b)(a^2+ab+b^2)",
"a^3 + b^3 = (a+b)(a^2-ab+b^2)",
"$a^n-b^n = (a-b)(a^(n-1)+...+b^(n-1))$",
"$a^n-b^n = (a+b)(a^(n-1)-...-b^(n-1))$ (n even)",
"$a^n+b^n=(a+b)(a^(n-1)-...+b^(n-1))$ (n odd)",
"$x^4+a^4=(x^2-\\sqrt 2ax+a^2)(x^2+\\sqrt 2ax+a^2)$",
"$x^4+(2p-q^2)x^2+p^2=(x^2-qx+p)(x^2+qx+p)$",
"computer makes a substitution",
"guess a factor",
"search for linear factor",
"factor by grouping",
"write it as a polynomial in ?"
},
{ /* solve_equations */
"switch sides",
"change signs of both sides",
"add ? to both sides",
"subtract ? from both sides",
"transfer ? left to right",
"transfer ? right to left",
"multiply both sides by ?",
"divide both sides by ?",
"square both sides",
"cancel $\\pm $ term from both sides",
"cancel common factor of sides",
"subtract to put in form u=0",
"equation is identically true",
"a=-b becomes $a^2=-b^2$ if $a,b\\ge 0$",
"a=-b becomes a=0 if $a,b\\ge 0$",
"a=-b becomes b=0 if $a,b\\ge 0$"
},
{ /* quadratic_equations */
"if ab=0 then a=0 or b=0",
"quadratic formula",
"$x = -b/2a \\pm \\sqrt (b^2-4ac)/2a$",
"complete the square",
"take square root of both sides",
"cross multiply",
"$b^2-4ac < 0 implies no real roots$",
"[p=a,p=-a] becomes p=|a| (for $p\\ge 0$)",
arithstr
},
{ /* numerical_equations */
"evaluate numerically at a point",
"solve numerically"
},
{ /* advanced_equations */
"cross multiply (a/b=c/d => ad=bc)",
"if u=v then $u^n=v^n$",
"take $\\sqrt $ of both sides",
"take $^n\\sqrt $ of both sides",
"apply function ? to both sides",
"common denominator",
"if ab=0 then a=0 or b=0",
"if ab=ac then a=0 or b=c",
"display only the selected equation",
"show all equations again",
"collect multiple solutions",
"make a substitution, u = ?",
"eliminate defined variable",
"reject unsolvable equation",
"check root(s) in original eqn",
"solve linear equation at once",
},
{ /* cubic_equations */
"u=x+b/3 in ax^3+bx^2+cx+d=0",
"compute discriminant", /* if this changes change discriminant_line in cubics.c */
"show cubic equation again",
"Vieta's substitution x=y-a/3cy in cx^3+ax+b=0",
"cubic formula, 1 real root",
"cubic formula, 3 real roots",
"cubic formula, complex roots",
"substitute x = f(u)",
"eliminate defined variable",
"substitute n = ?-k",
"evaluate roots exactly",
"decimal calculation",
"simplify"
},
{ /* logarithmic_equations */
"if u=v then a^u = a^v",
"if ln u = v then u = e^v",
"if log u = v then u = 10^v",
"if log(b,u) = v then u = b^v",
"if a^u = a^v then u=v",
"take log of both sides",
"take ln of both sides",
"reject eqn--impossible log or ln"
},
{ /* cramers_rule */
"Cramer's rule",
"evaluate determinant"
},
{ /* several_linear_equations*/
"variables left, constants right",
"collect like terms", /* if position changes, change exec.c (search for "several_linear_equations") */
"line up variables nicely",
"add two equations",
"subtract two equations",
"multiply equation ? by ?",
"divide equation ? by ?",
"add multiple of eqn ? to eqn ?",
"subtract multiple of eqn ? from eqn ?",
"swap two equations",
"put solved equations in order",
"drop identity",
"regard a variable as constant",
"contradiction at hand: no soln"
},
{ /* selection_mode_only */
"a|b| = |ab| if $0 \\le a$",
"|b|/c = |b/c| if 0 < c",
"a|b|/c = |ab/c| if 0 <a/c",
"solve for ?" /* solvelinearfor */
},
{ /* linear_equations_by_selection */
"add selected equation to equation ?",
"subtract selected eqn from eqn ?",
"multiply selected eqn by ?",
"divide selected eqn by ?",
"add multiple of selected eqn to eqn ?",
"subtract multiple of selected eqn from eqn ?",
"swap selected equation with eqn ?",
"solve selected equation for ?",
"add selected row to row ?",
"subtract selected row from row ?",
"multiply selected row by ?",
"divide selected row by ?",
"add multiple of selected row to row ?",
"subtract multiple of selected row from row ?",
"swap selected row with row ?",
"A = IA"
},
{ /* linear_equations_by_substitution */
"collect like terms",
"solve equation ? for ?",
"simplify equations",
"cancel term from both sides",
"add ? to both sides of equation ?",
"subtract ? from both sides of equation ?",
"divide equation ? by ?",
"substitute for variable",
"contradiction at hand: no soln"
},
{ /* matrix_methods */
"write in matrix form",
"A = IA",
"swap two rows",
"add two rows",
"subtract one row from another",
"multiply row by constant",
"divide row by constant",
"add multiple of row to another",
"sub mult of row from another",
"multiply matrices",
"drop zero column",
"drop zero row",
"drop duplicate row",
"contradiction at hand: no soln",
"convert to system of equations"
},
{ /* advanced_matrix_methods */
"multiply matrices",
"AX = B becomes X = A^(-1)B",
"use formula for 2 by 2 inverse",
"compute exact matrix inverse",
"compute decimal matrix inverse"
},
{ /* absolute_value */
"|u| = u if $u\\ge 0$",
"Assume $u\\ge 0$ and set |u| = u",
"|u| = -u if $u\\le 0$",
"|cu| = c|u| if $c\\ge 0$",
"|u/c| = |u|/c if c>0",
"|u||v| = |uv|",
"|uv| = |u||v|",
"|u/v| = |u| / |v|",
"|u| / |v| = |u/v|",
"$|u|^2^n=u^2^n$ if u is real",
"$|u^n|=|u|^n$ if n is real",
"$|\\sqrt u| = \\sqrt |u|$",
"$|^n\\sqrt u| = ^n\\sqrt |u|$",
"|ab|/|ac| = |b|/|c|",
"|ab|/|a| = |b|",
"show common factor in |u|/|v|"
},
{ /* absolute_value_ineq1 */
"|u|=c iff u=c or u = -c ($c\\ge 0$)", /* abseqn */
"|u|/u = c iff c = $\\pm $1", /* abseqn2 */
"|u| < v iff -v < u < v", /* abslessthan */
"$|u| \\le v$ iff $-v \\le u \\le v$", /* absle */
"u < |v| iff v < -u or u < v", /* lessthanabs */
"$u \\le |v|$ iff $v \\le -u$ or $u \\le v$",/* leabs */
"|u| = u iff $0 \\le u$", /* abseqntoineq1 */
"|u| = -u iff $u \\le 0$", /* abseqntoineq2 */
"$0 \\le |u|$ is true", /* absineqtrue */
"|u| < 0 is false", /* absineqfalse */
"$-c \\le |u|$ is true ($c\\ge 0$)", /* absineqtrue2 */
"-c < |u| is true (c>0)", /* absineqtrue3 */
"|u| < -c is false ($c\\ge 0$)", /* abslessthanneg*/
"$|u| \\le -c$ is false (c>0)", /* absleneg */
"$|u| \\le -c$ iff u=0 assuming $c\\ge 0$", /* absleneg2 */
"|u| = -c iff u=0 assuming $c\\ge 0$" /* abseqnneg */
},
{ /* absolute_value_ineq2 */
"v > |u| iff -v < u < v", /* absgreaterthan */
"$v \\ge |u|$ iff $-v \\le u \\le v$", /* absge */
"|v| > u iff v < -u or v > u", /* greaterthanabs */
"$|v| \\ge u$ iff $v \\le -u$ or $v \\ge u$",/* geabs */
"$|u| \\ge 0$ is true", /* absineqtrueg */
"0 > |u| is false", /* absineqfalseg */
"-c > |u| is false ($c\\ge 0$)", /* absgreaterthanneg */
"$-c \\ge |u|$ is false (c>0)", /* absgeneg */
"$-c \\ge |u|$ iff u=0 assuming c=0", /* absgeneg2 */
"|u| > -c is true (c>0)", /* absineqtrue3g */
"$|u| \\ge -c$ is true ($c\\ge 0$)", /* absineqtrue2g */
"$-v \\le u \\le v$ iff $|u| \\le v$ ", /* intervalabs1 */
"v < -u or u < v iff u < |v| ", /* intervalabs2 */
"$u^(2n) = |u|^(2n)$ if u is real", /* absevenpowerrev */
"$|u|^n = |u^n|$ if n is real" /* abspowerrev */
},
{ /* less_than */
"change u < v to v > u",
"add ? to both sides",
"subtract ? from both sides",
"change -u < -v to v < u",
"change -u < -v to u > v",
"multiply both sides by ?",
"multiply both sides by ?^2",
"divide both sides by ?",
"evaluate numerical inequality",
"$a < x^2^n$ is true if $a < 0$",
"$x^2^n < a$ is false if $a \\le 0$",
"square both (non-negative) sides",
"square, if one side is $\\ge $ 0",
"u < v or u = v iff $u \\le v$",
"combine intervals",
"use assumptions"
},
{ /* greater_than */
"change x > y to y < x",
"change -u > -v to u < v",
"change -u > -v to v > u",
"$x^2^n > a$ is true if $a < 0$",
"$a > x^2^n$ is false if $a \\le 0$",
"square, if one side is $\\ge $ 0",
"u > v or u = v iff $u \\ge v$"
},
{ /* less_than_or_equals */
"change $x \\le y$ to $y \\ge x$",
"add ? to both sides",
"subtract ? from both sides",
"change $-u \\le -v$ to $v \\le u$",
"change $-u \\le -v$ to $u \\ge v$",
"multiply both sides by ?",
"multiply both sides by ?^2",
"divide both sides by ?",
"evaluate numerical inequality",
"$a \\le x^2^n$ is true if $a \\le 0$",
"$x^2^n \\le a$ is false if $a < 0$",
"square both sides",
"$u \\le v$ iff $u^2 \\le v^2$ or $u \\le 0$ provided $0 \\le v$",
"combine intervals",
"use assumptions"
},
{ /* greater_than_or_equals */
"change $x \\ge y$ to $y \\le x$",
"change $-u \\ge -v$ to $u \\le v$",
"change $-u \\ge -v$ to $v \\ge u$",
"$x^2^n \\ge a$ is true if $a \\le 0$",
"$a \\ge x^2^n$ is false if $a < 0$",
"$v \\ge u$ iff $v^2 \\ge u^2$ or $u \\le 0$ provided $0 \\le v$"
},
{ /* square_ineq1 */
"$u^2 < a$ iff $|u| < \\sqrt a$",
"$u^2 < a$ iff $-\\sqrt a < u < \\sqrt a$",
"$a < v^2$ iff $\\sqrt a < |v|$ provided $0\\le a$",
"$a < u^2$ iff $u < -\\sqrt a$ or $\\sqrt a < u$",
"$a < u^2 < b$ iff $-\\sqrt b<u<-\\sqrt a$ or $\\sqrt a<u<\\sqrt b$",
"$-a < u^2 < b$ iff $u^2 < b$ provided 0<a",
"$-a < u^2 \\le b$ iff $u^2 \\le b$ provided 0<a",
"$\\sqrt u < v$ iff $0 \\le u < v^2$",
"$0 \\le a\\sqrt u < v$ iff $0 \\le a^2u < v^2$",
"$a < \\sqrt v$ iff $a^2 < v$ provided $0\\le a$",
"$0 \\le u < v$ iff $\\sqrt u < \\sqrt v$",
"$a < x^2$ is true if $a < 0$",
"$x^2 < a$ is false if $a \\le 0$",
"$a < \\sqrt u$ iff $0 \\le u$ provided $a < 0$"
},
{ /* square_ineq2 */
"$u^2 \\le a$ iff $|u| \\le \\sqrt a$",
"$u^2 \\le a$ iff $-\\sqrt a \\le u \\le \\sqrt a$",
"$a \\le v^2$ iff $\\sqrt a \\le |v|$ provided $0\\le a$",
"$a \\le u^2$ iff $u \\le -\\sqrt a$ or $\\sqrt a \\le u$",
"$a \\le u^2 \\le b$ iff $-\\sqrt b\\le u\\le -\\sqrt a$ or $\\sqrt a\\le u\\le \\sqrt b$",
"$-a \\le u^2 \\le b$ iff $u^2 \\le b$ provided $0\\le a$",
"$-a \\le u^2 < b$ iff $u^2 < b$ provided $0\\le a$",
"$\\sqrt u \\le v$ iff $0 \\le u \\le v^2$",
"$0 \\le a\\sqrt u \\le v$ iff $0 \\le a^2u \\le v^2$",
"$a \\le \\sqrt v$ iff $a^2 \\le v$ provided $0\\le a$",
"$0 \\le u \\le v$ iff $\\sqrt u \\le \\sqrt v$",
"$x^2 > a$ is true if $a < 0$",
"$a > x^2$ is false if $a \\le 0$",
"$a \\le \\sqrt u$ iff $0 \\le u$ provided $a \\le 0$"
},
{ /* recip_ineq1 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"a < 1/x < b iff 1/b < x < 1/a, for a,b > 0",
"$a < 1/x \\le b$ iff $1/b \\le x < 1/a$, for a,b > 0",
"-a < 1/x < -b iff -1/b < x < -1/a, for a,b > 0",
"$-a < 1/x \\le -b$ iff $-1/b \\le x < -1/a$, for a,b > 0",
"-a < 1/x < b iff x < - 1/a or 1/b < x, for a,b > 0",
"$-a < 1/x \\le b$ iff x < -1/a or $1/b \\le x$, for a,b > 0"
},
{ /* recip_ineq2 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"$a \\le 1/x < b$ iff $1/b < x \\le 1/a$, for a,b > 0",
"$a \\le 1/x \\le b$ iff $1/b \\le x < 1/a$, for a,b > 0",
"$-a \\le 1/x < -b$ iff $-1/b < x \\le -1/a$, for a,b > 0",
"$-a \\le 1/x \\le -b$ iff $-1/b \\le x \\le -1/a$, for a,b > 0",
"$-a \\le 1/x < b$ iff $x \\le - 1/a$ or 1/b < x, for a,b > 0",
"$-a \\le 1/x \\le b$ iff $x \\le -1/a$ or $1/b \\le x$, for a,b > 0"
},
{ /* root_ineq1 */
"u < v iff $^n\\sqrt u < ^n\\sqrt v$ (n odd)",
"$u^2^n < a$ iff $|u| < ^2^n\\sqrt a$",
"$u^2^n < a$ iff $-^2^n\\sqrt a < u < ^2^n\\sqrt a$",
"$0 \\le a < u^2^n$ iff $^2^n\\sqrt a < |u|$",
"$a < u^2^n$ iff $u < -^2^n\\sqrt a$ or $^2^n\\sqrt a < u$",
"$a<u^2^n<b$ iff $-^2^n\\sqrt b<u<-^2^n\\sqrt a$ or $^2^n\\sqrt a<u<^2^n\\sqrt b$",
"$^2^n\\sqrt u < v$ iff $0 \\le u < v^2^n$",
"$^n\\sqrt u < v$ iff $u < v^n$ (n odd or $u\\ge 0$)",
"$a(^n\\sqrt u) < v$ iff $a^nu < v^n$ provided $0 \\le a(^n\\sqrt u)$",
"$u < ^n\\sqrt v$ iff $u^n < v$ provided $0 \\le u$",
"$u < v$ iff $u^n < v^n$ (n odd, n>0)",
"u < v iff $u^n < v^n$ (n > 0 and $0 \\le u$)",
"$a < ^2^n\\sqrt u$ iff $0 \\le u$ provided $a < 0$",
},
{ /* root_ineq2 */
"$u \\le v$ iff $^n\\sqrt u \\le ^n\\sqrt v$ (n odd)",
"$u^2^n \\le a$ iff $|u| \\le ^2^n\\sqrt a$",
"$u^2^n \\le a$ iff $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$",
"$0 \\le a \\le u^2^n$ iff $^2^n\\sqrt a \\le |u|$",
"$a \\le u^2^n$ iff $u \\le -^2^n\\sqrt a$ or $^2^n\\sqrt a \\le u$",
"$a\\le u^2^n\\le b$ iff $-^2^n\\sqrt b\\le u\\le -^2^n\\sqrt a$ or $^2^n\\sqrt a\\le u\\le ^2^n\\sqrt b$",
"$^2^n\\sqrt u \\le v$ iff $0 \\le u \\le v^2^n$",
"$^n\\sqrt u \\le v$ iff $u \\le v^n$ (n odd or $u\\ge 0$)",
"$a(^n\\sqrt u) \\le v$ iff $a^nu \\le v^n$ provided $0 \\le a(^n\\sqrt u)$",
"$u \\le ^n\\sqrt v$ iff $u^n \\le v$ provided $0 \\le u$",
"$u \\le v$ iff $u^n \\le v^n$ (n odd, $n \\ge 0$)",
"$u \\le v$ iff $u^n \\le v^n$ (n > 0 and $0 \\le u$)",
"$a \\le ^2^n\\sqrt u$ iff $0 \\le u$ provided $a \\le 0$"
},
{ /* zero_ineq1 */
"drop positive factors",
"0 < u/v iff 0 < v provided u > 0",
"change $0 < u/\\sqrt v$ to 0 < uv",
"0 < u/v iff 0 < uv",
"change $u/\\sqrt v < 0$ to uv < 0",
"u/v < 0 iff uv < 0",
"$ax \\pm b < 0$ iff $a(x\\pm b/a) < 0$",
"change u < v to v > u",
"(x-a)(x-b) < 0 iff a<x<b (where a<b)",
"0 < (x-a)(x-b) iff x<a or b<x (where a<b)"
},
{ /* zero_ineq2 */
"drop positive factors",
"$0 \\le u/v$ iff $0 \\le v$ provided $u \\ge 0$",
"$0 \\le u/\\sqrt v$ iff $0 \\le uv$",
"$0 \\le u/v$ iff 0 < uv or u = 0",
"$u/\\sqrt v \\le 0$ iff $uv \\le 0$",
"$u/v \\le 0$ iff uv < 0 or u = 0",
"$ax \\pm b \\le 0$ iff $a(x\\pm b/a) \\le 0$",
"change $u \\le v$ to $v \\ge u$",
"$(x-a)(x-b) \\le 0$ iff $a\\le x\\le b$ (where $a\\le b$)",
"$0\\le (x-a)(x-b)$ iff $x\\le a$ or $b\\le x$ (where $a\\le b$)"
},
{ /* square_ineq3 */
"$a > u^2$ iff $\\sqrt a > |u|$",
"$a > u^2$ iff $-\\sqrt a < u < \\sqrt a$",
"$v^2 > a$ iff $|v| > \\sqrt a$ provided $a\\ge 0$",
"$u^2 > a$ iff $u < -\\sqrt a$ or $u > \\sqrt a$",
"$v > \\sqrt u$ iff $0 \\le u < v^2$",
"$v>a\\sqrt u$ iff $0\\le a^2u<v^2$ provided $0\\le a$",
"$\\sqrt v > a$ iff $v > a^2$ provided $0\\le a$",
"v > u iff $\\sqrt v > \\sqrt u$ provided $u\\ge 0$",
"$x^2 > a$ is true if $a < 0$",
"$a > x^2$ is false if $a <= 0$",
"$\\sqrt u > a$ iff $u \\ge 0$ provided $a < 0$"
},
{ /* square_ineq4 */
"$a \\ge u^2$ iff $6\\sqrt a \\ge |u|$",
"$a \\ge u^2$ iff $-\\sqrt a \\le u \\le \\sqrt a$",
"$v^2 \\ge a$ iff $|v| \\ge \\sqrt a$ provided $0\\le a$",
"$u^2 \\ge a$ iff $u \\le -\\sqrt a$ or $\\sqrt a \\le u$",
"$v \\ge \\sqrt u$ iff $60 \\le u \\le v^2$",
"$v \\ge a\\sqrt u$ iff $0\\le a^2u\\le v^2$ provided $0\\le a$",
"$\\sqrt v \\ge a$ iff $v \\ge a^2$ provided $0\\le a$",
"$v \\ge u$ iff $\\sqrt v \\ge \\sqrt u$ provided $u\\ge 0$",
"$x^2 \\ge a$ is true if $a \\le 0$",
"$a \\ge x^2$ is false if $a < 0$",
"$\\sqrt u \\ge a$ iff $u \\ge 0$ provided $a \\le 0$"
},
{ /* recip_ineq3 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
},
{ /* recip_ineq4 */
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
"Take the reciprocal of both sides",
},
{ /* root_ineq3 */
"$u > v$ iff $^n\\sqrt u > ^n\\sqrt v$ (n odd)",
"$a > u^2^n$ iff $^2^n\\sqrt a > |u|$",
"$a > u^2^n$ iff $-^2^n\\sqrt a < u < ^2^n\\sqrt a$",
"$u^2^n > a$ iff $|u| > ^2^n\\sqrt a$ provided $a\\ge 0$",
"$u^2^n > a$ iff $u < -^2^n\\sqrt a$ or $u > ^2^n\\sqrt a$",
"$v > ^2^n\\sqrt u$ iff $0 \\le u < v^2^n$",
"$v > ^n\\sqrt u$ iff $v^n> u$ (n odd or $u\\ge 0$)",
"$v > a(^n\\sqrt u)$ iff $v^n > a^nu$ provided $0 \\le a(^n\\sqrt u)$",
"$^n\\sqrt v > a$ iff $v > a^n$ provided $a\\ge 0$",
"u > v iff $u^n > v^n$ (n odd, n>0)",
"u > v iff $u^n > v^n$ (n > 0 and $0 \\le u$)",
"$^2^n\\sqrt u > a$ iff $u \\ge 0$ provided $a < 0$"
},
{ /* root_ineq4 */
"$u \\ge v$ iff $^n\\sqrt u \\ge ^n\\sqrt v$ (n odd)",
"$a \\ge u^2^n$ iff $^2^n\\sqrt a \\ge |u|$",
"$a \\ge u^2^n$ iff $-^2^n\\sqrt a \\le u \\le ^2^n\\sqrt a$",
"$u^2^n \\ge a$ iff $|u| \\ge ^2^n\\sqrt a$ provided $a\\ge 0$",
"$u^2^n \\ge a$ iff $u \\le -^2^n\\sqrt a$ or $u \\ge ^2^n\\sqrt a$",
"$v \\ge ^2^n\\sqrt u$ iff $0 \\le u \\le v^2^n$",
"$v \\ge ^n\\sqrt u$ iff $v^n \\ge u$ (n odd or $u\\ge 0$)",
"$v \\ge a(^n\\sqrt u)$ iff $v^n \\ge a^nu$ provided $0 \\le a(^n\\sqrt u)$",
"$^n\\sqrt v \\ge a$ iff $a^n \\le v$ provided $a \\ge 0$",
"$u \\ge v$ iff $u^n \\ge v^n$ (n odd, $n \\ge 0$)",
"$u \\ge v$ iff $u^n \\ge v^n$ (n > 0 and $0 \\le u$)",
"$^2^n\\sqrt u \\ge a$ iff $u \\ge 0$ provided $a \\le 0$"
},
{ /* zero_ineq3 */
"u/v > 0 iff v > 0 provided u > 0",
"change $u/\\sqrt v > 0$ to uv > 0",
" u/v > 0 iff uv > 0",
"change $0 > u/\\sqrt v$ to 0 > uv",
"0 > u/v iff 0 > uv",
"$0 > ax \\pm b$ iff $0 > a(x\\pm b/a)$",
"0 > (x-a)(x-b) iff a<x<b (where a<b)",
"(x-a)(x-b) > 0 iff x<a or x>b (where a<b)"
},
{ /* zero_ineq4 */
"$u/v \\ge 0$ iff $v \\ge 0$ provided $u \\ge 0$",
"$u/\\sqrt v \\ge 0$ iff $uv \\ge 0$",
"$u/v \\ge 0$ iff uv > 0 or u = 0",
"$0 \\ge u/\\sqrt v$ iff $0 \\ge uv$",
"$0 \\ge u/v$ iff 0 > uv or u = 0",
"$0 \\ge ax \\pm b$ iff $0 \\ge a(x\\pm b/a)$",
"$0 \\ge (x-a)(x-b)$ iff $a\\le x\\le b$ (where $a\\le b$)",
"$(x-a)(x-b)\\ge 0$ iff $x\\le a$ or $b\\le x$ (where $a\\le b$)"
},
{ /* binomial_theorem */
"expand by binomial theorem",
"binomial theorem with (n k)",
"(n k) = n!/((n-k)!k!)",
"n! = n(n-1)(n-2)...1",
"compute factorial",
arithstr,
"evaluate binomial coefficient",
"expand $\\sum $ notation",
"evaluate $\\sum $ to rational",
"n! = n (n-1)!",
"n!/n = (n-1)!",
"n!/(n-1)! = n",
"n!/k! = n(n-1)...(n-k+1)",
"n/n! = 1/(n-1)!",
"(n-1)!/n! = 1/n",
"k!/n! =1/(n(n-1)...(n-k+1))"
},
{ /* factor_expansion */
"a^3+3a^2b+3ab^2+b^3 = (a+b)^3",
"a^3-3a^2b+3ab^2-b^3 = (a-b)^3",
"a^4+4a^3b+6a^2b^2+4ab^3+b^4 = (a+b)^4",
"a^4-4a^3b+6a^2b^2-4ab^3+b^4 = (a-b)^4",
"a^n+na^(n-1)b+...b^n = (a+b)^n",
"a^n-na^(n-1)b+...b^n = (a-b)^n"
},
{ /* sigma_notation */
"$\\sum $ 1 = number of terms",
"$\\sum $ -u = -$\\sum $ u",
"$\\sum $ cu = c$\\sum $ u (c const)",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"$\\sum (u-v) = \\sum u - \\sum v$",
"expand $\\sum $ using +",
"1+2+..+n = n(n+1)/2",
"$1^2+..+n^2 = n(n+1)(2n+1)/6$",
"$1+x+..+x^n=(1-x^(n+1))/(1-x)$",
"split off first few terms",
"evaluate $\\sum $ with parameter to rational",
"evaluate $\\sum $ with parameter to decimal",
"evaluate numerical $\\sum $ to rational",
"evaluate numerical $\\sum $ to decimal",
"express summand as polynomial",
"telescoping sum"
},
{ /* advanced_sigma_notation */
"shift sum limits",
"rename index variable",
"$(\\sum u)(\\sum v) = \\sum \\sum uv$",
"split off last term",
"$1^3+..+n^3 = n^2(n+1)^2/4$",
"$1^4+..+n^4=n(n+1)(2n+1)(3n^2+2n-1)/30$",
"$d/dx \\sum u = \\sum du/dx$",
"$\\sum du/dx = d/dx \\sum u$",
"$\\int \\sum u dx = \\sum \\int u dx$",
"$\\sum \\int u dx = \\int \\sum u dx$",
"c$\\sum $ u = $\\sum $ cu (c constant)",
"$$sum(t,i,a,b)=sum(t,i,0,b)-sum(t,i,0,a-1)$$",
"$$sum(t,i,a,b)=sum(t,i,c,b)-sum(t,i,c,a-1)$$"
},
{ /* prove_by_induction */
"select induction variable",
"start basis case",
"start induction step",
"use induction hypothesis",
"therefore as desired"
},
{ /* trig_ineq */
"$|sin u| \\le 1$",
"$|cos u| \\le 1$",
"$sin u \\le u$ if $u\\ge 0$",
"$1 - u^2/2 \\le cos u$",
"$|arctan u| \\le \\pi /2$",
"$arctan u \\le u$ if $u\\ge 0$",
"$u \\le tan u$ if $0\\le u\\le \\pi /2$"
},
{ /* log_ineq1 */
"Take the natural log of both sides",
"Take the log of both sides",
"u < ln v iff e^u < v",
"ln u < v iff u < e^v",
"u < log v iff 10^u < v",
"log u < v iff u < 10^v",
"u < v iff ?^u < ?^v"
},
{ /* log_ineq2 */
"Take the natural log of both sides",
"Take the log of both sides",
"$u \\le ln v$ iff $e^u \\le v$",
"$ln u \\le v$ iff $u \\le e^v$",
"$u \\le log v$ iff $10^u \\le v$",
"$log u \\le v$ iff $u \\le 10^v$",
"$u \\le v$ iff $?^u \\le ?^v$", /* takes arg in menu mode */
},
{ /* log_ineq3 */
"Take the natural log of both sides",
"Take the log of both sides",
"ln u > v iff u > e^v",
"u > ln v iff e^u > v",
"log u > v iff u > 10^v",
"u > log v iff 10^u > v",
"u > v iff ?^u > ?^v",
},
{ /* log_ineq4 */
"Take the natural log of both sides",
"Take the log of both sides",
"$ln u \\ge v$ iff $u \\ge e^v$",
"$u \\ge ln v$ iff $e^u \\ge v$",
"$log u \\ge v$ iff $u \\ge 10^v$",
"$u \\ge log v$ iff $10^u \\ge v$",
"$u \\ge v$ iff $?^u \\ge ?^v$", /* takes arg in menu mode */
},
{ /* logarithms_base10 */
"$10^(log a) = a$",
"$log 10^n = n$ ($n$ real)",
"log 1 = 0",
"log 10 = 1",
"$log a = (ln a)/(ln 10)$",
"u^v = 10^(v log u)",
"factor number completely",
"factor out powers of 10",
"10^(n log a) = a^n",
"log(a/b) = -log(b/a)",
"log(b,a/c) = -log(b,c/a)"
},
{ /* logarithms */
"$log a^n = n log a$",
"$log ab = log a + log b$",
"$log 1/a = -log a$",
"$log a/b = log a - log b$",
"$log a + log b = log ab$",
"$log a - log b = log a/b$",
"$log a + log b - log c =log ab/c$",
"$n log a = log a^n (n real)$",
"$log \\sqrt a = \\onehalf log a$",
"$log ^n\\sqrt a = (1/n) log a$",
"log 1 = 0",
"factor number completely",
"factor out powers of base",
"$log u = (1/?) log u^?$",
"evaluate logs numerically",
"$log a = (ln a)/(ln 10)$"
},
{ /* logarithms_base_e */
"e^(ln a) = a",
"ln e = 1",
"ln 1 = 0",
"ln e^n = n (n real)",
"u^v = e^(v ln u)",
"e^((ln c) a) = c^a"
},
{ /* natural_logarithms */
"ln a^n = n ln a",
"$ln ab = ln a + ln b$",
"ln 1/a = -ln a",
"$ln a/b = ln a - ln b$",
"ln 1 = 0",
"factor number completely",
"$ln a + ln b = ln ab$",
"$ln a - ln b = ln a/b$",
"$ln a + ln b - ln c = ln (ab/c)$",
"$n ln a = ln a^n (n real)$",
"$ln \\sqrt a = \\onehalf ln a$",
"$ln ^n\\sqrt a = (1/n) ln a$",
"ln u = (1/?) ln u^?", /* user supplies exponent; needed for diff(ln x,x) from defn */
"evaluate logarithm numerically",
"ln(a/b) = -ln(b/a)"
},
{ /* reverse_trig */
"sin u cos v + cos u sin v = sin(u+v)",
"sin u cos v - cos u sin v = sin(u-v)",
"cos u cos v - sin u sin v = cos(u+v)",
"cos u cos v + sin u sin v = cos(u-v)",
"(sin u)/(1+cos u) = tan(u/2)",
"(1-cos u)/sin u = tan(u/2)",
"(1+cos u)/(sin u) = cot(u/2)",
"sin u/(1-cos u) = cot(u/2)",
"(tan u+tan v)/(1-tan u tan v) = tan(u+v)",
"(tan u-tan v)/(1+tan u tan v) = tan(u-v)",
"(cot u cot v-1)/(cot u+cot v) = cot(u+v)",
"(1+cot u cot v)/(cot v-cot u) = cot(u-v)",
"1-cos u = 2 sin^2(u/2)"
},
{ /* complex_polar_form */
"polar form",
"$r e^(i\\theta ) = r (cos \\theta + i sin \\theta )$",
"$|e^(i\\theta )| = 1$",
"$|Re^(i\\theta )|=R$ if $R\\ge 0$",
"$|Re^(i\\theta )| = |R|$",
"$-a = ae^(\\pi i)$",
"$^n\\sqrt (-a) = e^(\\pi i/n) ^n\\sqrt a if a\\ge 0$",
"a/(ce^(ti)) = ae^(-ti)/c",
"de Moivre's theorem",
"substitute specific integers"
},
{ /* logs_to_any_base */
"b^(log(b,a)) = a",
"b^(n log(b,a)) = a^n",
"log(b,b) = 1",
"log(b,b^n) = n",
"log xy = log x + log y",
"log (1/x) = -log x",
"log x/y = log x-log y",
"log(b,1) = 0",
"factor base: log(4,x)=log(2^2,x)",
"log(b^n,x) = (1/n) log (b,x)",
"log x^n = n log x",
"factor out powers of base",
"log x + log y = log xy",
"log x - log y = log x/y",
"log x + log y - log z =log xy/z",
"n log x = log x^n (n real)"
},
{ /* change_base */
"log(b,x) = ln x / ln b",
"log(b,x) = log x / log b",
"log(b,x) = log(a,x) / log(a,b)",
"log(b^n,x) = (1/n) log (b,x)",
"log(10,x) = log x",
"log(e,x) = ln x",
"log x = ln x / ln 10",
"ln x = log x / log e",
"u^v = b^(v log(b,u))"
},
{ /* evaluate_trig_function */
"sin 0 = 0",
"cos 0 = 1",
"tan 0 = 0",
"$sin k\\pi = 0$", /* logically, these are needed to prove sin(x+2�)=sin x */
"$cos 2k\\pi = 1$", /* They have to be proved separately by induction */
"$tan k\\pi = 0$",
"find coterminal angle < $360\\deg $",
"find coterminal angle < $2\\pi $",
"angle is multiple of $90\\deg $",
"use 1-2-$\\sqrt 3$ triangle",
"use 1-1-$\\sqrt 2$ triangle",
"change radians to degrees",
"change degrees to radians",
"angle = $a 30\\deg + b 45\\deg $ etc.",
"evaluate numerically"
},
{ /* basic_trig */
"tan u = sin u / cos u",
"cot u = 1 / tan u",
"cot u = cos u / sin u",
"sec u = 1 / cos u",
"csc u = 1 / sin u",
"sin u / cos u = tan u",
"cos u / sin u = cot u"
},
{ /* trig_reciprocals */
"1 / sin u = csc u",
"1 / cos u = sec u",
"1 / tan u = cot u",
"1 / tan u = cos u / sin u",
"1 / cot u = tan u",
"1 / cot u = sin u / cos u",
"1 / sec u = cos u",
"1 / csc u = sin u",
"sin u = 1 / csc u",
"cos u = 1 / sec u",
"tan u = 1 / cot u"
},
{ /* trig_squares */
"$sin^2 u + cos^2 u = 1$",
"$1 - sin^2 u = cos^2 u$",
"$1 - cos^2 u = sin^2 u$",
"$sin^2 u = 1 - cos^2 u$",
"$cos^2 u = 1 - sin^2 u$",
"$sec^2 u - tan^2 u = 1$",
"$tan^2 u + 1 = sec^2 u$",
"$sec^2 u - 1 = tan^2 u$",
"$sec^2 u = tan^2 u + 1$",
"$tan^2 u = sec^2 u - 1$",
"$sin^(2n+1) u = sin u (1-cos^2 u)^n$",
"$cos^(2n+1) u = cos u (1-sin^2 u)^n$",
"$tan^(2n+1) u = tan u (sec^2 u-1)^n$",
"$sec^(2n+1) u = sec u (tan^2 u+1)^n$",
"(1-cos t)^n(1+cos t)^n = sin^(2n) t",
"(1-sin t)^n(1+sin t)^n = cos^(2n) t"
},
{ /* csc_and_cot_identities */
"$csc^2 u - cot^2 u = 1$",
"$cot^2 u + 1 = csc^2 u$",
"$csc^2 u - 1 = cot^2 u$",
"$csc^2 u = cot^2 u + 1$",
"$cot^2 u = csc^2 u - 1$",
"$csc(\\pi /2-\\theta ) = sec \\theta $",
"$cot(\\pi /2-\\theta ) = tan \\theta $",
"$cot^(2n+1) u = cot u (csc^2 u-1)^n$",
"$csc^(2n+1) u = csc u (cot^2 u+1)^n$"
},
{ /* trig_sum */
"sin(u+v)= sin u cos v + cos u sin v",
"sin(u-v)= sin u cos v - cos u sin v",
"cos(u+v)= cos u cos v - sin u sin v",
"cos(u-v)= cos u cos v + sin u sin v",
"tan(u+v)=(tan u+tan v)/(1-tan u tan v)",
"tan(u-v)=(tan u-tan v)/(1+tan u tan v)",
"cot(u+v)=(cot u cot v-1)/(cot u+cot v)",
"cot(u-v)=(1+cot u cot v)/(cot v-cot u)"
},
{ /* double_angle */
"$sin 2\\theta = 2 sin \\theta cos \\theta $",
"$cos 2\\theta = cos^2 \\theta - sin^2 \\theta $",
"$cos 2\\theta = 1 - 2 sin^2 \\theta $",
"$cos 2\\theta = 2 cos^2 \\theta - 1$",
"$cos 2\\theta + 1 = 2cos^2 \\theta $",
"$cos 2\\theta - 1 = - 2 sin^2 \\theta $",
"$tan 2\\theta = 2 tan \\theta /(1 - tan^2 \\theta )$",
"$cot 2\\theta = (cot^2 \\theta -1) / (2 cot \\theta )$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$2 sin \\theta cos \\theta = sin 2\\theta $",
"$cos^2 \\theta - sin^2 \\theta = cos 2\\theta $",
"$1 - 2 sin^2 \\theta = cos 2\\theta $",
"$2 cos^2 \\theta - 1 = cos 2\\theta $"
},
{ /* multiple_angles */
"$n\\theta = (n-1)\\theta + \\theta $",
"$n\\theta = ?\\theta +(n-?)\\theta $",
"$sin 3\\theta = 3 sin \\theta - 4 sin^3 \\theta $",
"$cos 3\\theta = -3 cos \\theta + 4 cos^3 \\theta $",
"expand $sin n\\theta $ in $sin \\theta $, $cos \\theta $",
"expand $cos n\\theta $ in $sin \\theta $, $cos \\theta $"
},
{ /* verify_identities */
"cross multiply",
"switch sides",
"transfer ? left to right",
"transfer ? right to left",
"add ? to both sides",
"subtract ? from both sides",
"multiply both sides by ?",
"cancel term from both sides",
"raise both sides to power",
"take square root of both sides",
"take root of both sides",
"apply function to both sides",
arithstr,
"check numerically",
"make a substitution, u = ?",
},
{ /* solve_by_30_60_90 */
"$sin(u)=1/2$ iff $u=\\pi /6$ or $5\\pi /6+2n\\pi $",
"$sin(u)=-1/2$ iff $u=-\\pi /6$ or $-5\\pi /6+2n\\pi $",
"$sin(u)=\\sqrt 3/2$ iff $u=\\pi /3$ or $2\\pi /3+2n\\pi $",
"$sin(u)=-\\sqrt 3/2$ iff $4u=-\\pi /3$ or $-2\\pi /3+2n\\pi $",
"$cos(u)=\\sqrt 3/2$ iff $u=\\pm \\pi /6 + 2n\\pi $",
"$cos(u)=-\\sqrt 3/2$ iff $u=\\pm 5\\pi /6 + 2n\\pi $",
"$cos(u)=1/2$ iff $u=\\pm \\pi /3+2n\\pi $",
"$cos(u)=-1/2$ iff $u=\\pm 2\\pi /3+2n\\pi $",
"$tan(u)=1/\\sqrt 3$ iff $u= \\pi /6 + n\\pi $",
"$tan(u)=-1/\\sqrt 3$ iff $u= -\\pi /6 + n\\pi $",
"$tan(u)=\\sqrt 3$ iff $u= \\pi /3 + n\\pi $",
"$tan(u)=-\\sqrt 3$ iff $u= 2\\pi /3 + n\\pi $"
},
{ /* solve_by_45_45_90 */
"$sin u = 1/\\sqrt 2$ if $u=\\pi /4$ or $3\\pi /4 + 2n\\pi $",
"$sin u=-1/\\sqrt 2$ if $u=5\\pi /4$ or $7\\pi /4 + 2n\\pi $2",
"$cos u = 1/\\sqrt 2$ if $u=\\pi /4$ or $7\\pi /4 + 2n\\pi $",
"$cos u=-1/\\sqrt 2$ if $u=3\\pi /4$ or $5\\pi /4 + 2n\\pi $",
"tan u = 1 if $u= \\pi /4$ or $5\\pi /4 + 2n\\pi $",
"tan u = -1 if $u=3\\pi /4$ or $7\\pi /4 + 2n\\pi $"
},
{ /* zeroes_of_trig_functions */
"sin u = 0 iff $u = n\\pi $",
"sin u = 1 iff $u = \\pi /2+2n\\pi $",
"sin u = -1 iff $u = 3\\pi /2+2n\\pi $",
"cos u = 0 iff $u = (2n+1)\\pi /2$",
"cos u = 1 iff $u = 2n\\pi $",
"cos u = -1 iff $u = (2n+1)\\pi $",
"tan u = 0 iff sin u = 0",
"cot u = 0 iff cos u = 0"
},
{ /* inverse_trig_functions */
"sin u=c iff $u= (-1)^narcsin c+n\\pi $",
"sin u=c iff $u=arcsin(c)+2n\\pi $ or $2n\\pi +\\pi -arcsin(c)$",
"cos u=c iff $u=\\pm arccos c+2n\\pi $",
"tan u=c iff $u=arctan c+n\\pi $", /* c not � i */
"evaluate arcsin exactly",
"evaluate arccos exactly",
"evaluate arctan exactly",
"arccot x = arctan (1/x)",
"arcsec x = arccos (1/x)",
"arccsc x = arcsin (1/x)",
"arcsin(-x) = -arcsin x",
"$arccos(-x) = \\pi -arccos x$",
"arctan(-x) = -arctan x",
"put solutions in periodic form",
"reject sin u = c if |c|>1",
"reject cos u = c if |c|>1"
},
{ /* invsimp */
"$tan(arcsin x) = x/\\sqrt (1-x^2)$",
"$tan(arccos x) = \\sqrt (1-x^2)/x$",
"tan(arctan x) = x",
"sin(arcsin x) = x",
"$sin(arccos x) = \\sqrt (1-x^2)$",
"$sin(arctan x) = x/\\sqrt (x^2+1)$",
"$cos(arcsin x) = \\sqrt (1-x^2)$",
"cos(arccos x) = x",
"$cos(arctan x) = 1/\\sqrt (x^2+1)$",
"$sec(arcsin x) = 1/\\sqrt (1-x^2)$",
"$sec(arccos x) = 1/x$",
"$sec(arctan x) = \\sqrt (x^2+1)$",
"$arctan(tan \\theta ) = \\theta $6 if $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arcsin(sin \\theta ) = \\theta $ if $-\\pi /2\\le \\theta \\le \\pi /2$",
"$arccos(cos \\theta ) = \\theta $ if $0\\le \\theta \\le \\pi $",
"arctan(tan x) = x + c1"
},
{ /* adding_arctrig_functions */
"arcsin x + arccos x = $\\pi /2$",
"$arctan x + arctan 1/x = \\pi x/2|x|$",
#if 0 /* Perhaps add these later */
"$arcsin x \\pm arcsin y = arcsin[x\\sqrt (1-y^2)\\pm y\\sqrt (1-x^2)]$",
"$arccos x + arccos y = arccos[xy-\\sqrt ((1-x^2)(1-y^2))]$",
"$arccos x - arccos y = arccos[xy+\\sqrt ((1-x^2)(1-y^2))]$",
"$arctan x + arctan y = arctan[(x+y)/(1-xy)]$",
"$arctan x - arctan y = arctan[(x-y)/(1+xy)]$",
#endif
},
{ /* complementary_trig */
"$sin(\\pi /2-\\theta ) = cos \\theta $",
"$cos(\\pi /2-\\theta ) = sin \\theta $",
"$tan(\\pi /2-\\theta ) = cot \\theta $",
"$cot(\\pi /2-\\theta ) = tan \\theta $",
"$sec(\\pi /2-\\theta ) = csc \\theta $",
"$csc(\\pi /2-\\theta ) = sec \\theta $",
"$sin \\theta = cos(\\pi /2-\\theta )$",
"$cos \\theta = sin(\\pi /2-\\theta )$",
"$tan \\theta = cot(\\pi /2-\\theta )$",
"$cot \\theta = tan(\\pi /2-\\theta )$",
"$sec \\theta = csc(\\pi /2-\\theta )$",
"$csc \\theta = sec(\\pi /2-\\theta )$"
},
{ /* complementary_degrees */
"$sin(90\\deg -\\theta ) = cos \\theta $",
"$cos(90\\deg -\\theta ) = sin \\theta $",
"$tan(90\\deg -\\theta ) = cot \\theta $",
"$cot(90\\deg -\\theta ) = tan \\theta $",
"$sec(90\\deg -\\theta ) = csc \\theta $",
"$csc(90\\deg -\\theta ) = sec \\theta $",
"$sin \\theta = cos(90\\deg -\\theta )$",
"$cos \\theta = sin(90\\deg -\\theta )$",
"$tan \\theta = cot(90\\deg -\\theta )$",
"$cot \\theta = tan(90\\deg -\\theta )$",
"$sec \\theta = csc(90\\deg -\\theta )$",
"$csc \\theta = sec(90\\deg -\\theta )$",
"$a\\deg + b\\deg = (a+b)\\deg $",
"$ca\\deg = (ca)\\deg $",
"$a\\deg /c = (a/c)\\deg $"
},
{ /* trig_odd_and_even */
"sin(-u) = - sin u",
"cos(-u) = cos u",
"tan(-u) = - tan u",
"cot(-u) = - cot u",
"sec(-u) = sec u",
"csc(-u) = - csc u",
"$sin^2(-u) = sin^2 u$",
"$cos^2(-u) = cos^2 u$",
"$tan^2(-u) = tan^2 u$",
"$cot^2(-u) = cot^2 u$",
"$sec^2(-u) = sec^2 u$",
"$csc^2(-u) = csc^2 u$"
},
{ /* trig_periodic */
"$sin(u+2\\pi ) = sin u$",
"$cos(u+2\\pi ) = cos u$",
"$tan(u+\\pi ) = tan u$",
"$sec(u+2\\pi ) = sec u$",
"$csc(u+2\\pi ) = csc u$",
"$cot(u+\\pi ) = cot u$",
"$sin^2(u+\\pi ) = sin^2 u$",
"$cos^2(u+\\pi ) = cos^2 u$",
"$sec^2(u+\\pi ) = sec^2 u$",
"$csc^2(u+\\pi ) = csc^2 u$",
"$sin u = -sin(u-\\pi )$",
"$sin u = sin(\\pi -u)$",
"$cos u = -cos(u-\\pi )$",
"$cos u = -cos(\\pi -u)$"
},
{ /* half_angle_identities */
"$sin^2(\\theta /2) = (1-cos \\theta )/2$",
"$cos^2(\\theta /2) = (1+cos \\theta )/2$",
"$sin^2(\\theta ) = (1-cos 2\\theta )/2$",
"$cos^2(\\theta ) = (1+cos 2\\theta )/2$",
"$sin \\theta cos \\theta = \\onehalf sin 2\\theta $",
"$tan(\\theta /2) = (sin \\theta )/(1+cos \\theta )$",
"$tan(\\theta /2) = (1-cos \\theta )/sin \\theta $",
"$cot(\\theta /2) = (1+cos \\theta )/(sin \\theta )$",
"$cot(\\theta /2) = sin \\theta /(1-cos \\theta )$",
"$sin(\\theta /2) = \\sqrt ((1-cos \\theta )/2) if sin(\\theta /2)\\ge 0$",
"$sin(\\theta /2) = -\\sqrt ((1-cos \\theta )/2) if sin(\\theta /2)\\le 0$",
"$cos(\\theta /2) = \\sqrt ((1+cos \\theta )/2) if cos(\\theta /2)\\ge 0$",
"$cos(\\theta /2) = -\\sqrt ((1+cos \\theta )/2) if cos(\\theta /2)\\le 0$",
"$\\theta = 2(\\theta /2)$"
},
{ /* product_and_factor_identities */
"$sin x cos x = \\onehalf sin 2x$",
"$sin x cos y = \\onehalf [sin(x+y)+sin(x-y)]$",
"$cos x sin y = \\onehalf [sin(x+y)-sin(x-y)]$",
"$sin x sin y = \\onehalf [cos(x-y)-cos(x+y)]$",
"$cos x cos y = \\onehalf [cos(x+y)+cos(x-y)]$",
"$sin x + sin y = 2 sin \\onehalf (x+y) cos \\onehalf (x-y)$",
"$sin x - sin y = 2 sin \\onehalf (x-y) cos \\onehalf (x+y)$",
"$cos x + cos y = 2 cos \\onehalf (x+y) cos \\onehalf (x-y)$",
"$cos x - cos y = -2 sin \\onehalf (x+y) sin \\onehalf (x-y)$",
"substitute u,v for expressions in trig functions"
},
{ /* limits */
"experiment numerically",
"$lim u\\pm v = lim u \\pm lim v$",
"$lim u-v = lim u - lim v$",
"lim(t\32a,c) = c (c constant)",
"lim(t\32a,t) = a",
"lim cu=c lim u (c const)",
"lim -u = -lim u",
"lim uv = lim u lim v",
"$lim u^n = (lim u)^n$",
"lim c^v=c^(lim v) (c constant > 0)",
"lim u^v=(lim u)^(lim v)",
"$lim \\sqrt u=\\sqrt (lim u)$ if lim u>0",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ if n is odd",
"$lim ^n\\sqrt u = ^n\\sqrt (lim u)$ if lim u > 0",
"lim(t\32a,f(t))=f(a) (polynomial f)",
"lim |u| = |lim u|"
},
{ /* limits_of_quotients */
"lim cu/v = c lim u/v (c const)",
"lim c/v = c/lim v (c const)",
"lim u/v = lim u/lim v",
"factor out (x-a)^n in limit as x\32a",
"limit of rational function",
"$a^n/b^n = (a/b)^n$",
"rationalize fraction",
"pull out nonzero finite limits", /* lim uv = lim u lim v where lim u is finite nonzero */
"factor out constant",
"mult num and denom by ?",
"divide num and denom by ?",
"lim u/v = lim (u/?) / lim (v/?)",
"(ab+ac+d)/q = a(b+c)/q + d/q", /* limapartandfactor */
/* example : (sin x cos h + cos x sin h - sin x)/h */
},
{ /* quotients_of_roots */
"$\\sqrt a/b = \\sqrt (a/b^2)$ if b>0",
"$\\sqrt a/b = -\\sqrt (a/b^2)$ if b<0",
"$^n\\sqrt a/b = ^n\\sqrt (a/b^n)$ (b>0 or n odd)",
"$^n\\sqrt a/b = -^n\\sqrt (a/b^n)$ (b<0, n even)",
"$a/\\sqrt b = \\sqrt (a^2/b)$ if $a\\ge 0$",
"$a/\\sqrt b = -\\sqrt (a^2/b)$ if $a\\le 0$",
"$a/^n\\sqrt b = ^n\\sqrt (a^n/b)$ ($a\\ge 0$ or n odd)",
"$a/^n\\sqrt b = -^n\\sqrt (a^n/b)$ ($a\\le 0$, n even)"
},
{ /* lhopitalmenu */
"L'Hospital's rule",
"evaluate derivative in one step",
"lim u ln v = lim (ln v)/(1/u)",
"$lim u (ln v)^n = lim (ln v)^n/(1/u)$",
"$lim x^(-n) u = lim u/x^n$",
"lim u e^x = lim u/e^(-x)",
"move trig function to denominator",
"lim ?v = lim v/(1/?)",
"common denom and simplify numerator"
},
{ /* special_limits */
"(sin t)/t \32 1 as t\32""0",
"(tan t)/t \32 1 as t\32""0",
"(1-cos t)/t \32 0 as t\32""0",
"$(1-cos t)/t^2\32""\\onehalf $ as t\32""0",
"lim(t\32""0,(1+t)^(1/t)) = e",
"$(ln(1\\pm t))/t \32 \\pm 1$ as t\32""0",
"(e^t-1)/t \32 1 as t\32""0",
"(e^(-t)-1)/t \32 -1 as t\32""0",
"$lim(t\32""0,t^nln |t|)=0 (n > 0)$",
"lim(t\32""0,cos(1/t))=undefined",
"lim(t\32""0,sin(1/t))=undefined",
"lim(t\32""0,tan(1/t))=undefined",
"lim(t\32""$\\pm \\infty $,cos t)=undefined",
"lim(t\32""$\\pm \\infty $,sin t)=undefined",
"lim(t\32""$\\pm \\infty $,tan t)=undefined"
},
{ /* hyper_limits */
"(sinh t)/t \32 1 as t\32""0",
"(tanh t)/t \32 1 as t\32""0",
"(cosh t - 1)/t \32 0 as t\32""0",
"(cosh t - 1)/t^2\32""1/2 as t\32""0",
},
{ /* advanced_limits */
"lim ln u=ln lim u (if lim u > 0)",
"lim f(u)=f(lim u), f continuous",
"change limit variable", /* lim(t\32a,f(g(t)))=lim(u\32g(a),f(u)) */
"evaluate limit in one step",
"lim u^v = lim e^(v ln u)",
"lim ?v = lim v/(1/?)",
"limit undefined due to domain",
"lim u = e^(lim ln u)",
"squeeze theorem: uv\32""0 if v\32""0 & $|u|\\le c$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/(\\sqrt u+v)$",
"lim u/v = limit of leading terms",
"leading term: lim(u+a)=lim(u) if a/u\32""0",
"replace sum by leading term",
"f(undefined) = undefined",
"lim(e^u) = e^(lim u)",
"lim(ln u) = ln(lim u)"
},
{ /* logarithmic_limits */
"$lim(t\32""0+,t ln t) = 0$",
"$lim(t\32""0+,t^n ln t) = 0 if n\\ge 1$",
"$lim(t\32""0+,t (ln t)^n) = 0 if n\\ge 1$",
"$lim(t\32""0+,t^k (ln t)^n) = 0 if k,n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t) = 0$",
"$lim(t\32\\infty ,ln(t)^n/t) = 0 if n\\ge 1$",
"$lim(t\32\\infty ,ln(t)/t^n) = 0 if n\\ge 1$",
"$lim(t\32\\infty ,ln(t)^k/t^n) = 0 if k,n\\ge 1$",
"$lim(t\32\\infty ,t/ln(t)) = \\infty $",
"$lim(t\32\\infty ,t/ln(t)^n) = \\infty if n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)) = \\infty if n\\ge 1$",
"$lim(t\32\\infty ,t^n/ln(t)^k) = \\infty if k,n\\ge 1$"
},
{ /* limits_at_infinity */
"$lim(t\32\\infty ,1/t^n) = 0 if n\\ge 1$",
"$lim(t\32\\infty ,t^n) = \\infty if n\\ge 1$",
"$lim(t\32\\infty ,e^t) = \\infty $",
"$lim(t\32-\\infty ,e^t) = 0$",
"$lim(t\32\\infty ,ln t) = \\infty $",
"$lim(t\32\\infty ,\\sqrt t) = \\infty $",
"$lim(t\32\\infty ,^n\\sqrt t) = \\infty $",
"$lim(t\32\\pm \\infty ,arctan t) = \\pm \\pi /2$",
"$lim(t\32\\infty ,arccot t) = 0$",
"$lim(t\32-\\infty ,arccot t) = \\pi $",
"$lim(t\32\\pm \\infty ,tanh t) = \\pm 1$",
"$lim \\sqrt u-v=lim (\\sqrt u-v)(\\sqrt u+v)/\\sqrt u+v)$",
"lim sin u = sin(lim u)",
"lim cos u = cos(lim u)",
"change limit at $\\infty $ to limit at 0",
"lim u/v = limit of leading terms"
},
{ /* infinite_limits */
"$lim(1/u^2^n) = \\infty $ if u\32""0",
"lim(1/u^n) undef if u\32""0, n odd",
"lim(t\32a+,1/u^n) = $\\infty $ if u\32""0",
"lim(t\32a-,1/u^n)=-$\\infty $, u\32""0, n odd",
"lim u/v undef if lim v =0, lim u #0",
"lim(t\32""0+,ln t) = -$\\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,tan t) = \\pm \\infty $",
"$lim(t\32n\\pi \\pm ,cot t) = \\pm \\infty $",
"$lim(t\32(2n+1)\\pi /2\\pm ,sec t) = \\pm \\infty $",
"$lim(t\32n\\pi \\pm ,csc t) = \\pm \\infty $",
"lim(uv) = lim(u/?) lim(?v)",
"lim(uv) = lim(?u) lim(v/?)"
},
{ /* infinities */
"$\\pm \\infty $/positive = $\\pm \\infty $",
"nonzero/$\\pm \\infty $ = 0",
"positive$\\times \\pm \\infty = \\pm \\infty $",
"$\\pm \\infty \\times \\infty = \\pm \\infty $",
"$\\pm \\infty $ + finite = $\\pm \\infty $",
"$\\infty + \\infty = \\infty $",
"$u^\\infty = \\infty $ if u > 1",
"$u^\\infty = 0$ if 0 < u < 1",
"$u^(-\\infty ) = 0$ if u > 1",
"$u^(-\\infty ) = \\infty $ if 0 < u < 1",
"$\\infty ^n = \\infty $ if n > 0",
"$\\infty - \\infty =$ undefined"
},
{ /* zero_denom */
"$a/0+ = \\infty $ if $a>0$",
"$a/0- = -\\infty $ if $a>0$",
"a/0 = undefined",
"$\\infty /0+ = \\infty $",
"$\\infty /0- = -\\infty $",
"$\\infty /0$ = undefined",
"$\\infty /0^2 = \\infty $",
"$\\infty /0^2^n = \\infty $",
"$a/0^2 = \\infty $ if $a > 0$",
"$a/0^2 = -\\infty $ if $a < 0$",
"$a/0^2^n = \\infty $ if $a > 0$",
"$a/0^2^n = -\\infty $ if $a < 0$"
},
{ /* more_infinities */
"$ln \\infty = log \\infty = \\infty $",
"$\\sqrt \\infty = \\infty $",
"$^n\\sqrt \\infty = \\infty $",
"$arctan \\pm \\infty = \\pm \\pi /2$",
"$arccot \\infty = 0$",
"$arccot -\\infty = \\pi $",
"$arcsec \\pm \\infty = \\pi /2$",
"$arccsc \\pm \\infty = 0$",
"trig limits at $\\infty $ undefined",
"$cosh \\pm \\infty = \\infty $",
"$sinh \\pm \\infty = \\pm \\infty $",
"$tanh \\pm \\infty = \\pm 1$",
"$ln 0 = -\\infty $"
},
{ /* polynomial_derivs */
"dc/dx=0 (c not dependent on x)",
"dx/dx = 1",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx(cu)=c du/dx (c indep of x)",
"d/dx x^n = n x^(n-1)",
"differentiate polynomial",
"f'(x) = d/dx f(x)"
},
{ /* derivatives */
"$$diff(f,x) = lim(h->0,(f(x+h)-f(x))/h)$$",
"differentiate polynomial",
"$d/dx (u \\pm v) = du/dx \\pm dv/dx$",
"d/dx (-u) = -du/dx",
"d/dx (cu) = c du/dx (c indep of x)",
"d/dx (u/c)=(1/c)du/dx (c ind of x)",
"d/dx x^n = n x^(n-1)",
"d/dx (uv) = u (dv/dx) + v (du/dx)",
"d/dx (1/v) = -(dv/dx)/v^2",
"d/dx (u/v)=[v(du/dx)-u(dv/dx)]/v^2",
"$d/dx \\sqrt x = 1/(2\\sqrt x)$",
"$d/dx ^n\\sqrt x = d/dx x^(1/n)$",
"$d/dx (c/x^n) = -nc/x^(n+1)$",
"d/dx |x| = x/|x|",
"f'(x) = d/dx f(x)"
},
{ /* dif_trig */
"d/dx sin x = cos x",
"d/dx cos x = - sin x",
"d/dx tan x = sec^2 x",
"d/dx sec x = sec x tan x",
"d/dx cot x = - csc^2 x",
"d/dx csc x = - csc x cot x"
},
{ /* dif_explog */
"d/dx e^x = e^x",
"d/dx c^x = (ln c) c^x, c constant",
"d/dx u^v= (d/dx) e^(v ln u)",
"d/dx ln x = 1/x",
"d/dx ln |x| = 1/x",
"dy/dx = y (d/dx) ln y",
"d/dx e^u = e^u du/dx",
"d/dx c^u=(ln c)c^u du/dx, c const",
"d/dx ln u = (1/u)(du/dx)",
"d/dx ln |u| = (1/u) du/dx",
"d/dx ln(cos x) = -tan x",
"d/dx ln(sin x) = cot x"
},
{ /* dif_inverse_trig */
"$d/dx arctan x = 1/(1+x^2)$",
"$d/dx arcsin x = 1/\\sqrt (1-x^2)$",
"$d/dx arccos x = -1/\\sqrt (1-x^2)$",
"$d/dx arccot x = -1/(1+x^2)$",
"$d/dx arcsec x = 1/(|x|\\sqrt (x^2-1))$",
"$d/dx arccsc x = -1/(|x|\\sqrt (x^2-1))$",
"$d/dx arctan u = (du/dx)/(1+u^2)$",
"$d/dx arcsin u = (du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccos u = -(du/dx)/\\sqrt (1-u^2)$",
"$d/dx arccot u = -(du/dx)/(1+u^2)$",
"$d/dx arcsec u=(du/dx)/(|u|\\sqrt (u^2-1))$",
"$d/dx arccsc u=-(du/dx)/(|u|\\sqrt (u^2-1))$"
},
{ /* chain_rule */
"d/dx u^n = nu^(n-1) du/dx",
"$d/dx \\sqrt u = (du/dx)/(2\\sqrt u)$",
"d/dx sin u = (cos u) du/dx",
"d/dx cos u = -(sin u) du/dx",
"$d/dx tan u = (sec^2 u) du/dx$",
"d/dx sec u=(sec u tan u) du/dx",
"$d/dx cot u = -(csc^2 u) du/dx$",
"d/dx csc u=-(csc u cot u) du/dx",
"d/dx |u| = (u du/dx)/|u|",
"d/dx f(u) = f'(u) du/dx",
"make a substitution, $u = ?$",
"eliminate defined variable"
},
{ /* minima_and_maxima */
"experiment numerically",
"consider points where f'(x)=0",
"consider endpoints of interval",
"points where f'(x) undefined",
"consider limits at open ends",
"reject point outside interval",
"make table of decimal y-values",
"make table of exact y-values",
"choose maximum value(s)",
"choose minimum value(s)",
"evaluate derivative in one step",
"solve simple equation",
"evaluate limit in one step",
"eliminate integer parameter",
"function is constant",
"eliminate open endpoints"
},
{ /* implicit_diff */
"evaluate derivative",
"simplify",
"solve simple equation"
},
{ /* related_rates */
"differentiate the equation",
"evaluate derivative in one step",
"eliminate derivative by substitution",
"solve simple equation"
},
{ /* simplify */
"simplify sums and products",
"eliminate compound fractions",
"common denominator and simplify",
"factor out common term",
"factor expression (not integer)",
"multiply out and simplify", /* meaning either collect or cancel or both */
"show common factor in u/v",
"solve simple equation",
"write as polynomial (in ?)",
"express as polynomial",
"make the leading coeffient 1",
"$x^(1/2) = \\sqrt x$", /* backtosqrts */
"convert fractional exponents to roots",
"convert roots to fractional exponents"
},
{ /* higher_derivatives */
"u=v => du/dx = dv/dx",
"$d^2u/dx^2 = (d/dx)(du/dx)$",
"$d^nu/dx^n= d/dx d^(n-1)u/dx^(n-1)$",
"$d/dx du/dx = d^2u/dx^2$",
"$d/dx d^nu/dx^n = d^(n+1)/dx^(n+1)$",
"evaluate derivative in one step",
"evaluate numerically at a point"
},
{ /* basic_integration */
"$\\int 1 dt = t$",
"$\\int c dt = ct$ (c constant)",
"$\\int t dt = t^2/2$",
"$\\int cu dt = c\\int u dt$ (c constant)",
"$\\int (-u)dt = -\\int u dt$",
"$\\int u+v dt = \\int u dt + \\int v dt$",
"$\\int u-v dt = \\int u dt - \\int v dt$",
"$\\int au\\pm bv dt = a\\int u dt \\pm b\\int v dt$",
"$\\int t^n dt=t^(n+1)/(n+1) (n # -1)$",
"$\\int 1/t^(n+1) dt= -1/(nt^n) (n # 0)$",
"integrate polynomial",
"$\\int (1/t) dt = ln |t|$",
"$\\int 1/(t\\pm a) dt = ln |t\\pm a|$",
"multiply out integrand",
"expand $(a+b)^n$ in integrand",
"$\\int |t| dt = t|t|/2$"
},
{ /* trig_integration */
"$\\int sin t dt = -cos t$",
"$\\int cos t dt = sin t$",
"$\\int tan t dt = -ln |cos t|$",
"$\\int cot t dt = ln |sin t|$",
"$\\int sec t dt = ln |sec t + tan t|$",
"$\\int csc t dt = ln |csc t - cot t|$",
"$\\int sec^2 t dt = tan t$",
"$\\int csc^2 t dt = -cot t$",
"$\\int tan^2 t dt = tan t - t$",
"$\\int cot^2 t dt = -cot t - t$",
"$\\int sec t tan t dt = sec t$",
"$\\int csc t cot t dt = -csc t$"
},
{ /* trig_integration2 */
"$\\int sin ct dt = -(1/c) cos ct$",
"$\\int cos ct dt = (1/c) sin ct$",
"$\\int tan ct dt = -(1/c) ln |cos ct|$",
"$\\int cot ct dt = (1/c) ln |sin ct|$",
"$\\int sec ct dt = (1/c) ln |sec ct + tan ct|$",
"$\\int csc ct dt = (1/c) ln |csc ct - cot ct|$",
"$\\int sec^2 ct dt = (1/c) tan ct$",
"$\\int csc^2 ct dt = -(1/c) cot ct$",
"$\\int tan^2 ct dt = (1/c) tan ct - t$",
"$\\int cot^2 ct dt = -(1/c) cot ct - t$",
"$\\int sec ct tan ct dt = (1/c) sec ct$",
"$\\int csc ct cot ct dt = -(1/c) csc ct$"
},
{ /* integrate_exp */
"$\\int e^t dt = e^t$",
"$\\int e^ct dt =(1/c) e^(ct)$",
"$\\int e^(-t)dt = -e^(-t)$",
"$\\int e^(-ct)dt = -(1/c) e^(-ct)$",
"$\\int e^(t/c)dt = c e^(t/c)$",
"$\\int c^t dt = (1/ln c) c^t$",
"$\\int u^v dt = \\int (e^(v ln u) dt$",
"$\\int ln t = t ln t - t$",
"$$integral(e^(-t^2),t) = sqrt(pi)/2 Erf(t)$$"
},
{ /* integrate_by_substitution */
"select substitution u = ?",
"computer selects substitution u",
"differentiate the equation",
"evaluate derivative in one step",
"show integral again",
"integrand = $f(u) \\times du/dx$",
"$\\int f(u) (du/dx) dx = \\int f(u) du$",
"eliminate defined variable",
"integrate by subst (u = ?)",
"integrate by substitution"
},
{ /* integrate_by_parts */
"$\\int u dv = uv - \\int v du (u = ?)$",
"$\\int u dv = uv - \\int v du$",
"set current line = original",
"original integral to left side",
"evaluate derivative in one step",
"integrate by subst (u = ?)",
"integrate by substitution",
"evaluate simple integral"
},
{ /* fundamental_theorem */
"$$integral(f'(x),x,a,b)=f(b)-f(a)$$",
"$$diff(integral(f(t),t,a,x),x) = f(x)$$"
},
{ /* definite_integration */
"$$eval(f(t),t,a,b) = f(b) - f(a)$$",
"$$eval(ln f(t),t,a,b) = ln(f(b)/f(a))$$",
"$$integral(u,t,a,b) = - integral(u,t,b,a)$$",
"$$integral(u,t,a,b) + integral(u,t,b,c) = integral(u,t,a,c)$$",
"$$integral(u,t,a,c) = integral(u,t,a,?) + integral(u,t,?,c)$$",
"break $\\int |f(t)| dt$ at zeroes of f",
"calculate integral with parameter numerically",
"calculate integral numerically",
"$$integral(u,t,a,a) = 0$$"
},
{ /* improper_integrals */
"$$integral(u,x,a,infinity) = lim(t->infinity,integral(u,x,a,t))$$",
"$$integral(u,x,-infinity,b) = lim(t->-infinity,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->a+,integral(u,x,t,b))$$",
"$$integral(u,x,a,b) = lim(t->b-,integral(u,x,a,t))$$",
"limit of integrand is not zero at $\\infty $",
"limit of integrand is not zero at $-\\infty $"
},
{ /* oddandeven */
"$$integral(u,t,-a,a) = 0$$ (u odd)",
"$$integral(u,t,-a,a) = 2 integral(u,t,0,a)$$ (u even)"
},
{ /* trig_substitutions */
"$x = a sin \\theta {for \\sqrt (a^2-x^2)}$",
"$x = a tan \\theta {for \\sqrt (a^2+x^2)}$",
"$x = a sec \\theta {for \\sqrt (x^2-a^2)}$",
"$x = a sinh \\theta {for \\sqrt (a^2+x^2)}$",
"$x = a cosh \\theta {for \\sqrt (x^2-a^2)}$",
"$x = a tanh \\theta {for \\sqrt (a^2-x^2)}$",
"define inverse substitution x = ?",
"evaluate derivative",
"simple integral in one step"
},
{ /* trigonometric_integrals */
"$sin^2 t = (1-cos 2t)/2$ in integral",
"$cos^2 t = (1+cos 2t)/2$ in integral",
"u=cos x after using $sin^2=1-cos^2$",
"u=sin x after using $cos^2=1-sin^2$",
"u=tan x after using $sec^2=1+tan^2$",
"u=cot x after using $csc^2=1+cot^2$",
"u=sec x after using $tan^2=sec^2-1$",
"u=csc x after using $cot^2=csc^2-1$",
"$tan^2 x = sec^2 x - 1$ in integrand",
"$2cot^2 x = csc^2 x - 1$ in integrand",
"reduce $\\int sec^n x dx$",
"reduce $\\int csc^n x dx$",
"u = tan(x/2) (Weierstrass subst.)"
},
{ /* trigrationalize */
"multiply num and denom by 1+cos x",
"multiply num and denom by 1-cos x",
"multiply num and denom by 1+sin x",
"multiply num and denom by 1-sin x",
"mult num and denom by sin x+cos x",
"mult num and denom by cos x-sin x"
},
{ /* integrate_rational*/
"polynomial division",
"factor denominator (if easy)",
"show common factor in u/v",
"square-free factorization",
"factor polynomial numerically",
"expand in partial fractions",
"complete the square",
"$\\int 1/(ct\\pm b) dt = (1/c) ln |ct\\pm b|$",
"$\\int 1/(ct\\pm b)^(n+1) dt = -1/nc(ct\\pm b)^n$",
"$\\int 1/(t^2+a^2)dt=(1/a)arctan(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/a)arccoth(t/a)$",
"$\\int 1/(t^2-a^2)dt=(1/2a)ln|(t-a)/(t+a)|$",
"$\\int 1/(a^2-t^2)dt=(1/a)arctanh(t/a)$",
"$\\int 1/(a^2-t^2)dt=(1/2a)ln|(t+a)/(a-t)|$"
},
{ /* integrate_sqrtdenom */
"complete the square",
"$\\int 1/\\sqrt (a^2-t^2)dt = arcsin(t/a)$",
"$\\int 1/\\sqrt (t^2\\pm a^2)dt)=ln|t+\\sqrt (t^2\\pm a^2)|$",
"$\\int 1/(t\\sqrt (t^2-a^2))dt=(1/a)arccos(t/a)$",
"make a rationalizing substitution"
},
{ /* integrate_arctrig */
"$\\int arcsin z dz = z arcsin z + \\sqrt (1-z^2)$",
"$\\int arccos z dz = z arccos z - \\sqrt (1-z^2)$",
"$\\int arctan z dz = z arctan z - (1/2)ln(1+z^2)$",
"$\\int arccot z dz = z arccot z + (1/2)ln(1+z^2)$",
"$\\int arccsc z dz = z arccsc z+ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arccsc z dz = z arccsc z-ln(z + \\sqrt (z^2-1)) (z<0)$",
"$\\int arcsec z dz = z arcsec z-ln(z + \\sqrt (z^2-1)) (z>0)$",
"$\\int arcsec z dz = z arcsec z+ln(z + \\sqrt (z^2-1)) (z<0)$"
},
{ /* simplify_calculus */
"simplify",
"eliminate compound fractions",
"common denominator and simplify",
"factor out common term",
"factor expression (not integer)",
"multiply out and simplify", /* meaning either collect or cancel or both */
"show common factor in u/v",
"solve simple equation",
"evaluate derivative in one step",
"evaluate limit in one step",
"change integral by substitution",
"simple integral in one step",
"absorb number in const of int"
},
{ /* integrate_hyperbolic */
"$\\int sinh u du = cosh u$",
"$\\int cosh u du = sinh u$",
"$\\int tanh u du = ln cosh u$",
"$\\int coth u du = ln sinh u$",
"$\\int csch u du = ln tanh(u/2)$",
"$\\int sech u du = arctan (sinh u)$"
},
{ /* series_geom1 */
"$$1/(1-x) = sum(x^n,n,0,infinity)$$",
"$1/(1-x) = 1+x+x^2+...$",
"$1/(1-x) = 1+x+x^2+...x^n...$",
"$$1/(1+x) = sum((-1)^n x^n,n,0,infinity)$$",
"$1/(1+x) = 1-x+x^2+...$",
"$1/(1+x) = 1-x+x^2+...(-1)^nx^n...$",
"$$sum(x^n,n,0,infinity)=1/(1-x)$$",
"$1+x+x^2+... = 1/(1-x)$",
"$1+x+x^2+...x^n...= 1/(1-x)$",
"$$sum((-1)^n x^n,n,0,infinity) = 1/(1+x)$$",
"$1-x+x^2+... = 1/(1+x)$",
"$1-x+x^2+...(-1)^nx^n... = 1/(1+x)$"
},
{ /* series_geom2 */
"$$x/(1-x) = sum(x^n,n,1,infinity)$$",
"$x/(1-x) = x+x^2+x^3+...$",
"$x/(1-x) = x+x^2+...x^n...$",
"$$x/(1+x) = sum((-1)^(n+1) x^n,n,1,infinity)$$",
"$x/(1+x) = x-x^2+x^3+...$",
"$x/(1+x) = x-x^2+...(-1)^(n+1)x^n...$",
"$$sum(x^n,n,1,infinity)=x/(1-x)$$",
"$x+x^2+x^3+...=x/(1-x)$",
"$x+x^2+...x^n...=x/(1-x)$",
"$$sum((-1)^(n+1) x^n,n,1,infinity)=x/(1+x) $$",
"$x-x^2+x^3+...=x/(1+x) $",
"$x-x^2+...(-1)^(n+1)x^n...=x/(1+x) $"
},
{ /* series_geom3 */
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,-3)$$",
"$$1/(1-x^k) = sum(x^(kn),n,0,infinity,2)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,2)$$",
"$$sum(x^(kn),n,0,infinity)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,-3)=1/(1-x^k)$$",
"$$sum(x^(kn),n,0,infinity,2)=1/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,-3)=x^m/(1-x^k)$$",
"$$sum(x^(m+kn),n,0,infinity,2)=x^m/(1-x^k)$$"
},
{ /* series_geom4 */
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,-3)$$",
"$$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,2)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,-3)$$",
"$$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,2)$$",
"$$sum((-1)^nx^(kn),n,0,infinity)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,-3)=1/(1+x^k)$$",
"$$sum((-1)^nx^(kn),n,0,infinity,2)=1/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,-3)=x^m/(1+x^k)$$",
"$$sum((-1)^nx^(m+kn),n,0,infinity,2)=x^m/(1+x^k)$$"
},
{ /* series_geom5 */
"$$x^k/(1-x) = sum(x^n,n,k,infinity)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,-3)$$",
"$$x^k/(1-x) = sum(x^n,n,k,infinity,2)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,-3)$$",
"$$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,2)$$",
"$$sum(x^n,n,k,infinity) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,-3) = x^k/(1-x)$$",
"$$sum(x^n,n,k,infinity,2) = x^k/(1-x)$$",
"$$sum((-1)^nx^n,n,k,infinity) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,-3) = x^k/(1+x)$$",
"$$sum((-1)^nx^n,n,k,infinity,2) = x^k/(1+x)$$"
},
{ /* series_ln */
"$$ln(1-x) = sum(x^n/n,n,1,infinity)$$",
"$$ln(1-x) = sum(x^n/n,n,1,infinity,-3)$$",
"$$ln(1-x) = sum(x^n/n,n,1,infinity,2)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,-3)$$",
"$$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,2)$$",
"$$sum(x^n/n,n,1,infinity) = ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,-3)=ln(1-x)$$",
"$$sum(x^n/n,n,1,infinity,2)=ln(1-x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,-3)=ln(1+x)$$",
"$$sum((-1)^(n+1) x^n/n,n,1,infinity,2)=ln(1+x)$$"
},
{ /* series_trig */
"$$ sin x = sum( (-1)^n x^(2n+1)/(2n+1)!,n,0,infinity)$$",
"$sin x = x-x^3/3!+x^5/5!+...$",
"$sin x = x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+...$",
"$$cos x = sum( (-1)^n x^(2n)/(2n)!,n,0,infinity)$$",
"$cos x = 1-\\onehalf x^2+x^4/4! + ...$",
"$cos x = 1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+...$",
"$$sum((-1)^n x^(2n+1)/(2n+1)!,n,0,infinity) = sin x$$",
"$x-x^3/3!+x^5/5!+... = sin x$",
"$x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+... = sin x$",
"$$sum( (-1)^n x^(2n)/(2n)!,n,0,infinity) = cos x$$",
"$1-\\onehalf x^2+x^4/4! + ... = cos x$",
"$1-\\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+... = cos x$"
},
{ /* series_exp */
"$$e^x = sum(x^n/n!,n,0,infinity)$$",
"$e^x = 1+x+x^2/2!+...$",
"$e^x = 1+x+...+x^n/n!...$",
"$$sum(x^n/n!,n,0,infinity)= e^x$$",
"$1+x+x^2/2!+ x^3/3!+... = e^x$",
"$1+x+...+x^n/n!... = e^x$",
"$$e^(-x) = sum((-x)^n x^n/n!,n,0,infinity)$$",
"$e^(-x) = 1-x+x^2/2!+...$",
"$e^(-x) = 1-x+...(-1)^nx^n/n!...$",
"$$sum((-1)^nx^n/n!,n,0,infinity)= e^(-x)$$",
"$1-x+x^2/2!+ x^3/3!+... = e^(-x)$",
"$1-x+...+(-1)^nx^n/n!... = e^(-x)$"
},
{ /* series_atan */
"$$arctan x = sum(x^(2n+1)/(2n+1),n,0,infinity)$$",
"$arctan x = x -x^3/3 + x^5/5 ...$",
"$arctan x = x -x^3/3 +...+ x^(2n+1)/(2n+1)+...$",
"$$sum(x^(2n+1)/(2n+1),n,0,infinity) = arctan x$$",
"$x -x^3/3 + x^5/5 ...=arctan x$",
"$x -x^3/3 +...+ x^(2n+1)/(2n+1)+...=arctan x$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,-3)$$",
"$$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,2)$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,-3)= (1+x)^alpha$$",
"$$sum(binomial(alpha,n) x^n,n,0,infinity,2)= (1+x)^alpha$$"
},
{ /* series_appearance */
"express series as $a_0 + a_1 + ...$",
"express series as $a_0 + a_1 + a_2 + ... $",
"express series using ... and general term",
"express series using sigma notation",
"show another term before ...",
"show ? more terms before ...",
"show terms with factorials evaluated",
"do not evaluate factorials in terms",
"show the coefficients in decimal form",
"do not use decimal form for coefficients"
},
{ /* series_algebra */
"telescoping series",
"multiply series",
"multiply power series",
"divide power series by polynomial",
"divide polynomial by power series",
"divide power series",
"square series",
"square power series",
"express $(\\sum a_k x^k)^n$ as a series",
"add series",
"subtract series",
},
{ /* series_manipulations */
"split off first few terms",
"decrease lower limit by subtracting terms",
"add ? to index variable",
"subtract ? from index variable",
"rename index variable",
"$\\sum (u\\pm v) = \\sum u \\pm \\sum v$",
"differentiate power series term by term",
"$\\sum du/dx = d/dx \\sum u$",
"integrate power series term by term",
"$\\sum \\int u dx = \\int \\sum u dx$",
"calculate sum of first few terms",
"$$u = integral(diff(u,x),x)$$",
"$$u = integral(diff(u,t),t,0,x) + u0$$",
"$$u = diff(integral(u,x),x)$$",
"solve for constant of integration",
"$\\sum a_k = \\sum a_(2k) + \\sum a_(2k+1)$"
},
{ /* series_convergence_tests */
"$\\sum u$ diverges if $lim u$ is not zero",
"integral test",
"ratio test",
"root test",
"comparison test for convergence",
"comparison test for divergence",
"limit comparison test",
"condensation test",
"finish divergence test",
"finish integral test",
"finish ratio test",
"finish root test",
"finish comparison test",
"finish comparison test",
"finish limit comparison test",
"finish condensation test"
},
{ /* series_convergence2 */
"positive result of comparison test",
"negative result of comparison test",
"$$sum(1/k,k,1,infinity) = infinity$$",
"$$sum(1/k^2,k,1,infinity) = pi^2/6$$"
},
{ /* complex_functions */
"$ln(u+iv) = ln(re^(i\\theta ))$",
"$ln(re^(i\\theta ))=ln r + i\\theta (-\\pi <\\theta \\le \\pi )$",
"$ln i = i\\pi /2$",
"$ln(-1) = i\\pi $",
"$ln(-a) = ln a + i\\pi (a > 0)$",
"$cos \\theta = [e^(i\\theta ) + e^(-i\\theta )]/2$",
"$sin \\theta = [e^(i\\theta ) - e^(-i\\theta )]/2i$",
"$$sqrt(re^(i theta))=sqrt(r) e^(i theta/2)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$$root(n,re^(i theta))=root(n,r) e^(i theta/n)$$ $ (-\\pi < \\theta \\le \\pi )$",
"$e^(i\\theta ) = cos \\theta + i sin \\theta $",
"$e^(x+iy) = e^x cos y + i e^x sin y$",
"$e^(i\\pi ) = -1$",
"$e^(-i\\pi ) = -1$",
"$e^(2n\\pi i) = 1$",
"$e^((2n\\pi + \\theta )i) = e^(i\\theta )$",
"$u^v = e^(v ln u)$"
},
{ /* complex_hyperbolic */
"sin(it) = i sinh t",
"cos(it) = cosh t",
"cosh(it) = cos t",
"sinh(it) = i sin t",
"tan(it) = i tanh t",
"cot(it) = -i coth t",
"tanh(it) = i tan t",
"coth(it) = -i cot t",
"cos t + i sin t = e^(it)",
"cos t - i sin t = e^(-it)",
"$[e^(i\\theta ) + e^(-i\\theta )]/2 = cos \\theta $",
"$[e^(i\\theta ) - e^(-i\\theta )]/2i = sin \\theta $",
"$e^(i\\theta ) + e^(-i\\theta ) = 2 cos \\theta $",
"$e^(i\\theta ) - e^(-i\\theta ) = 2i sin \\theta $"
},
{ /* hyperbolic_functions */
"cosh u = (e^u+e^(-u))/2",
"e^u + e^-u = 2 cosh u",
"sinh u = (e^u-e^(-u))/2",
"e^u-e^(-u) = 2 sinh u",
"[e^u + e^-u]/2 = cosh u",
"[e^u-e^(-u)]/2 = sinh u",
"cosh(-u) = cosh u",
"sinh(-u) = -sinh u",
"cosh u + sinh u = e^u",
"cosh u - sinh u = e^(-u)",
"cosh 0 = 1",
"sinh 0 = 0",
"e^x = cosh x + sinh x",
"e^(-x) = cosh x - sinh x"
},
{ /* hyperbolic2 */
"$sinh^2u + 1 = cosh^2 u$",
"$cosh^2 u - 1 = sinh^2u $",
"$cosh^2 u - sinh^2u = 1$",
"$cosh^2 u = sinh^2u + 1$",
"$sinh^2u = cosh^2 u - 1$",
"$1 - tan^2u = sech^2u$",
"$1 - sech^2u = tan^2u$"
},
{ /* more_hyperbolic */
"tanh u = sinh u / cosh u",
"sinh u / cosh u = tanh u",
"coth u = cosh u / sinh u",
"cosh u / sinh u = coth u",
"sech u = 1 / cosh u",
"1 / cosh u = sech u",
"csch u = 1 / sinh u",
"1 / sinh u = csch u",
"$tanh^2 u + sech^2 u = 1$",
"$tanh^2 u = 1 - sech^2 u$",
"$sech^2 u = 1 - tanh^2 u $",
"$sinh(u\\pm v)=sinh u cosh v \\pm cosh u sinh v$",
"$cosh(u\\pm v)=cosh u cosh v \\pm sinh u sinh v$",
"sinh 2u = 2 sinh u cosh u",
"$cosh 2u = cosh^2 u + sinh^2 u$",
"$tanh(ln u) = (1-u^2)/(1+u^2)$"
},
{ /* inverse_hyperbolic */
"$arcsinh x = ln(x + \\sqrt (x^2+1))$",
"$arccosh x = ln(x + \\sqrt (x^2-1))$",
"$arctanh x = (1/2) ln((1+x)/(1-x))$"
},
{ /* dif_hyperbolic */
"d/du sinh u = cosh u",
"d/du cosh u = sinh u",
"$d/du tanh u = sech^2 u$",
"$d/du coth u = -csch^2 u$",
"d/du sech u = -sech u tanh u",
"d/du csch u = -csch u coth u",
"d/du ln sinh u = coth u",
"d/du ln cosh u = tanh u"
},
{ /* dif_inversehyperbolic */
"$d/du arcsinh u = 1/\\sqrt (u^2+1)$",
"$d/du arccosh u = 1/\\sqrt (u^2-1)$",
"$d/du arctanh u = 1/(1-u^2)$",
"$d/du arccoth u = 1/(1-u^2)$",
"$d/du arcsech u= -1/(u\\sqrt (1-u^2))$",
"$d/du arccsch u= -1/(|u|\\sqrt (u^2+1))$"
},
{ /* sg_function1 */
"sg(x) = 1 if x > 0", /* sgpos */
"sg(x) = -1 if x < 0", /* sgneg */
"sg(0) = 0", /* sgzero */
"sg(-x) = -sg(x)", /* sgodd */
"-sg(x) = sg(-x)", /* sgodd2 */
"sg(x) = |x|/x (x nonzero)", /* sgabs1 */
"sg(x) = x/|x| (x nonzero)", /* sgabs2 */
"abs(x) = x sg(x)", /* abssg */
"$sg(x)^(2n) = 1$", /* also sg(x)^(even/odd) sgevenpower */
"sg(x)^(2n+1) = sg(x)", /* also sg(x)^odd/odd sgoddpower */
"1/sg(x) = sg(x)", /* sgrecip */
"d/dx sg(u) = 0 (u nonzero)", /* difsg */
"$\\int sg(x) = x sg(x)$", /* intsg */
"$\\int sg(u)v dx = sg(u)\\int v dx$ (u nonzero)", /* sgint */
"sg(x) = 1 assuming x > 0", /* sgassumepos */
"sg(x) = -1 assuming x < 0" /* sgassumeneg */
},
{ /* sg_function2 */
"$sg(au) = sg(u)$ if $a > 0$",
"$sg(au) = -sg(u)$ if a < 0",
"sg(au/b) = sg(u) if a/b > 0",
"sg(au/b) = - sg(u) if a/b < 0",
"sg(x^(2n+1)) = sg(x)",
"sg(1/u) = sg(u)",
"sg(c/u) = sg(u) if c > 0",
"u sg(u) = |u|",
"|u| sg(u) = u"
},
{ /* bessel_functions */
"d/dx J0(x) = -J1(x)",
"d/dx J1(x) = J0(x) - J1(x)/x",
"d/dx J(n,x)=J(n-1,x)-(n/x)J(n,x)",
"d/dx Y0(x) = -Y1(x)",
"d/dx Y1(x) = Y0(x) - Y1(x)/x",
"d/dx Y(n,x)=Y(n-1,x)-(n/x)Y(n,x)"
},
{ /* modified_bessel_functions */
"d/dx I0(x) = -I1(x)",
"d/dx I1(x) = I0(x) - I1(x)/x",
"d/dx I(n,x)=I(n-1,x)-(n/x)I(n,x)",
"d/dx K0(x) = -K1(x)",
"d/dx K1(x) = -K0(x) - K1(x)/x",
"d/dx K(n,x)= -K(n-1,x)-(n/x)K(n,x)"
},
{ /* functions_menu -- user-defined functions */
"" /* definitions of user-defined functions appear here. */
},
{"expand", /* automode_only, this menu never appears! */
"multiply if cancels" /* but model.c uses corresponding entries in optable */
},
{"cancel square roots" /* automode_only2, also never appears */
},
{"" /* automode_only3, also never appears */
}
};
/*_______________________________________________________________*/
/* array of titles of the command menus */
const char *const menutitles[MAXMENUS] =
{
/* first the algebra menus */
"Numerical Calculation",
"Express Number in Different Form",
"Complex Arithmetic",
"Simplify Sums",
"Simplify Products",
"Expand",
"Fractions",
"Signed Fractions",
"Compound Fractions",
"Common Denominators",
"Exponents",
"Expand Powers",
"Negative Exponents",
"Square Roots",
"Advanced Square Roots",
"Fractional Exponents",
"N-th Roots",
"Roots of Roots",
"Roots and Fractions",
"Complex Numbers",
"Factoring",
"Advanced Factoring",
"Solve Equations",
"Quadratic Equations",
"Study Equations Numerically",
"Advanced Equations",
"Cubic Equations",
"Log Or Exponential Equations",
"Cramer's Rule",
"Several Linear Equations",
"Selection Mode Only", /* This title is never shown */
"Linear Equations by Term Selection", /* This title is never shown */
"Equations by Substitution",
"Matrix Methods",
"Advanced Matrix Methods",
"Absolute Value",
"Absolute Value Inequalities", /* absolute_value_ineq1 */
"Absolute Value Inequalities", /* absolute_value_ineq2 */
"Strict Inequalities", /* less_than */
"Strict Inequalities", /* greater_than */
"Inequalities", /* less_than_or_equal */
"Inequalities", /* greater_than_or_equal */
"Inequalities involving Squares",
"Inequalities involving Squares",
"Inequalities involving Reciprocals",
"Inequalities involving Reciprocals",
"Root and Power Inequalities",
"Root and Power Inequalities",
"Inequalities--One Side Zero",
"Inequalities--One Side Zero",
"Inequalities involving Squares", /* Now repeat the last six for > and GE */
"Inequalities involving Squares",
"Inequalities involving Reciprocals",
"Inequalities involving Reciprocals",
"Root and Power Inequalities",
"Root and Power Inequalities",
"Inequalities--One Side Zero",
"Inequalities--One Side Zero",
"Binomial Theorem",
"Factoring Binomial Expansions",
"Sigma Notation",
"Advanced Sigma Notation",
"Prove by Induction",
"Trig Inequalities",
"Log and Power Inequalities",
"Log and Power Inequalities",
"Log and Power Inequalities",
"Log and Power Inequalities",
"Logarithms Base 10",
"Logarithms",
"Natural Logarithms and e",
"Natural Logarithms",
"Reverse Trig Sum Formulas",
"Complex Polar Form",
"Logarithms to any Base",
"Change Base of Logarithms",
"Evaluate Trig Functions",
"Basic Trig",
"Trig Reciprocals",
"Trig Square Identities",
"Csc and Cot Identities",
"Trig Sum Formulas",
"Double Angle Formulas",
"Expand sin nx or cos nx",
"Verify Identities",
"Solve by 30-60-90",
"Solve by 45-45-90",
"Zeroes of Trig Functions",
"Inverse Trig Functions",
"Simplify Inverse Trig",
"Adding Inverse Trig Functions",
"Complementary Trig Functions",
"Complementary Angles in Degrees",
"Odd and Even Trig Functions",
"Periodicity of Trig Functions",
"Half-Angle Identities",
"Product and Factor Identities",
"Limits",
"Limits of Quotients",
"Limits of Quotients of Roots",
"L'Hospital's Rule",
"Special Limits",
"Limits of Hyperbolic Functions",
"Advanced Limits",
"Logarithmic Limits",
"Limits at Infinity",
"Infinite Limits",
"Infinity",
"Zero Denominator",
"Functions at Infinity",
"Differentiate Polynomials",
"Derivatives",
"Differentiate Trig Functions",
"Differentiate Exp and Log",
"Diff Inverse Trig Functions",
"Chain Rule",
"Minima and Maxima",
"Implicit Differentiation",
"Related Rates",
"Simplify",
"Higher Derivatives",
"Basic Integration",
"Integrate Trig Functions",
"Integrate Trig Functions of ct",
"Integrate Exponentials and Ln",
"Integrate by Substitution",
"Integrate by Parts",
"Fundamental Theorem",
"Definite Integration",
"Improper Integrals",
"Odd and Even Integrands",
"Inverse Substitutions",
"Trigonometric Integrals",
"Simplify Trig Integrand",
"Integrate Rational Functions",
"Integrate Square Root In Denom",
"Integrate Inverse Trig Functions",
"Simplify",
"Integrate Hyperbolic Functions",
"Geometric Series",
"Geometric Series 2",
"Geometric Series 3",
"Geometric Series 4",
"Geometric Series 5",
"Infinite Series for the Logarithm",
"Infinite Series for sin and cos",
"Infinite Series for the Exponential Function",
"Infinite series for arctan",
"Appearance of Series",
"Algebraic Operations on Series",
"Manipulating Infinite Series",
"Convergence Tests",
"Finish Convergence Tests",
"Complex Functions",
"Complex Function Identities",
"Hyperbolic Sine and Cosine",
"Hyperbolic Trig Identities",
"Hyperbolic Functions",
"Inverse Hyperbolic Functions",
"Differentiate Hyperbolics",
"Differentiate Inverse Hyperbolics",
"Sg Function",
"Simplify Sg Function",
"Bessel Functions",
"Modified Bessel Functions",
"User-Defined Functions",
"Invisible", /* automode_only operators */
"Invisible Too", /* Automode_only2 */
"and This Too" /* Automode_only3 */
};
/*_____________________________________________________________*/
MEXPORT_ENGLISH const char **cmdmenu(int i)
/* returns an array of strings for the i-th menu */
{ return (const char **) menutext[i];
}
MEXPORT_ENGLISH const char *menutitle(int i)
{ return (const char *)menutitles[i];
}
Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists