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�� J P� S P H H J �C �� ���� � � ��C ��_�C �� ���� � � yi��C ��_� arithmetic 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 evaluate Bernoulli number exactly evaluate Euler number exactly decimal to fraction express as square express as cube express as ?-th power express as power of ? write integer as a^n x = ? + (x-?) $i^2 = -1$ 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) cancel double minus $-(-a)=a$ push minus in -(a+b) = -a-b -a-b = -(a+b) 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) $x\times 0 = 0\times x = 0$ $x\times 1 = 1\times x = x$ 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 multiply out product of sums multiply out numerator multiply out denominator $na = a +...+ a$ $0/a = 0$ $a/1 = a$ $a(1/a) = 1$ 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 $a,b$) $ab/cd = (a/c)(b/d)$ $ab/c = (a/c) b$ cancel minus $(-a)/(-b) = a/b$ $-(a/b) = (-a)/b$ $-(a/b) = a/(-b)$ $(-a)/b = -(a/b)$ $a/(-b)= -a/b$ $(-a-b)/c = -(a+b)/c$ $a/(-b-c) = -a/(b+c)$ $a/(b-c) = -a/(c-b)$ $-a/(-b-c) = a/(b+c)$ $-a/(b-c) = a/(c-b)$ $-(-a-b)/c = (a+b)/c$ $$(a-b)/(c-d) = (b-a)/(d-c)$$ $ab/c = a (b/c)$ $a/bc = (1/b) (a/c)$ $(a/c)/(b/c) = a/b$ $a/(b/c)=ac/b$ (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 find common denominator find common denom (fracts only) multiply fractions (a/b)(c/d)=ac/bd multiply fractions a(c/d)= ac/d 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 ? 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 a^(b+c) = a^b a^c $a^n/b^n = (a/b)^n$ b^n/b^m = b^(n-m) ab^n/b^m = a/b^(m-n) 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 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) $(a/b)^(-n) = (b/a)^n$ a^(b-c) = a^b/a^c $\sqrt x\sqrt y = \sqrt (xy)$ $\sqrt (xy) = \sqrt x\sqrt y$ $\sqrt (x^2y) = x\sqrt y$ 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 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 $a ^ \onehalf = \sqrt a$ $a^(n/2) = \sqrt (a^n)$ $a^(b/n) = ^n\sqrt (a^b)$ $\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) $$a^(p/q)/b = (a^p/b^q)^(1/q)$$ $a^(n/2) = (\sqrt a)^n$ $a^(m/n) = (^n\root a)^m$ $^n\sqrt x^n\sqrt y = ^n\sqrt (xy)$ $^n\sqrt (xy) = ^n\sqrt x ^n\sqrt y$ $^n\sqrt x^m = (^n\sqrt x)^m$ if $x\ge 0$ or n odd $^n\sqrt (x^ny) = x ^n\sqrt y$ or $|x|^n\sqrt y$ $^n\sqrt (x^n) = x$ if $x\ge 0$ or n odd $^n\sqrt (x^(nm))=x^m$ if $x\ge 0$ or n odd $^2^n\sqrt (x^n) = \sqrt x$ $^m^n\sqrt x^m) = ^n\sqrt x$ $(^n\sqrt x)^n = x$ $(^n\sqrt a)^m = ^n\sqrt (a^m)$ $(^n\sqrt a)^(qn+r) = a^q ^n\sqrt (a^r)$ factor integer x in $^n\sqrt x$ $^n\sqrt (-a) = -^n\sqrt a$, n odd evaluate to rational factor polynomial under $^n\sqrt $ multiply out under $^n\sqrt $ $\sqrt (\sqrt x) = ^4\sqrt x$ $\sqrt (^n\sqrt x) = ^2^n\sqrt x$ $^n\sqrt (\sqrt x) = ^2^n\sqrt x$ $^n\sqrt (^m\sqrt x) = ^n^m\sqrt x$ $^n\sqrt (x/y) = ^n\sqrt x/^n\sqrt y$ $^n\sqrt x/^n\sqrt y = ^n\sqrt (x/y)$ $x/^n\sqrt x = (^n\sqrt x)^(n-1)$ $^n\sqrt x/x = 1/(^n\sqrt x)^(n-1)$ 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 $a\ge 0$ $(^m^n\sqrt a)^n = ^m\sqrt a$ $(^2^n\sqrt a)^n = \sqrt a$ 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 $c^2=a^2+b^2$ 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 make a substitution, u = ? eliminate defined variable regard a variable as constant 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 ? 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 $a,b\ge 0$ 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 $p\ge 0$) solve numerically 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 if ab=ac then a=0 or b=c display only the selected equation show all equations again collect multiple solutions reject unsolvable equation check root(s) in original eqn solve linear equation at once u=x+b/3 in ax^3+bx^2+cx+d=0 compute discriminant 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) substitute n = ?-k evaluate roots exactly simplify 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 Cramer's rule evaluate determinant variables left, constants right collect like terms 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 contradiction at hand: no soln 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 ? 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 solve equation ? for ? simplify equations cancel term from both sides add ? to both sides of equation ? subtract ? from both sides of equation ? substitute for variable write in matrix form 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 convert to system of equations AX = B becomes X = A^(-1)B use formula for 2 by 2 inverse compute exact matrix inverse compute decimal matrix inverse |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| |u|=c iff u=c or u = -c ($c\ge 0$) |u|/u = c iff c = $\pm $1 |u| < v iff -v < u < v $|u| \le v$ iff $-v \le u \le v$ u < |v| iff v < -u or u < v $u \le |v|$ iff $v \le -u$ or $u \le v$ |u| = u iff $0 \le u$ |u| = -u iff $u \le 0$ $0 \le |u|$ is true |u| < 0 is false $-c \le |u|$ is true ($c\ge 0$) -c < |u| is true (c>0) |u| < -c is false ($c\ge 0$) $|u| \le -c$ is false (c>0) $|u| \le -c$ iff u=0 assuming $c\ge 0$ |u| = -c iff u=0 assuming $c\ge 0$ v > |u| iff -v < u < v $v \ge |u|$ iff $-v \le u \le v$ |v| > u iff v < -u or v > u $|v| \ge u$ iff $v \le -u$ or $v \ge u$ $|u| \ge 0$ is true 0 > |u| is false -c > |u| is false ($c\ge 0$) $-c \ge |u|$ is false (c>0) $-c \ge |u|$ iff u=0 assuming c=0 |u| > -c is true (c>0) $|u| \ge -c$ is true ($c\ge 0$) $-v \le u \le v$ iff $|u| \le v$ v < -u or u < v iff u < |v| $u^(2n) = |u|^(2n)$ if u is real $|u|^n = |u^n|$ if n is real change u < v to v > u change -u < -v to v < u change -u < -v to u > v multiply both sides by ?^2 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 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 $a \le 0$ u > v or u = v iff $u \ge v$ change $x \le y$ to $y \ge x$ change $-u \le -v$ to $v \le u$ change $-u \le -v$ to $u \ge v$ $a \le x^2^n$ is true if $a \le 0$ $x^2^n \le a$ is false if $a < 0$ $u \le v$ iff $u^2 \le v^2$ or $u \le 0$ provided $0 \le v$ 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 $0 \le v$ $u^2 < a$ 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 $a < 0$ $u^2 \le a$ 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 $a \le 0$ 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 $a \le 1/x < b$ 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 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 $a < 0$ $u \le v$ 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 $a \le 0$ 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 $a(x\pm b/a) < 0$ (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) $0 \le u/v$ 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 $a\le b$) $a > u^2$ 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 $u\ge 0$ $a > x^2$ is false if $a <= 0$ $\sqrt u > a$ iff $u \ge 0$ provided $a < 0$ $a \ge u^2$ 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 $a \le 0$ $u > v$ 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 $a < 0$ $u \ge v$ 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 $a \le 0$ 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) $u/v \ge 0$ 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 with (n k) $$binomial(n,k) = factorial(n)/ factorial(k) * factorial(n-k)$$ n! = n(n-1)(n-2)...1 compute factorial 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)) 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 $\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 shift sum limits rename index variable $(\sum u)(\sum v) = \sum \sum uv$ split off last term $1^3+..+n^3 = n^2(n+1)^2/4$ $1^4+..+n^4=n(n+1)(2n+1)(3n^2+2n-1)/30$ $d/dx \sum u = \sum du/dx$ $\sum du/dx = d/dx \sum u$ $\int \sum u dx = \sum \int u dx$ $\sum \int u dx = \int \sum u dx$ $c\sum u = \sum cu$ $$sum(t,i,a,b)=sum(t,i,0,b)-sum(t,i,0,a-1)$$ $$sum(t,i,a,b)=sum(t,i,c,b)-sum(t,i,c,a-1)$$ select induction variable start basis case start induction step use induction hypothesis therefore as desired $|sin u| \le 1$ $|cos u| \le 1$ $sin u \le u$ 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 $0\le u\le \pi /2$ 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 $u \le ln v$ 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 $?^u \le ?^v$ 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 $ln u \ge v$ 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 $?^u \ge ?^v$ exponentials dominate polynomials algebraic functions dominate logarithms $10^(log a) = a$ $log 10^n = n$ ($n$ 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) $log a^n = n log a$ $log ab = log a + log b$ $log 1/a = -log a$ $log a/b = log a - log b$ $log a + log b = log ab$ $log a - log b = log a/b$ $log a + log b - log c =log ab/c$ $n log a = log a^n (n real)$ $log \sqrt a = \onehalf log a$ $log ^n\sqrt a = (1/n) log a$ factor out powers of base $log u = (1/?) log u^?$ evaluate logs numerically 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 ln a^n = n ln a $ln ab = ln a + ln b$ ln 1/a = -ln a $ln a/b = ln a - ln b$ $ln a + ln b = ln ab$ $ln a - ln b = ln a/b$ $ln a + ln b - ln c = ln (ab/c)$ $n ln a = ln a^n (n real)$ $ln \sqrt a = \onehalf ln a$ $ln ^n\sqrt a = (1/n) ln a$ ln u = (1/?) ln u^? evaluate logarithm numerically ln(a/b) = -ln(b/a) 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) 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 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 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) log(b,x) = ln x / ln b log(b,x) = log x / log b log(b,x) = log(a,x) / log(a,b) 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)) sin 0 = 0 cos 0 = 1 tan 0 = 0 $sin k\pi = 0$ $cos 2k\pi = 1$ $tan k\pi = 0$ 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 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 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 $sin^2 u + cos^2 u = 1$ $1 - sin^2 u = cos^2 u$ $1 - cos^2 u = sin^2 u$ $sin^2 u = 1 - cos^2 u$ $cos^2 u = 1 - sin^2 u$ $sec^2 u - tan^2 u = 1$ $tan^2 u + 1 = sec^2 u$ $sec^2 u - 1 = tan^2 u$ $sec^2 u = tan^2 u + 1$ $tan^2 u = sec^2 u - 1$ $sin^(2n+1) u = sin u (1-cos^2 u)^n$ $cos^(2n+1) u = cos u (1-sin^2 u)^n$ $tan^(2n+1) u = tan u (sec^2 u-1)^n$ $sec^(2n+1) u = sec u (tan^2 u+1)^n$ (1-cos t)^n(1+cos t)^n = sin^(2n) t (1-sin t)^n(1+sin t)^n = cos^(2n) t $csc^2 u - cot^2 u = 1$ $cot^2 u + 1 = csc^2 u$ $csc^2 u - 1 = cot^2 u$ $csc^2 u = cot^2 u + 1$ $cot^2 u = csc^2 u - 1$ $csc(\pi /2-\theta ) = sec \theta $ $cot(\pi /2-\theta ) = tan \theta $ $cot^(2n+1) u = cot u (csc^2 u-1)^n$ $csc^(2n+1) u = csc u (cot^2 u+1)^n$ 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) $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 $ $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 $ raise both sides to power take root of both sides apply function to both sides check numerically $sin(u)=1/2$ 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 $ $sin u = 1/\sqrt 2$ 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 $ 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 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 $ 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 $tan(arcsin x) = x/\sqrt (1-x^2)$ $tan(arccos x) = \sqrt (1-x^2)/x$ 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 arcsin x + arccos x = $\pi /2$ $arctan x + arctan 1/x = \pi x/2|x|$ $sin(\pi /2-\theta ) = cos \theta $ $cos(\pi /2-\theta ) = sin \theta $ $tan(\pi /2-\theta ) = cot \theta $ $sec(\pi /2-\theta ) = csc \theta $ $sin \theta = cos(\pi /2-\theta )$ $cos \theta = sin(\pi /2-\theta )$ $tan \theta = cot(\pi /2-\theta )$ $cot \theta = tan(\pi /2-\theta )$ $sec \theta = csc(\pi /2-\theta )$ $csc \theta = sec(\pi /2-\theta )$ $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 $ sin(-u) = - sin u cos(-u) = cos u tan(-u) = - tan u cot(-u) = - cot u sec(-u) = sec u csc(-u) = - csc u $sin^2(-u) = sin^2 u$ $cos^2(-u) = cos^2 u$ $tan^2(-u) = tan^2 u$ $cot^2(-u) = cot^2 u$ $sec^2(-u) = sec^2 u$ $csc^2(-u) = csc^2 u$ $sin(u+2\pi ) = sin u$ $cos(u+2\pi ) = cos u$ $tan(u+\pi ) = tan u$ $sec(u+2\pi ) = sec u$ $csc(u+2\pi ) = csc u$ $cot(u+\pi ) = cot u$ $sin^2(u+\pi ) = sin^2 u$ $cos^2(u+\pi ) = cos^2 u$ $sec^2(u+\pi ) = sec^2 u$ $csc^2(u+\pi ) = csc^2 u$ $sin u = -sin(u-\pi )$ $sin u = sin(\pi -u)$ $cos u = -cos(u-\pi )$ $cos u = -cos(\pi -u)$ $sin^2(\theta /2) = (1-cos \theta )/2$ $cos^2(\theta /2) = (1+cos \theta )/2$ $sin^2(\theta ) = (1-cos 2\theta )/2$ $cos^2(\theta ) = (1+cos 2\theta )/2$ $tan(\theta /2) = (sin \theta )/(1+cos \theta )$ $tan(\theta /2) = (1-cos \theta )/sin \theta $ $cot(\theta /2) = (1+cos \theta )/(sin \theta )$ $cot(\theta /2) = sin \theta /(1-cos \theta )$ $sin(\theta /2) = \sqrt ((1-cos \theta )/2) if sin(\theta /2)\ge 0$ $sin(\theta /2) = -\sqrt ((1-cos \theta )/2) if sin(\theta /2)\le 0$ $cos(\theta /2) = \sqrt ((1+cos \theta )/2) if cos(\theta /2)\ge 0$ $cos(\theta /2) = -\sqrt ((1+cos \theta )/2) if cos(\theta /2)\le 0$ $\theta = 2(\theta /2)$ $sin x cos x = \onehalf sin 2x$ $sin x cos y = \onehalf [sin(x+y)+sin(x-y)]$ $cos x sin y = \onehalf [sin(x+y)-sin(x-y)]$ $sin x sin y = \onehalf [cos(x-y)-cos(x+y)]$ $cos x cos y = \onehalf [cos(x+y)+cos(x-y)]$ $sin x + sin y = 2 sin \onehalf (x+y) cos \onehalf (x-y)$ $sin x - sin y = 2 sin \onehalf (x-y) cos \onehalf (x+y)$ $cos x + cos y = 2 cos \onehalf (x+y) cos \onehalf (x-y)$ $cos x - cos y = -2 sin \onehalf (x+y) sin \onehalf (x-y)$ substitute u,v for expressions in trig functions experiment numerically $lim u\pm v = lim u \pm lim v$ $lim u-v = lim u - lim v$ lim(t\toa,c) = c (c constant) lim(t\toa,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\toa,f(t))=f(a) (polynomial f) lim |u| = |lim u| 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\toa limit of rational function rationalize fraction pull out nonzero finite limits 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 $\sqrt a/b = \sqrt (a/b^2)$ 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) 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/?) (sin t)/t \to 1 as t\to0 (tan t)/t \to 1 as t\to0 (1-cos t)/t \to 0 as t\to0 $(1-cos t)/t^2\to \onehalf $ as t\to0 lim(t\to0,(1+t)^(1/t)) = e $(ln(1\pm t))/t \to \pm 1$ as t\to0 (e^t-1)/t \to 1 as t\to0 (e^(-t)-1)/t \to -1 as t\to0 $lim(t\to0,t^nln |t|)=0 (n > 0)$ lim(t\to0,cos(1/t))=undefined lim(t\to0,sin(1/t))=undefined lim(t\to0,tan(1/t))=undefined lim(t\to$\pm \infty $,cos t)=undefined lim(t\to$\pm \infty $,sin t)=undefined lim(t\to$\pm \infty $,tan t)=undefined (sinh t)/t \to 1 as t\to0 (tanh t)/t \to 1 as t\to0 (cosh t - 1)/t \to 0 as t\to0 (cosh t - 1)/t^2\to1/2 as t\to0 lim ln u=ln lim u (if lim u > 0) lim f(u)=f(lim u), f continuous change limit variable evaluate limit in one step lim u^v = lim e^(v ln u) limit undefined due to domain lim u = e^(lim ln u) squeeze theorem: uv\to0 if v\to0 & $|u|\le c$ $lim \sqrt u-v=lim (\sqrt u-v)(\sqrt u+v)/(\sqrt u+v)$ lim u/v = limit of leading terms leading term: lim(u+a)=lim(u) if a/u\to0 replace sum by leading term f(undefined) = undefined lim(e^u) = e^(lim u) lim(ln u) = ln(lim u) $lim(t\to0+,t ln t) = 0$ $lim(t\to0+,t^n ln t) = 0 if n\ge 1$ $lim(t\to0+,t (ln t)^n) = 0 if n\ge 1$ $lim(t\to0+,t^k (ln t)^n) = 0 if k,n\ge 1$ $lim(t\to\infty ,ln(t)/t) = 0$ $lim(t\to\infty ,ln(t)^n/t) = 0 if n\ge 1$ $lim(t\to\infty ,ln(t)/t^n) = 0 if n\ge 1$ $lim(t\to\infty ,ln(t)^k/t^n) = 0 if k,n\ge 1$ $lim(t\to\infty ,t/ln(t)) = \infty $ $lim(t\to\infty ,t/ln(t)^n) = \infty if n\ge 1$ $lim(t\to\infty ,t^n/ln(t)) = \infty if n\ge 1$ $lim(t\to\infty ,t^n/ln(t)^k) = \infty if k,n\ge 1$ $lim(t\to\infty ,1/t^n) = 0 if n\ge 1$ $lim(t\to\infty ,t^n) = \infty if n\ge 1$ $lim(t\to\infty ,e^t) = \infty $ $lim(t\to-\infty ,e^t) = 0$ $lim(t\to\infty ,ln t) = \infty $ $lim(t\to\infty ,\sqrt t) = \infty $ $lim(t\to\infty ,^n\sqrt t) = \infty $ $lim(t\to\pm \infty ,arctan t) = \pm \pi /2$ $lim(t\to\infty ,arccot t) = 0$ $lim(t\to-\infty ,arccot t) = \pi $ $lim(t\to\pm \infty ,tanh t) = \pm 1$ $lim \sqrt u-v=lim (\sqrt u-v)(\sqrt u+v)/\sqrt u+v)$ lim sin u = sin(lim u) lim cos u = cos(lim u) change limit at $\infty $ to limit at 0 $lim(1/u^2^n) = \infty $ if u\to0 lim(1/u^n) undef if u\to0, n odd lim(t\toa+,1/u^n) = $\infty $ if u\to0 lim(t\toa-,1/u^n)=-$\infty $, u\to0, n odd lim u/v undef if lim v =0, lim u #0 lim(t\to 0+,ln t) = -$\infty $ $lim(t\to(2n+1)\pi /2\pm ,tan t) = \pm \infty $ $lim(t\to n\pi \pm ,cot t) = \pm \infty $ $lim(t\to(2n+1)\pi /2\pm ,sec t) = \pm \infty $ $lim(t\to n\pi \pm ,csc t) = \pm \infty $ lim(uv) = lim(u/?) lim(?v) lim(uv) = lim(?u) lim(v/?) $\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 $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 $a < 0$ $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 $ 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) $$diff(f,x) = lim(h->0,(f(x+h)-f(x))/h)$$ d/dx (cu) = c du/dx (c indep of x) d/dx (u/c)=(1/c)du/dx (c ind of x) 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| 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 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 $d/dx arctan x = 1/(1+x^2)$ $d/dx arcsin x = 1/\sqrt (1-x^2)$ $d/dx arccos x = -1/\sqrt (1-x^2)$ $d/dx arccot x = -1/(1+x^2)$ $d/dx arcsec x = 1/(|x|\sqrt (x^2-1))$ $d/dx arccsc x = -1/(|x|\sqrt (x^2-1))$ $d/dx arctan u = (du/dx)/(1+u^2)$ $d/dx arcsin u = (du/dx)/\sqrt (1-u^2)$ $d/dx arccos u = -(du/dx)/\sqrt (1-u^2)$ $d/dx arccot u = -(du/dx)/(1+u^2)$ $d/dx arcsec u=(du/dx)/(|u|\sqrt (u^2-1))$ $d/dx arccsc u=-(du/dx)/(|u|\sqrt (u^2-1))$ 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, $u = ?$ 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) solve simple equation eliminate integer parameter function is constant evaluate derivative differentiate the equation eliminate derivative by substitution simplify sums and products eliminate compound fractions common denominator and simplify factor out common term factor expression (not integer) multiply out and simplify show common factor in u/v write as polynomial (in ?) express as polynomial make the leading coeffient 1 $x^(1/2) = \sqrt x$ convert fractional exponents to roots convert roots to fractional exponents u=v => du/dx = dv/dx $d^2u/dx^2 = (d/dx)(du/dx)$ $d^nu/dx^n= d/dx d^(n-1)u/dx^(n-1)$ $d/dx du/dx = d^2u/dx^2$ $d/dx d^nu/dx^n = d^(n+1)/dx^(n+1)$ $\int 1 dt = t$ $\int c dt = ct$ (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 $\int |t| dt = t|t|/2$ $\int sin t dt = -cos t$ $\int cos t dt = sin t$ $\int tan t dt = -ln |cos t|$ $\int cot t dt = ln |sin t|$ $\int sec t dt = ln |sec t + tan t|$ $\int csc t dt = ln |csc t - cot t|$ $\int sec^2 t dt = tan t$ $\int csc^2 t dt = -cot t$ $\int tan^2 t dt = tan t - t$ $\int cot^2 t dt = -cot t - t$ $\int sec t tan t dt = sec t$ $\int csc t cot t dt = -csc t$ $\int sin ct dt = -(1/c) cos ct$ $\int cos ct dt = (1/c) sin ct$ $\int tan ct dt = -(1/c) ln |cos ct|$ $\int cot ct dt = (1/c) ln |sin ct|$ $\int sec ct dt = (1/c) ln |sec ct + tan ct|$ $\int csc ct dt = (1/c) ln |csc ct - cot ct|$ $\int sec^2 ct dt = (1/c) tan ct$ $\int csc^2 ct dt = -(1/c) cot ct$ $\int tan^2 ct dt = (1/c) tan ct - t$ $\int cot^2 ct dt = -(1/c) cot ct - t$ $\int sec ct tan ct dt = (1/c) sec ct$ $\int csc ct cot ct dt = -(1/c) csc ct$ $\int e^t dt = e^t$ $\int e^ct dt =(1/c) e^(ct)$ $\int e^(-t)dt = -e^(-t)$ $\int e^(-ct)dt = -(1/c) e^(-ct)$ $\int e^(t/c)dt = c e^(t/c)$ $\int c^t dt = (1/ln c) c^t$ $\int u^v dt = \int (e^(v ln u) dt$ $\int ln t = t ln t - t$ $$integral(e^(-t^2),t) = sqrt(pi)/2 Erf(t)$$ select substitution u = ? computer selects substitution u show integral again integrand = $f(u) \times du/dx$ $\int f(u) (du/dx) dx = \int f(u) du$ integrate by subst (u = ?) integrate by substitution $\int u dv = uv - \int v du (u = ?)$ $\int u dv = uv - \int v du$ set current line = original original integral to left side evaluate simple integral $$integral(f'(x),x,a,b)=f(b)-f(a)$$ $$diff(integral(f(t),t,a,x),x) = f(x)$$ $$eval(f(t),t,a,b) = f(b) - f(a)$$ $$eval(ln f(t),t,a,b) = ln(f(b)/f(a))$$ $$integral(u,t,a,b) = - integral(u,t,b,a)$$ $$integral(u,t,a,b) + integral(u,t,b,c) = integral(u,t,a,c)$$ $$integral(u,t,a,c) = integral(u,t,a,?) + integral(u,t,?,c)$$ break $\int |f(t)| dt$ at zeroes of f calculate integral with parameter numerically calculate integral numerically $$integral(u,t,a,a) = 0$$ $$integral(u,x,a,infinity) = lim(t->infinity,integral(u,x,a,t))$$ $$integral(u,x,-infinity,b) = lim(t->-infinity,integral(u,x,t,b))$$ $$integral(u,x,a,b) = lim(t->a+,integral(u,x,t,b))$$ $$integral(u,x,a,b) = lim(t->b-,integral(u,x,a,t))$$ limit of integrand is not zero at $\infty $ limit of integrand is not zero at $-\infty $ $$integral(u,t,-a,a) = 0$$ (u odd) $$integral(u,t,-a,a) = 2 integral(u,t,0,a)$$ (u even) $x = a sin \theta {for \sqrt (a^2-x^2)}$ $x = a tan \theta {for \sqrt (a^2+x^2)}$ $x = a sec \theta {for \sqrt (x^2-a^2)}$ $x = a sinh \theta {for \sqrt (a^2+x^2)}$ $x = a cosh \theta {for \sqrt (x^2-a^2)}$ $x = a tanh \theta {for \sqrt (a^2-x^2)}$ define inverse substitution x = ? simple integral in one step $sin^2 t = (1-cos 2t)/2$ 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.) 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 factor denominator (if easy) square-free factorization expand in partial fractions $\int 1/(ct\pm b) dt = (1/c) ln |ct\pm b|$ $\int 1/(ct\pm b)^(n+1) dt = -1/nc(ct\pm b)^n$ $\int 1/(t^2+a^2)dt=(1/a)arctan(t/a)$ $\int 1/(t^2-a^2)dt=(1/a)arccoth(t/a)$ $\int 1/(t^2-a^2)dt=(1/2a)ln|(t-a)/(t+a)|$ $\int 1/(a^2-t^2)dt=(1/a)arctanh(t/a)$ $\int 1/(a^2-t^2)dt=(1/2a)ln|(t+a)/(a-t)|$ $\int 1/\sqrt (a^2-t^2)dt = arcsin(t/a)$ $\int 1/\sqrt (t^2\pm a^2)dt)=ln|t+\sqrt (t^2\pm a^2)|$ $\int 1/(t\sqrt (t^2-a^2))dt=(1/a)arccos(t/a)$ make a rationalizing substitution $\int arcsin z dz = z arcsin z + \sqrt (1-z^2)$ $\int arccos z dz = z arccos z - \sqrt (1-z^2)$ $\int arctan z dz = z arctan z - (1/2)ln(1+z^2)$ $\int arccot z dz = z arccot z + (1/2)ln(1+z^2)$ $\int arccsc z dz = z arccsc z+ln(z + \sqrt (z^2-1)) (z>0)$ $\int arccsc z dz = z arccsc z-ln(z + \sqrt (z^2-1)) (z<0)$ $\int arcsec z dz = z arcsec z-ln(z + \sqrt (z^2-1)) (z>0)$ $\int arcsec z dz = z arcsec z+ln(z + \sqrt (z^2-1)) (z<0)$ change integral by substitution absorb number in const of int $\int sinh u du = cosh u$ $\int cosh u du = sinh u$ $\int tanh u du = ln cosh u$ $\int coth u du = ln sinh u$ $\int csch u du = ln tanh(u/2)$ $\int sech u du = arctan (sinh u)$ $$1/(1-x) = sum(x^n,n,0,infinity)$$ $1/(1-x) = 1+x+x^2+...$ $1/(1-x) = 1+x+x^2+...x^n...$ $$1/(1+x) = sum((-1)^n x^n,n,0,infinity)$$ $1/(1+x) = 1-x+x^2+...$ $1/(1+x) = 1-x+x^2+...(-1)^nx^n...$ $$sum(x^n,n,0,infinity)=1/(1-x)$$ $1+x+x^2+... = 1/(1-x)$ $1+x+x^2+...x^n...= 1/(1-x)$ $$sum((-1)^n x^n,n,0,infinity) = 1/(1+x)$$ $1-x+x^2+... = 1/(1+x)$ $1-x+x^2+...(-1)^nx^n... = 1/(1+x)$ $$x/(1-x) = sum(x^n,n,1,infinity)$$ $x/(1-x) = x+x^2+x^3+...$ $x/(1-x) = x+x^2+...x^n...$ $$x/(1+x) = sum((-1)^(n+1) x^n,n,1,infinity)$$ $x/(1+x) = x-x^2+x^3+...$ $x/(1+x) = x-x^2+...(-1)^(n+1)x^n...$ $$sum(x^n,n,1,infinity)=x/(1-x)$$ $x+x^2+x^3+...=x/(1-x)$ $x+x^2+...x^n...=x/(1-x)$ $$sum((-1)^(n+1) x^n,n,1,infinity)=x/(1+x) $$ $x-x^2+x^3+...=x/(1+x) $ $x-x^2+...(-1)^(n+1)x^n...=x/(1+x) $ $$1/(1-x^k) = sum(x^(kn),n,0,infinity)$$ $$1/(1-x^k) = sum(x^(kn),n,0,infinity,-3)$$ $$1/(1-x^k) = sum(x^(kn),n,0,infinity,2)$$ $$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity)$$ $$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,-3)$$ $$x^m/(1-x^k) = sum(x^(kn+m),n,0,infinity,2)$$ $$sum(x^(kn),n,0,infinity)=1/(1-x^k)$$ $$sum(x^(kn),n,0,infinity,-3)=1/(1-x^k)$$ $$sum(x^(kn),n,0,infinity,2)=1/(1-x^k)$$ $$sum(x^(m+kn),n,0,infinity)=x^m/(1-x^k)$$ $$sum(x^(m+kn),n,0,infinity,-3)=x^m/(1-x^k)$$ $$sum(x^(m+kn),n,0,infinity,2)=x^m/(1-x^k)$$ $$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity)$$ $$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,-3)$$ $$1/(1+x^k) = sum((-1)^n x^(kn),n,0,infinity,2)$$ $$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity)$$ $$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,-3)$$ $$x^m/(1+x^k) = sum((-1)^n x^(kn+m),n,0,infinity,2)$$ $$sum((-1)^nx^(kn),n,0,infinity)=1/(1+x^k)$$ $$sum((-1)^nx^(kn),n,0,infinity,-3)=1/(1+x^k)$$ $$sum((-1)^nx^(kn),n,0,infinity,2)=1/(1+x^k)$$ $$sum((-1)^nx^(m+kn),n,0,infinity)=x^m/(1+x^k)$$ $$sum((-1)^nx^(m+kn),n,0,infinity,-3)=x^m/(1+x^k)$$ $$sum((-1)^nx^(m+kn),n,0,infinity,2)=x^m/(1+x^k)$$ $$x^k/(1-x) = sum(x^n,n,k,infinity)$$ $$x^k/(1-x) = sum(x^n,n,k,infinity,-3)$$ $$x^k/(1-x) = sum(x^n,n,k,infinity,2)$$ $$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity)$$ $$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,-3)$$ $$x^k/(1+x) = sum((-1)^nx^n,n,k,infinity,2)$$ $$sum(x^n,n,k,infinity) = x^k/(1-x)$$ $$sum(x^n,n,k,infinity,-3) = x^k/(1-x)$$ $$sum(x^n,n,k,infinity,2) = x^k/(1-x)$$ $$sum((-1)^nx^n,n,k,infinity) = x^k/(1+x)$$ $$sum((-1)^nx^n,n,k,infinity,-3) = x^k/(1+x)$$ $$sum((-1)^nx^n,n,k,infinity,2) = x^k/(1+x)$$ $$ln(1-x) = -sum(x^n/n,n,1,infinity)$$ $$ln(1-x) = -sum(x^n/n,n,1,infinity,-3)$$ $$ln(1-x) = -sum(x^n/n,n,1,infinity,2)$$ $$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity)$$ $$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,-3)$$ $$ln(1+x) = sum((-1)^(n+1) x^n/n,n,1,infinity,2)$$ $$sum(x^n/n,n,1,infinity) = -ln(1-x)$$ $$sum(x^n/n,n,1,infinity,-3)=-ln(1-x)$$ $$sum(x^n/n,n,1,infinity,2)=-ln(1-x)$$ $$sum((-1)^(n+1) x^n/n,n,1,infinity)=ln(1+x)$$ $$sum((-1)^(n+1) x^n/n,n,1,infinity,-3)=ln(1+x)$$ $$sum((-1)^(n+1) x^n/n,n,1,infinity,2)=ln(1+x)$$ $$ sin x = sum( (-1)^n x^(2n+1)/(2n+1)!,n,0,infinity)$$ $sin x = x-x^3/3!+x^5/5!+...$ $sin x = x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+...$ $$cos x = sum( (-1)^n x^(2n)/(2n)!,n,0,infinity)$$ $cos x = 1-\onehalf x^2+x^4/4! + ...$ $cos x = 1-\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+...$ $$sum((-1)^n x^(2n+1)/(2n+1)!,n,0,infinity) = sin x$$ $x-x^3/3!+x^5/5!+... = sin x$ $x-x^3/3!+x^5/5!+...+ (-1)^nx^(2n+1)/(2n+1)!+... = sin x$ $$sum( (-1)^n x^(2n)/(2n)!,n,0,infinity) = cos x$$ $1-\onehalf x^2+x^4/4! + ... = cos x$ $1-\onehalf x^2+...+(-1)^nx^(2n)/(2n)!+... = cos x$ $$e^x = sum(x^n/n!,n,0,infinity)$$ $e^x = 1+x+x^2/2!+...$ $e^x = 1+x+...+x^n/n!...$ $$sum(x^n/n!,n,0,infinity)= e^x$$ $1+x+x^2/2!+ x^3/3!+... = e^x$ $1+x+...+x^n/n!... = e^x$ $$e^(-x) = sum((-x)^n x^n/n!,n,0,infinity)$$ $e^(-x) = 1-x+x^2/2!+...$ $e^(-x) = 1-x+...(-1)^nx^n/n!...$ $$sum((-1)^nx^n/n!,n,0,infinity)= e^(-x)$$ $1-x+x^2/2!+ x^3/3!+... = e^(-x)$ $1-x+...+(-1)^nx^n/n!... = e^(-x)$ $$arctan x = sum(x^(2n+1)/(2n+1),n,0,infinity)$$ $arctan x = x -x^3/3 + x^5/5 ...$ $arctan x = x -x^3/3 +...+ x^(2n+1)/(2n+1)+...$ $$sum(x^(2n+1)/(2n+1),n,0,infinity) = arctan x$$ $x -x^3/3 + x^5/5 ...=arctan x$ $x -x^3/3 +...+ x^(2n+1)/(2n+1)+...=arctan x$ $$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity)$$ $$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,-3)$$ $$(1+x)^alpha = sum(binomial(alpha,n) x^n,n,0,infinity,2)$$ $$sum(binomial(alpha,n) x^n,n,0,infinity)= (1+x)^alpha$$ $$sum(binomial(alpha,n) x^n,n,0,infinity,-3)= (1+x)^alpha$$ $$sum(binomial(alpha,n) x^n,n,0,infinity,2)= (1+x)^alpha$$ $$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$ $$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$ $$tan x = sum((-1)^(n-1) (2^(2n)(2^(2n)-1) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$ $$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity)$$ $$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,-3)$$ $$x cot x = sum((-1)^n (2^(2n) bernoulli(2n))/(2n)! x^(2n-1),n,0,infinity,2)$$ $$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity)$$ $$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$ $$x/(e^x-1) = 1-x/2 + sum(( bernoulli(2n))/((2n)!) x^(2n),n,1,infinity,2)$$ $$sec x = sum( (-1)^n (eulernumber(2n))/((2n)!) x^(2n),n,1,infinity)$$ $$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,-3)$$ $$sec x = sum(( eulernumber(2n))/((2n)!) x^(2n),n,1,infinity,2)$$ $$zeta(s) = sum(1/n^s,n,1,infinity)$$ $$zeta(s) = sum(1/n^s,n,1,infinity,-3)$$ $$zeta(s) = sum(1/n^s,n,1,infinity,-2)$$ $$sum((-1)^n/n,n,1,infinity) = ln 2$$ 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 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 decrease lower limit by subtracting terms add ? to index variable subtract ? from index variable differentiate power series term by term integrate power series term by term 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 $\sum a_k = \sum a_(2k) + \sum a_(2k+1)$ $\sum u$ 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 limit comparison test finish condensation test positive result of comparison test negative result of comparison test $$sum(1/k,k,1,infinity) = infinity$$ $$sum(1/k^2,k,1,infinity) = pi^2/6$$ $$sum(1/k^s,k,1,infinity) = zeta(s)$$ $$zeta(2k) = (2^(2k-1) abs(bernoulli(2k)) pi^(2k))/factorial(2k)$$ $ln(u+iv) = ln(re^(i\theta ))$ $ln(re^(i\theta ))=ln r + i\theta (-\pi <\theta \le \pi )$ $ln i = i\pi /2$ $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 $ $e^(x+iy) = e^x cos y + i e^x sin y$ $e^(i\pi ) = -1$ $e^(-i\pi ) = -1$ $e^(2n\pi i) = 1$ $e^((2n\pi + \theta )i) = e^(i\theta )$ $u^v = e^(v ln u)$ 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 $ 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 $sinh^2u + 1 = cosh^2 u$ $cosh^2 u - 1 = sinh^2u $ $cosh^2 u - sinh^2u = 1$ $cosh^2 u = sinh^2u + 1$ $sinh^2u = cosh^2 u - 1$ $1 - tan^2u = sech^2u$ $1 - sech^2u = tan^2u$ 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 $cosh 2u = cosh^2 u + sinh^2 u$ $tanh(ln u) = (1-u^2)/(1+u^2)$ $arcsinh x = ln(x + \sqrt (x^2+1))$ $arccosh x = ln(x + \sqrt (x^2-1))$ $arctanh x = (1/2) ln((1+x)/(1-x))$ $sinh(asinh x) = x$ $cosh(acosh x) = x$ $tanh(atanh x) = x$ $coth(acoth x) = x$ $sech(asech x) = x$ $csch(acsch x) = x$ 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 $d/du arcsinh u = 1/\sqrt (u^2+1)$ $d/du arccosh u = 1/\sqrt (u^2-1)$ $d/du arctanh u = 1/(1-u^2)$ $d/du arccoth u = 1/(1-u^2)$ $d/du arcsech u= -1/(u\sqrt (1-u^2))$ $d/du arccsch u= -1/(|u|\sqrt (u^2+1))$ sg(x) = 1 if x > 0 sg(x) = -1 if x < 0 sg(0) = 0 sg(-x) = -sg(x) -sg(x) = sg(-x) sg(x) = |x|/x (x nonzero) sg(x) = x/|x| (x nonzero) abs(x) = x sg(x) $sg(x)^(2n) = 1$ sg(x)^(2n+1) = sg(x) 1/sg(x) = sg(x) d/dx sg(u) = 0 (u nonzero) $\int sg(x) = x sg(x)$ $\int sg(u)v dx = sg(u)\int v dx$ (u nonzero) sg(x) = 1 assuming x > 0 sg(x) = -1 assuming x < 0 $sg(au) = sg(u)$ 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 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) 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) expand multiply if cancels cancel square roots 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 Linear Equations by Term Selection Equations by Substitution Matrix Methods Advanced Matrix Methods Absolute Value Absolute Value Inequalities Strict Inequalities Inequalities Inequalities involving Squares Inequalities involving Reciprocals Root and Power Inequalities Inequalities--One Side Zero Binomial Theorem Factoring Binomial Expansions Sigma Notation Advanced Sigma Notation Prove by Induction Trig 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 Integrate Hyperbolic Functions Geometric Series Geometric Series 2 Geometric Series 3 Geometric Series 4 Geometric Series 5 Power Series for the Logarithm Power Series for sin and cos Power Series for the Exponential Function Power series for arctan Power series for tan and cot 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 Invisible Too and This Too %�|� 4 I: ; I ! 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Sindbad File Manager Version 1.0, Coded By Sindbad EG ~ The Terrorists