;Computing the slope of x^3 via a limit LIM(((x+h)^3-x^3)/h,h,0) 3*x^2 ;A famous limit of an indeterminate form 0/0 LIM(SIN(x)/x,x,0) 1 ;A limit as x approaches infinity LIM(x^(1/x),x,inf) 1 ;A limit involving extra variables LIM((a^x-b^x)/x,x,0) LN(a)-LN(b) ;The limit of 1/x as x approaches 0 from the left LIM(1/x,x,0,-1) -inf ;A limit for which L'Hopital's rule would be slow LIM(SIN(x)^249*LN(1-x)^251/(x^100*ATAN(x)^400),x,0) -1 ;The formula for the slope of x^3 DIF(x^3,x) 3*x^2 ;An illustration of the chain rule DIF(SIN(x^3),x) 3*x^2*COS(x^3) ;A partial derivative DIF(x^2*y^3,y) 3*x^2*y^2 ;A 5th-order Taylor polynomial expanded about x=0 TAYLOR(#e^x,x,0,5) 0.00833333*x^5+0.0416666*x^4+0.166666*x^3+0.5*x^2+x+1 ;A 7th-order Taylor polynomial TAYLOR(LN(COS(a*x)),x,0,7) -0.0222222*a^6*x^6-0.0833333*a^4*x^4-0.5*a^2*x^2 ;An antiderivative of x^2 with respect to x INT(x^2,x) 0.333333*x^3 ;An antiderivative of cosine(x) with respect to x INT(COS(x),x) SIN(x) ;An antiderivative obtained by substitution INT(x^2*COS(a*x^3+b),x) 0.333333*SIN(b)*COS(a*x^3)/a+0.333333*COS(b)*SIN(a*x^3)/a ;A definite integral for x going from a to b INT(x^2,x,a,b) 0.333333*b^3-0.333333*a^3 ;An integral having an infinite integration limit INT(1/x^2,x,a^2,inf) 1/a^2 ;An integral having an endpoint singularity INT(1/SQRT(x),x,0,b^2) 2*ABS(b) ;A 2-dimensional integral over a quarter disk INT(INT(x*y,y,0,SQRT(r^2-x^2)),x,0,r) 0.125*r^4 ;The formula for the sum of an arithmetic series SUM(k,k,0,n) 0.5*n*(n+1) ;The formula for the sum of successive cubes SUM(k^3,k,0,n) 0.25*n^2*(n+1)^2 ;The formula for the sum of a geometric series SUM(a^k,k,0,n) a^(n+1)/(a-1)-1/(a-1) ;The sum of an infinite series SUM(2^(-k),k,0,inf) 2 ;A sum for which iteration would be slow SUM(k/370370367,k,-123456788,123456789) 0.333333 ;The product of successive even integers PRODUCT(2*k,k,1,n) 2^n*n!