Computing the slope of x3 via a limit
 lim h® 0 (x+h)3-x3h = 3 x2

A famous limit of an indeterminate form 0/0

 lim x® 0 sin(x)x = 1

A limit as x approaches infinity

 lim x® ¥ x1/x = 1

A limit involving extra variables

 lim x® 0 ax-bxx = ln(a)-ln(b)

The limit of 1/x as x approaches 0 from the left

 lim x® 0 1x = -¥

A limit for which L'Hopital's rule would be slow

 lim x® 0 (sin(x))249 (ln(1-x))251x100 (arctan(x))400 = -1

The formula for the slope of x3

 dd x x3 = 3 x2

An illustration of the chain rule

 dd x sin(x3) = 3 x2 cos(x3)

A partial derivative

 dd y x2 y3 = 3 x2 y2

A 5th-order Taylor polynomial expanded about x=0

 TAYLOR(ex,x,0,5) = 0.00833333 x5+0.0416666 x4+0.166666 x3+0.5 x2+x+1

A 7th-order Taylor polynomial

 TAYLOR(ln(cos(a x)),x,0,7) = -0.0222222 a6 x6-0.0833333 a4 x4-0.5 a2 x2

An antiderivative of x2 with respect to x

 óõ x2 dx = 0.333333 x3

An antiderivative of cosine(x) with respect to x

 óõ cos(x) dx = sin(x)

An antiderivative obtained by substitution

 óõ x2 cos(a x3+b) dx = 0.333333 sin(b) cos(a x3)a + 0.333333 cos(b) sin(a x3)a

A definite integral for x going from a to b

 óõ b a x2 dx = 0.333333 b3-0.333333 a3

An integral having an infinite integration limit

 óõ ¥ a2 1x2 dx = 1a2

An integral having an endpoint singularity

 óõ b2 0 1Öx dx = 2 |b|

A 2-dimensional integral over a quarter disk

 óõ r 0 óõ Ö[(r2-x2)] 0 x y dy dx = 0.125 r4

The formula for the sum of an arithmetic series

 n å k = 0 k = 0.5 n (n+1)

The formula for the sum of successive cubes

 n å k = 0 k3 = 0.25 n2 (n+1)2

The formula for the sum of a geometric series

 n å k = 0 ak = an+1a-1 - 1a-1

The sum of an infinite series

 ¥ å k = 0 2-k = 2

A sum for which iteration would be slow

 123456789 å k = -123456788 k370370367 = 0.333333

The product of successive even integers

 n Õ k = 1 2 k = 2n n

File translated from TEX by TTH, version 2.30.
On 29 Jun 1999, 18:34.