Computing the slope of x3 via a limit

lim
h® 0 
 (x+h)3-x3
h
 = 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-bx
x
 = ln(a)-ln(b)

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


lim
x® 0 
 1
x
 = -¥

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


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

The formula for the slope of x3

d
d x
 x3 = 3 x2

An illustration of the chain rule

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

A partial derivative

d
d 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

ó
õ 		x2 dx = 0.333333 x3

An antiderivative of cosine(x) with respect to x

ó
õ 		cos(x) dx = sin(x)

An antiderivative obtained by substitution

ó
õ 		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

ó
õ 		b

a 
 x2 dx = 0.333333 b3-0.333333 a3

An integral having an infinite integration limit

ó
õ 		¥

a2 
 1
x2
  dx = 1
a2

An integral having an endpoint singularity

ó
õ 		b2

0 
 1
Öx
  dx = 2 |b|

A 2-dimensional integral over a quarter disk

ó
õ 		r

0 
 ó
õ 		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+1
a-1
 - 1
a-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 
 k
370370367
 = 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.