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


x2 dx = 0.333333 x3

An antiderivative of cosine(x) with respect to x


cos(x) dx = sin(x)

An antiderivative obtained by substitution


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


b

a 
x2 dx = 0.333333 b3-0.333333 a3

An integral having an infinite integration limit




a2 
1
x2
 dx = 1
a2

An integral having an endpoint singularity


b2

0 
1
x
 dx = 2 |b|

A 2-dimensional integral over a quarter disk


r

0 

[(r2-x2)]

0 
x y dy dx = 0.125 r4

The formula for the sum of an arithmetic series

n

k = 0 
k = 0.5 n (n+1)

The formula for the sum of successive cubes

n

k = 0 
k3 = 0.25 n2 (n+1)2

The formula for the sum of a geometric series

n

k = 0 
ak = an+1
a-1
- 1
a-1

The sum of an infinite series



k = 0 
2-k = 2

A sum for which iteration would be slow

123456789

k = -123456788 
k
370370367
= 0.333333

The product of successive even integers

n

k = 1 
2 k = 2n n


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