Computing the slope of x^{3} via a limit

lim
h® 0


(x+h)^{3}x^{3} h

= 3 x^{2} 

A famous limit of an indeterminate form 0/0
A limit as x approaches infinity
A limit involving extra variables

lim
x® 0


a^{x}b^{x} x

= ln(a)ln(b) 

The limit of 1/x as x approaches 0 from the left
A limit for which L'Hopital's rule would be slow

lim
x® 0


(sin(x))^{249} (ln(1x))^{251} x^{100} (arctan(x))^{400}

= 1 

The formula for the slope of x^{3}
An illustration of the chain rule

d d x

sin(x^{3}) = 3 x^{2} cos(x^{3}) 

A partial derivative

d d y

x^{2} y^{3} = 3 x^{2} y^{2} 

A 5thorder 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 7thorder 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

ó õ

x^{2} dx = 0.333333 x^{3} 

An antiderivative of cosine(x) with respect to x
An antiderivative obtained by substitution

ó õ

x^{2} cos(a x^{3}+b) dx = 
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

ó õ

b
a

x^{2} dx = 0.333333 b^{3}0.333333 a^{3} 

An integral having an infinite integration limit

ó õ

¥
a^{2}


1 x^{2}

dx = 
1 a^{2}



An integral having an endpoint singularity

ó õ

b^{2}
0


1 Öx

dx = 2 b 

A 2dimensional integral over a quarter disk

ó õ

r
0


ó õ

Ö[(r^{2}x^{2})]
0

x y dy dx = 0.125 r^{4} 

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

k^{3} = 0.25 n^{2} (n+1)^{2} 

The formula for the sum of a geometric series

n å
k = 0

a^{k} = 
a^{n+1} a1

 
1 a1



The sum of an infinite series
A sum for which iteration would be slow

123456789 å
k = 123456788


k 370370367

= 0.333333 

The product of successive even integers
File translated from T_{E}X by T_{T}H, version 2.30.
On 29 Jun 1999, 18:34.