Innity and Limits
Innity is a symbol not a number. We denote it by . In terms of limits it says that a quantity is
1
becoming as large as we want, like lim 2 = . Keeping this in mind we have the following rules (c is a
x0 x
constant, n positive integer):
Limits
Denition
We write
lim f (x) = L
xa
and say the limit of f (x) as x approaches a, equals L if we can make the values of f (x) arbitrarily
close to L (as close to L as we like) by taking x to be suciently close to a (on either side of a) but not
equa
Dierentiation Rules
d
(c) = 0
dx
Derivative of constant function
c=
The Power Rule
(xn ) =
The Constant Multiple Rule
(cf (x) = cf (x)
The Sum Rule
[f (x) + g (x)] = f (x) + g (x)
The Dierence Rule
[f (x) g (x)] = f (x) g (x)
Derivative of Natural Exponen
DIFFERENTIATION RULES
c =0
(xn ) = nxn1
(ex ) = ex
(cf ) = cf
(f + g ) = f + g
(f g ) = f g
(f g ) = f g + f g
f
g
=
f g fg
g2
(sin x) = cos x
(cos x) = sin x
(tan x) = sec2 x
(csc x) = csc x cot x
(sec x) = sec x tan x
(cot x) = csc2 x
[f g ] = (f (g ) =
The Derivative Function
Denition:
The function f (x) dened as
f (x) = lim
h0
f (x + h) f (x)
h
is called the derivative of f , whenever the limit exists.
Notations: Let y = f (x) be a function. Then its derivative is
f (x) = y =
df
d
dy
=
=
f (x) = Df (x)
The Fundamental Theorem of Calculus
The FTC, Part 1
If f is continuous on [a, b], then the function g dened by
x
f (t) dt,
g ( x) =
axb
a
is an antiderivative of f , that is, g (x) = f (x) for a < x < b.
The FTC, Part 2
If f is continuous on the interval
Continuity
Denition A function f is continuous at a number a if lim f (x) = f (a).
xa
If f is not continuous at a we say that f is discontinous at a.
Denition A function f is continous from the right at a if lim+ f (x) = f (a) and
x a
f is continuous from
Tangents, Rates of Change, and Derivatives
Denition:
slope
The tangent line to the curve y = f (x) at the point P (a, f (a) is the line through P with
f (x) f (a)
f (a + h) f (a)
= lim
xa
h0
xa
h
m = lim
provided that the limit exists.
If an object moves
Evaluating Denite Integrals
by
Denition The antiderivative of f (x) is called the indenite integral and it is denoted
f (x) dx. Thus
f (x) dx = F (x) means F (x) = f (x).
Table of Indenite Integrals
xn+1
+ C (n = 1)
xn dx =
n+1
1
dx = ln |x| + C
x
e dx =
Improper Integrals
Type I: Innite Intervals
t
f (x)dx exists for every number t a, then
(a) If
a
t
f (x)dx = lim
t
a
f (x)dx
a
provided the limit exists (as a nite number).
b
f (x)dx exists for every number t b, then
(b) If
t
b
b
f (x)dx = lim
t
f (x)dx
t