1
SOME FORMULAS FOR REFERENCES FOR BUSINESS CALCULUS AND CALCULUS I, II, II
DIFFERENTIAL CALCULUS
(A) EXISTENCE OF LIMIT OF A FUNCTION
The limit of a function f (x ) at
x  c , i.e., lim f ( x ) exists if and only if the left-hand limit
x c
lim f ( x ) and the right-hand limit lim f ( x ) exist and are equal.
x c
x c
(B) CONTINUITY OF A FUNCTION AT A POINT
A function f (x ) is continuous at
x  c if the following conditions are satisfied:
(i) f (c ) is defined;
(ii) lim f ( x ) exists;
x c
(iii) lim f ( x)  f (c) .
x c
(C) THE DERIVATIVE OF A FUNCTION
(1) Definition: The derivative of the function y  f (x) with respect to
given by
f / ( x)  lim
h 0
x is the function f / ( x)
f ( x  h)  f ( x )
h
(1.1)
(read as “ f prime of
x ”). The process of computing the derivative is called differentiation, and
f (x ) is said to be differentiable at x  c if f / ( x) exists; i.e., if the above limit (1.1) that defines
f / ( x) exists when x  c .
Notation: The derivative of y  f (x) is denoted as: f ( x ) ,
/
df
dy
or
.
dx
dx
(2) Slope (m ) of the tangent line to the graph of y  f (x) at
given by the derivative of the function y  f (x) at
x0 , i.e., by m  f / ( x0 ) .
(3) Equation the tangent line to the graph of y  f (x) at
y  y0  m( x  x0 ) .
( x0 , y 0 ) , where y 0  f ( x0 ) , is
( x0 , y 0 ) is given by
2
(D) BASIC RULES OF DERIVATIVE
(i) The constant rule:
d
(c )  0
dx
(ii) The derivative of x is one:
(ii) The power rule:
d
( x)  1
dx
d n
( x )  nx n 1
dx
(iii) The constant multiple rule:
d
df
[cf ( x)]  c
dx
dx
(iv) The sum and difference rules:
d
df dg
[ f ( x)  g ( x)] 

dx
dx dx
d
df
dg
[ f ( x) g ( x)]  g ( x)
 f ( x)
dx
dx
dx
(v) The product rule:
(vi) The quotient rule:
d  f ( x) 

dx  g ( x) 
g ( x)
(vii) The chain rule for y  f [u ( x)] :
(viii) The general power rule:
df
dg
 f ( x)
dx
dx , g ( x)  0.
[ g ( x)] 2
dy dy du

.
dx du dx
d
h( x)n  nh( x)n 1 dh
dx
dx
(ix) Derivatives of exponential functions:
 
(a)
d x
e  e x ; and
dx
(b)
d u( x)
e
dx

  e 
u( x)
du
dx
(x) Derivatives of logarithmic functions:
(a)
or
d
ln x  1 , ( x  0) ; and
dx
x
 f (u ( x))/
 f / (u ) . u / ( x)
3
d
ln u( x)   1  du
dx
 u ( x)  dx
(b)
x
(xi) Derivatives of b and
Let
log b x for base b  e :
b  0 be a positive real number. Let b  1 . Then we have
 
(a)
d x
b  (ln b) . b x ; and
dx
(b)
d
log b x   1  . 1 , ( x  0)
dx
 ln b  x


/
//
/
(xii) The second derivative of y  f (x) denoted as: f ( x)  f ( x) 
of change of
d2 f
gives the rate
dx 2
f / ( x) .
(xiii) Notation for the
nth derivative of y  f (x) :
dny
dn f
( n)

f
(
x
)

dx n
dx n
(xiv) Demand Function, Supply Function and Market Equilibrium:
Demand Function: A demand function p  D (x ) is a function that relates the unit price p for
a particular commodity to the number of units x demanded by consumers at that price or sold in
the marketplace.
Supply Function: A supply function denoted by S (x ) is defined as a function which gives the
corresponding price p  S (x) at which producers are willing to supply
x units.
Law of Supply and Demand: In a competitive market environment, if S ( x )  D ( x ) , the market
is said to be in “equilibrium.” Thus, if
xe denote the production level at which the market
equilibrium occurs, and if p e denote the corresponding unit price, then pe  S ( xe )  D( xe ) .
Here pe is called the “equilibrium price” and xe , pe  is called the “equilibrium point.”
Shortage and Surplus: When the market is not in equilibrium, it has a
(i)
a “shortage” when D( x)  S ( x) ; and
(ii)
a “surplus” when S ( x)  D( x) .
4
(xv) Cost Function, Revenue Function, Profit Function and Break Even:
x represent the number of units produced and sold in the marketplace. Let
p c denote a variable cost per unit. Then the total cost of producing the x units denoted by
C (x ) is defined as a function given by
Cost Function: Let
C ( x)  (var iable cos t )  ( fixed cos t ) , where var iable cos t  x . pc .
Revenue Function: Let p  D (x ) denote a demand function that relates the unit price p for a
particular commodity to the number of units x demanded by consumers at that price or sold in
the marketplace. Then the total revenue from the sale of x units, denoted by R (x ) , is defined as
a function given by
R( x)  (number of items sold ) . ( price per item)  x . p  x . D( x) .
Profit Function: The profit from the production and sale of
x units at the unit price p , denoted
by P (x ) , is defined as a function given by
P( x)  R( x)  C ( x)  x . D( x)  C ( x) .
Break Even: When total revenue equals total cost, that is, R( x)  C ( x) , that is, P ( x )  0 , we
say that the manufacturer “breaks even,” experiencing neither a profit nor a loss. The point at
which the graphs of the two functions, Revenue Function: y  R(x) and Cost Function:
y  C (x) intersect is called the “break- even point.”
(xvi) Marginal Cost, Marginal Revenue and Marginal Profit: Let C (x)  the total cost of
x units of a particular commodity, R (x)  the revenue generated when x units of a
particular commodity are produced, and P (x )  the corresponding profit. Then, if x  x0 denote
producing
the number of units being produced, the marginal cost, marginal revenue and marginal profit of
producing x 0 units are defined as follows:
/
(a) Marginal Cost: It is given by the derivative C ( x 0 ) of the cost function C (x ) at
x  x0 ,
which approximates the additional cost (that is, extra cost)
C ( x0  1)  C ( x0 ) incurred
when the level of production is increased by one unit, from
x0 to x0  1 , that is, of
producing the
( x0  1) st unit.
/
(b) Marginal Revenue: It is given by the derivative R ( x 0 ) of the revenue function R (x ) at
x  x0 , which approximates R( x0  1)  R( x0 ) , the additional revenue from producing
the
( x0  1) st unit.
/
(c) Marginal Profit: It is given by the derivative P ( x 0 ) of the profit function P (x ) at
x  x0 , which approximates P( x0  1)  P( x0 ) , the additional profit from producing
the ( x0  1) st unit.
5
(xvii) Approximation by increments: Let y  f (x) be a function of
x which is differentiable at
x  x0 , and let x denote a small change in x , then we have
f ( x0  x)  f ( x0 )  f / ( x0 ) . x .
If
f  f ( x0  x)  f ( x0 ) in the above formula, then we have f  f / ( x0 ) . x .
(xviii) Differentials: Sometimes the increment x is called the “differential” of x and is denoted
by dx . Thus, if the differential of x be dx  x , and if y  f (x) be a differentiable function of
x , then dy  f / ( x) . dx is called the “differential” of y .
(ix) Approximation Formula for Percentage Change: Let y  f (x) be a differentiable function
of x . Let x denote a small change in
function f (x ) is defined as
x . Then the corresponding “percentage change” in the
Percentage change in the function f ( x)  100
f / ( x) . x
f
%  100
%.
f ( x)
f ( x)
INTEGRAL CALCULUS
BASIC RULES OF INTEGRATION
(A) INDEFINITE INTEGRAL:
(i) Antiderivative; indefinite integral:
 f ( x)dx  F ( x)  C iff
F / ( x)  f ( x) , where C is a
constant of integration.
(ii) Power rule:
n
 x dx 
(iii) Logarithmic rule:
x n 1
 C for n  1
n 1
1
 x dx  ln x  C
(iv) Exponential rule: (a)
e
(v) Constant multiple rule:
x
dx  e x  C ; (b)  e k x dx 
1 kx
e C
k
 k f ( x) dx  k  f ( x) dx
(vi) Sum and difference rules:
  f ( x)  g ( x) dx   f ( x) dx   g ( x) dx
6
(vii) Integration by substitution:
 g (u( x))u
(viii) Integration by parts:
/
( x) dx   g (u ) du where u  u ( x) and du  u / ( x) dx
 u dv  u v   v du
(B) DEFINITE INTEGRAL:
(i) The Definite Integral: Let y  f (x) be a function of
x that is continuous on the interval
a  x  b . Let the interval a  x  b be subdivided into n equal parts, each of width
ba
x 
. Let a number x k be chosen from the kth subinterval for k  1, 2 ,  , n . Then
n
the sum  f ( x 1 )  f ( x 2 )   f ( x n ) x is formed, which is called a “Riemann sum.” The
b
“definite Integral” of f (x ) on the interval
Riemann sum as
a  x  b , denoted by
a
n    ; that is,
b
 f ( x) dx 
a
 f ( x) dx , is the limit of the
lim
n  
 f ( x1 )  f ( x2 )   f ( xn )x .
The function f (x ) is called the “integrand,” and the numbers a and b are called the “lower
and upper limits of integration,” respectively. The process of finding a definite integral is called
“definite integration.”
(ii) Area as a Definite Integral: If the function y  f (x) is continuous and f ( x )  0 on the
interval
a  x  b , then the region R under the curve y  f (x) over the interval a  x  b
b
has area A given by the definite Integral
A   f ( x) dx .
a
(iii) The Fundamental Theorem of Calculus: If the function y  f (x) is continuous on the
b
interval
a
x  b , then
 f ( x) dx  F (b)  F (a) , where F (x) is any antiderivative (i.e.,
a
indefinite integral) of f (x ) over the interval
a  x  b.
(iv) The Area Between Two Curves: If f (x ) and g (x ) are continuous with f ( x)  g ( x) on the
interval
a  x  b , then the area A between the curves y  f (x) and y  g (x) over the
interval
a xb
b
is given by the definite Integral
A    f ( x)  g ( x)dx .
a
7
(v) The Average Value of a Function: Let y  f (x) be a function of
x that is continuous on
the interval a  x  b . Then the " average value V of f ( x) over a  x  b" is given by
b
V
1
 f ( x)dx .
b  a a
(vi) RULES FOR DEFINITE INTEGRALS:
a
(a)
 f ( x) dx  0
a
a
(b)

b
f ( x) dx    f ( x) dx
b
a
(c) Constant multiple rule:
(d) Sum rule:
b
b
a
a
 k f ( x) dx  k  f ( x) dx for constant k
b
b
b
a
a
a
  f ( x)  g ( x)dx    f ( x)dx   g ( x)dx
(e) Difference rule:
b
b
b
a
a
a
  f ( x)  g ( x)dx    f ( x)dx   g ( x)dx
(f) Subdivision rule:
b
c
b
a
a
c
  f ( x)dx    f ( x)dx    f ( x)dx
8
TABLE OF INTEGRALS FOR QUICK REFERENCES
(From World Wide Web Site At The Address:
http://www.jessschwartz.com/~swalker/calculus/General_Integration_rules.htm)
9