Applied Mathematics for Business and Economics

Lecture Note Differentiation

d. The actual change in the worker’s rate of production between 11:00 and

11:10 A.M. is the difference between the values of the rate when

and when

Qt′()

t= (^316)

19

3

6

t==. That is,

()

Actual change in 19

3 1.08 units per hour

rate of production 6

QQ

⎛⎞⎛⎞

⎜⎟=−=−′′⎜⎟

⎝⎠⎝⎠

6.2 The nth Derivative ...................................................................................

For any positive integer n, the nth derivative of a function is obtained from the

function by differentiating successively n times. If the original function isy=fx(),

the nth derivative is denoted by

or ()()

n

n

n

dy
f x
dx
Example 3
Find the 5th derivative of each of the following functions:

a. fx( )=++− 52 x^642 x x 3 b.

1

y

x

=.

7 Concavity and the Second Derivative Test

Concavity

Suppose that f is differentiable on the interval (a,b).

a. Iff′is increasing on (a,b),then the graph of f is concave upward

on (a,b).

b. If f′ is decreasing on (a,b), then the graph of f is concave downward on

(a,b).

To Determine Concavity

Suppose that f is a function and f′ and f′′ both exist on the interval (a,b).

a. If fx′′()> 0 for all x in(ab, ), then f′ is increasing and f is concave

upward on (a,b).

b. If fx′′() 0< for all x in (a,b), then f′ is decreasing and f is concave

downward on (a,b).

Critical Points

A critical point of a function is a point on its graph where either:

+ The derivative is zero, or

+ The derivative is undefined

The relative maxima and minima of the function can occur only at critical points.

• 
  

To Determine Points of Inflection

A point on the graph of a function at which the concavity of the function changes is

called an inflection point.

• 
  
  
  
  • 
    
    
    
    • 
      
      
      
      • Slope
        is negative
      
      
      
      
    
    
    
    
  
  
  
  

Slope

is 0

Slope

is positive

Slope

is negative

Slope

is 0

()

Concave upward

holds water

()

Concave downward

spills water