Applied Mathematics for Business and Economics

Lecture Note Differentiation

whent= 3 , p=p(3) 3.1 0.1 3=+×=^24 and so

2

0.1 4 3 1.2

0.24 parts per million per year

0.5 4 17 25

dc

dt

××

===

×+

6 Higher-Order Derivatives

.1 The Second Derivative

the derivative of its derivative. If , the

6

The second derivative of a function is yfx= ()

second derivative is denoted by:

(^) ()
2
2 or
dy
f x
dx

′′

The second derivative gives the rate of change of the rate change of the original
function.
Example 1

Find both the first and second derivatives of the functions:

a. f()xx=^3 − + 12 x 1 b. fx xxx( )= 53342 −−+ 7 c. ()
()

2

32

1

x

fx

x

−

=

−

Example 2
An efficiency study o he mof t rning shift at a certain factory indicates that an average

worker who arrives on the job at 8:00AM. Will have producedQt()=− + +t^32624 t t

units t hours later.
a. Compute the worker’s rate of production at 11:00A.M
b. At what rate is the worker’s rate of production changing
at 11:00

with respect to time

A.M?

rate of production

ctual change in the worker’s rate of production between

Solution

c. Use calculus to estimate the change in the worker’s

between 11:00 and 11:10A.M.

d.Compute the a

11:00 and 11:10A.M.

a. The worker’s rate of production is the first derivative

Q′(t)=−+ + 3122 t^2 t 4

At 11:00 A.M., t= 3 and the rate of production is

Q′() 3 =− × + × + =3 3^2 12 3 24 3 units per hour 3

b. The rate of change of the rate of production is the second derivative

Qt′′()=−+ 612 t

At 11:00 A.M., the rate is

Q′′() 3 =− × + =−6 3 12 6 unit per hour per hour

c. Note that 10 minutes is 1/6 hours, and hence Δt= 16 hour.

Change in rate of production is ()

1

=− × 6 1unit per hour

6

ΔΔQQtt′′′

=−