Applied Mathematics for Business and Economics

Lecture Note Differentiation

Solution
a. The rate of change of the population with respect to time is the derivative of the
population function. That is,
Rate of change=Px′( )=+ 2 x 20

The rate of change of the population 15 months from now will be

P′( (^15) )=2 15 20 50×+ = people per month
b. The actual change in the population during the 16th month is the difference
between the population at the end of 16 month and the population at the end of
15 months. That is,
Change in population=−=−=PP( (^16) ) ( (^15) ) 8, 576 8, 525 51 People

3.2 Percentage Rate of Change .....................................................................

rate of change of quantity

Percentage rate of change 100

size of quantity

=

Ifyfx= (), the percentage rate of change of y with respect to x is given by the

formula

( )

()

Percentage rate of change 100 100

fx dy dx

f xy

′

==

Example 2
The gross national product (GNP) of a certain country wasNt t( )=++^25106 t billion

dollars years after 1980.
a. At what rate was the GNP changing with respect to time in 1988?
b. At what percentage rate was the GNP changing with respect to time in 1988?

Solution

a. The rate of change of the GNP is the derivativeNt′()= 25 t+. The rate of change in

1988 was N′( (^82852) )=×+= 1 billion dollars per year.
b. The percentage rate of change of the GNP in 1988 was
()
()
(^821)
100 100 10 percent per year
8210

N

N

′

==

4 Approximation by Differentials; Marginal Analysis .......................................

If y is a function of x, then we have basic formula

()

Change rate of change of

change in

in with respect to

y

x

yx

⎛⎞⎛ ⎞

⎜⎟⎜≈×⎟

⎝⎠⎝ ⎠

Approximation Formula

If y=fx() and Δx is a small change in x, then the corresponding change in y is

dy

yx

dx

Δ ≈Δ

or, in functional notation, the corresponding change in f is

Δ= +Δ − ≈ Δf fx x fx f x x( ) ( ) ′( )

That is, the change in the function is approximately that derivative of the

function times the change in its variable.