Applied Mathematics for Business and Economics

Lecture Note Differentiation

a. The elasticity of demand is

()

2

22

2

2

pdq p p

η== −=−p

qdp 300 p 300 −p

−

The demand is of unit elasticity when η = 1 , that is, when

2

2

2

2

1

300

100

10

p

p

p

p

=

−

=

=±

of which only p= 10 is in the relevent interval 03 ≤≤p 00

0

If 01 ≤<p

22

22

2210

1

300 300 10

p

p

η

×

= <=

−−

and hence the demand is inelastic.

If 10 <≤p 300

22

22

2210

1

300 300 10

p

p

η

×

= >=

−−

and hence the demand is elastic.

b. The total revenue is an increasing function of p when demand is inelastic, that

is, on the interval 0 and a decreasing function of p when demand is

elastic, that is, on the interval

≤<p 10

10 <≤p 300. At the price ൌ݌ of unit

elasticity, the revenue function has a relative maximum.

10

c. The revenue function is ൌ݌ݍ݌ or

R()pp= ( 300 −= −p^23 ) 300 pp

Its derivative is

R′()ppp=300 3−= − +^2 3 10( )( 10 p)

which is zero when p=± 10 , of which only ൌ݌ 10 is in the relevant interval

0 ≤≤p 300.

On the interval 01 ≤<p 0 , R′(p)is positive and so R(p)is increasing. On the

interval 10 <≤p 300 , R′(p)is negative and so R(p) is decreasing. At the

critical value݌ ൌ10, R()p stops increasing and starts decreasing and hence

has a relative maximum.

Exercises

1 KNnaedrIev:énGnuKmn_xageRkam rYcsRmYllT§pl[enAgaybMput (Differentiate the

following function, do as much computation as possible to simplify the results)

a.yx x=++^233 b.fx x( )=^98 +++ 5 x x 12

c. (^2)

11 1

y

xx x

=+ − d. (^) ()^33

1

fx x

x

=+

e. fx()=+ −( 2132 x)( x ) f.f(xx)=− −(^22512 )( x)

g. f() ( )( )xx=+−100 2 1 1 5x