Elasticity and Its Application
Chapter 5
Elasticity…
… is a measure of how much buyers and
sellers respond to changes in market
conditions
… allows us to analyze supply and
demand with greater precision.
Price Elasticity of Demand
Price elasticity of demand is the
percentage change in quantity demanded
given a percent change in the price,
It is a measure of how much the quantity
demanded of a good responds to a change
in the price of that good.
Determinants of
Price Elasticity of Demand
Necessities versus Luxuries
Availability of Close Substitutes
Definition of the Market
Time Horizon
Determinants of
Price Elasticity of Demand
Demand tends to be more elastic,
if the good is a luxury.
the longer the time period.
the larger the number of close substitutes.
the more narrowly defined the market.
Computing the
Price Elasticity of Demand
Example,If the price of an ice cream cone
increases from $2.00 to $2.20 and the amount
you buy falls from 10 to 8 cones then your
elasticity of demand would be calculated as:
P r i c e E l a s ti c i ty o f D e m a n d =
P e r c e n ta g e C h a n g e
i n Q u a n ti t y D e m a n d e d
P e r c e n ta g e C h a n g e
i n P r i c e
P r i c e E l a s ti c i ty o f D e a n d =
P e r c e n ta g e C h a n g e
i n Q u a n ti t y D e a n d e d
P e r c e n ta g e C h a n g e
i n P r i c e
2
p e r c e n t10
p e r c e n t20
100
002
002202
100
10
810
.
)..(
)(
Computing the Price Elasticity of
Demand Using the Midpoint Formula
The midpoint formula is preferable when
calculating the price elasticity of demand
because it gives the same answer regardless of
the direction of the change.
) / 2 ]P) / [ ( PP(P
) / 2 ]Q) / [ ( QQ(Q=D e m a n d of E l a s t i c i t y P r i c e
1212
1212
Computing the Price Elasticity of
Demand Using the Midpoint Formula
Example,If the price of an ice cream cone
increases from $2.00 to $2.20 and the amount
you buy falls from 10 to 8 cones the your
elasticity of demand,using the midpoint
formula,would be calculated as:
32.2
5.9
22
2/)20.200.2(
)00.220.2(
2/)810(
)810(
p e r c e n t
p e r c e n t
Ranges of Elasticity
Inelastic Demand
– Price elasticity of demand is less than one
Elastic Demand
– Price elasticity of demand is greater than one
Perfect Inelastic
– Quantity demanded does not respond to price changes
Perfect Elastic
– Quantity demanded changes infinitely with any change in price.
Unit Elastic
– Quantity demanded changes by the same percentage as
the price
Perfectly Inelastic Demand
- Elasticity equals 0
Quantity
Price
4
$5
Demand
100
2.,..leaves the quantity demanded unchanged.
1,An
increase
in price...
Inelastic Demand
- Elasticity is less than 1
Quantity
Price
4
$51,A 22%
increase
in price...
Demand
10090
2.,..leads to a 11% decrease in quantity.
Unit Elastic Demand
- Elasticity equals 1
Quantity
Price
4
$51,A 22%
increase
in price...
Demand
10080
2.,..leads to a 22% decrease in quantity.
Elastic Demand
- Elasticity is greater than 1
Quantity
Price
4
$
51,A 22%increase
in price...
Demand
10050
2.,..leads to a 67% decrease in quantity.
Perfectly Elastic Demand
- Elasticity equals infinity
Quantity
Price
Demand$4
1,At any price
above $4,quantity
demanded is zero.
2,At exactly $4,
consumers will
buy any quantity.
3,At a price below $4,
quantity demanded is infinite.
Elasticity and Total Revenue
Total revenue is the amount paid by
buyers and received by sellers of a good.
Computed as the price of the good
times the quantity sold.
TR = P x Q
$4
Demand
Quantity
P
0
Price
P x Q = $400
(total revenue)
100
Q
Elasticity and Total Revenue
With an inelastic demand
curve,an increase in price
leads to a decrease in
quantity that is
proportionately smaller,
Thus,total revenue increases.
Elasticity and Total Revenue
$3
Quantity0
Price
80
Revenue = $240
Demand$1 Demand
Quantity0
Revenue = $100
100
Price
An increase in price
from $1 to $3...
…leads to an increase
in total revenue
from$100 to $240
Elasticity and Total Revenue,
Inelastic Demand
Elasticity and Total Revenue
With an elastic demand curve,
an increase in the price leads to
a decrease in quantity
demanded that is
proportionately larger,Thus,
total revenue decreases.
Demand
Quantity0
Price
$4
50
Demand
Quantity0
Price
Revenue = $100
$5
20
Revenue = $200
An increase in price
from $4 to $5...
…leads to a decrease
in total revenue
from$200 to $100
Elasticity and Total Revenue,
Elastic Demand
Questions
If you are the curator of the National
Museum,and your financial manager tells
you that you have not enough money to
renew the facilities and advise you to raise
the ticket price to make more money,what
will you do?
If you are a manager of a cinema,then
what will you do in the same situation?
Some Price Elasticities
in the U.S,Economy
Elastic demands
– Purchased meals 2.27
– Metals 1.52
– Furnitures,timber 1.25
– Motor,Vehicles 1.14
– Transportation 1.03
Inelastic demands
– Gas,electricity,water 0.92
– Oil 0.91
– Chemicals 0.89
Some Price Elasticities
in the U.S,Economy
Inelastic demands (contd.)
– Beverages 0.78
– Tobacco 0.61
– Food 0.58
– Housing service 0.55
– Clothing 0.49
– Books,magazines,newspapers 0.34
– Meat 0.2
P r i c e Q ua nt i t y
Tot a l R e v e nue
( P r i c e x
Q ua nt i t y )
P e r c e nt C ha nge
i n P r i c e
P e r c e nt
C ha nge
i n Q ua nt i t y E l a s t i c i t y D e s c r i pt i on
$0 14 $0 200% 15% 0,1 I ne l a s t i c
1 12 12 67 18 0,3 I ne l a s t i c
2 10 20 40 22 0,6 I ne l a s t i c
3 8 24 29 29 1 U ni t e l a s t i c
4 6 24 22 40 1,8 e l a s t i c
5 4 20 18 67 3,7 e l a s t i c
6 2 12 15 200 13 e l a s t i c
7 0 0
Computing the Elasticity of a
Linear Demand Curve
Income Elasticity of Demand
Income elasticity of demand measures
how much the quantity demanded of a
good responds to a change in consumers’
income,
It is computed as the percentage change in
the quantity demanded divided by the
percentage change in income.
Computing Income Elasticity
Income Elasticity
of Demand
Percentage Change
in Quantity Demanded
Percentage Change
in Income
=
Income Elasticity
- Types of Goods -
Normal Goods
Inferior Goods
Higher income raises the quantity
demanded for normal goods but lowers
the quantity demanded for inferior
goods.
Goods consumers regard as necessities
tend to be income inelastic
Examples include food,fuel,clothing,
utilities,and medical services.
Goods consumers regard as luxuries tend
to be income elastic.
Examples include sports cars,furs,and
expensive foods.
Income Elasticity
- Types of Goods -
Application
Whose expenditure weight will rise when
a economy develops and people get rich?
Necessities or Luxuries?
Luxuries!
Example in China
– 80s,TV set,Refrigerator,Washer
– 90s,Air conditioner,Car,PC
– NOW,House,Tourism
1992年去丽江的游客为 20万人,2005年超过 400万。
九寨沟 1990年 15万游客,2006年超过 230万人
Cross-price elasticity of demand
A measure of how much the quantity demanded
of one good responds to a change in the price of
another good.
computed as the percentage change in quantity
demanded of the first good divided by the
percentage change in the price of the second good
Price Elasticity of Supply
Price elasticity of supply is the
percentage change in quantity supplied
resulting from a percent change in price.
It is a measure of how much the
quantity supplied of a good responds to
a change in the price of that good.
Ranges of Elasticity
Perfectly Elastic
ES =?
Relatively Elastic
ES > 1
Unit Elastic
ES = 1
Relatively Inelastic
ES < 1
Perfectly Inelastic
ES = 0
Perfectly Inelastic Supply
- Elasticity equals 0
Quantity
Price
4
$5
Supply
100
2.,..leaves the quantity supplied unchanged.
1,An
increase
in price...
Quantity
Price
4
$51,A 22%
increase
in price...
110100
Supply
2.,..leads to a 10% increase in quantity.
Inelastic Supply
- Elasticity is less than 1
Quantity
Price
4
$51,A 22%
increase
in price...
125100
Supply
2.,..leads to a 22% increase in quantity.
Unit Elastic Supply
- Elasticity equals 1
Quantity
Price
4
$51,A 22%
increase
in price...
200100
Supply
2.,..leads to a 67% increase in quantity.
Elastic Supply
- Elasticity is greater than 1
Quantity
Price
Supply$4
1,At any price
above $4,quantity
supplied is infinite.
2,At exactly $4,
producers will
supply any quantity.
3,At a price below $4,
quantity supplied is zero.
Perfect Elastic Supply
- Elasticity equals infinity
Ability of sellers to change the amount
of the good they produce.
– Beach-front land is inelastic.
– Books,cars,or manufactured goods are
elastic.
Time period,
– Supply is more elastic in the long run.
Determinants of
Elasticity of Supply
Computing the Price
Elasticity of Supply
The price elasticity of supply is computed
as the percentage change in the quantity
supplied divided by the percentage
change in price.E l a s t i c i t y o f S u p p l y =
P e r c e n t a g e C h a n g e i n
Q u a n t i t y S u p p l i e d
P e r c e n t a g e C h a n g e
i n P r i c e
E l a s t i c i t y o f S u p p l y =
P e r c e n t a g e C h a n g e i n
Q u a n t i t y S u p p l i e d
P e r c e n t a g e C h a n g e
i n P r i c e
Application of Elasticity
Can good news for farming be bad news
for farmers?
What happens to wheat farmers and the
market for wheat when university
agronomists discover a new wheat hybrid
that is more productive than existing
varieties?
Application of Elasticity
Examine whether the supply or demand
curve shifts.
Determine the direction of the shift of the
curve.
Use the supply-and-demand diagram to
see how the market equilibrium changes.
3.,..and a proportionately smaller
increase in quantity sold,As a result,
revenue falls from $300 to $220.
An Increase in Supply in the
Market for Wheat
$3
Quantity of Wheat1000
Price of
Wheat 1,When demand is inelastic,an increase in supply...
Demand
S1 S2
2
110
2.,..leads
to a large
fall in
price...
Compute Elasticity
- 0,2 4
0,4
0,0 9 5-
2,0 0 ) / 2( 3,0 0
2,0 0-3,0 0
1 1 0 ) / 2( 1 0 0
1 1 0-1 0 0
=E
D
Demand is inelastic
Policy Implications
Increase in supply of agriculture products
and decrease in quantity of farmers are
inevitable with the progress of agriculture
technology (or institution)
17% of American are farmers in 1948,but
only 2% in 1993,and the production are
doubled.
84% of labor force worked on agriculture in
China in 1952,only 50% in 1999 and the
production are tripled.
Why Did OPEC Fail to Keep the
Price of Oil High?
Supply and Demand can behave
differently in the short run and the long
run
In the short run,both supply and demand
for oil are relatively inelastic
But in the long run,both are elastic
– Production outside of OPEC
– More conservation by consumers
Does Drug Interdiction Increase or
Decrease Drug-Related Crime?
Drug interdiction impacts sellers rather
than buyers.
– Demand is unchanged.
– Equilibrium price rises although quantity falls.
Drug education impacts the buyers rather
than sellers.
– Demand is shifted.
– Equilibrium price and quantity are lowered.
Policies to Reduce the Use of Illegal
Drugs
Price of Drugs
Quantity of Drugs
Price of Drugs
Quantity of Drugs
Drug Interdiction Drug Education
D2D1
D1
S2
S1S1
The demand for illegal
drugs is inelastic.
Interdiction shifts the supply,while education shifts the demand.In each case,the change in pr ce is the same,But in one market the price goes up.
And in the other it goes down.
The changes in quantities (and TR) are
remarkable.
It is amazing
how useful
knowledge of
elasticities can be!
Summary
Price elasticity of demand measures how
much the quantity demanded responds to
changes in the price,
If a demand curve is elastic,total revenue
falls when the price rises,
If it is inelastic,total revenue rises as the
price rises,
Summary
The price elasticity of supply measures
how much the quantity supplied responds
to changes in the price,
In most markets,supply is more elastic in
the long run than in the short run,
The tools of supply and demand can be
applied in many different types of markets.
Exercise #5
Problems and Applications
– #2,#3,#4,#6,#10
Chapter 5
Elasticity…
… is a measure of how much buyers and
sellers respond to changes in market
conditions
… allows us to analyze supply and
demand with greater precision.
Price Elasticity of Demand
Price elasticity of demand is the
percentage change in quantity demanded
given a percent change in the price,
It is a measure of how much the quantity
demanded of a good responds to a change
in the price of that good.
Determinants of
Price Elasticity of Demand
Necessities versus Luxuries
Availability of Close Substitutes
Definition of the Market
Time Horizon
Determinants of
Price Elasticity of Demand
Demand tends to be more elastic,
if the good is a luxury.
the longer the time period.
the larger the number of close substitutes.
the more narrowly defined the market.
Computing the
Price Elasticity of Demand
Example,If the price of an ice cream cone
increases from $2.00 to $2.20 and the amount
you buy falls from 10 to 8 cones then your
elasticity of demand would be calculated as:
P r i c e E l a s ti c i ty o f D e m a n d =
P e r c e n ta g e C h a n g e
i n Q u a n ti t y D e m a n d e d
P e r c e n ta g e C h a n g e
i n P r i c e
P r i c e E l a s ti c i ty o f D e a n d =
P e r c e n ta g e C h a n g e
i n Q u a n ti t y D e a n d e d
P e r c e n ta g e C h a n g e
i n P r i c e
2
p e r c e n t10
p e r c e n t20
100
002
002202
100
10
810
.
)..(
)(
Computing the Price Elasticity of
Demand Using the Midpoint Formula
The midpoint formula is preferable when
calculating the price elasticity of demand
because it gives the same answer regardless of
the direction of the change.
) / 2 ]P) / [ ( PP(P
) / 2 ]Q) / [ ( QQ(Q=D e m a n d of E l a s t i c i t y P r i c e
1212
1212
Computing the Price Elasticity of
Demand Using the Midpoint Formula
Example,If the price of an ice cream cone
increases from $2.00 to $2.20 and the amount
you buy falls from 10 to 8 cones the your
elasticity of demand,using the midpoint
formula,would be calculated as:
32.2
5.9
22
2/)20.200.2(
)00.220.2(
2/)810(
)810(
p e r c e n t
p e r c e n t
Ranges of Elasticity
Inelastic Demand
– Price elasticity of demand is less than one
Elastic Demand
– Price elasticity of demand is greater than one
Perfect Inelastic
– Quantity demanded does not respond to price changes
Perfect Elastic
– Quantity demanded changes infinitely with any change in price.
Unit Elastic
– Quantity demanded changes by the same percentage as
the price
Perfectly Inelastic Demand
- Elasticity equals 0
Quantity
Price
4
$5
Demand
100
2.,..leaves the quantity demanded unchanged.
1,An
increase
in price...
Inelastic Demand
- Elasticity is less than 1
Quantity
Price
4
$51,A 22%
increase
in price...
Demand
10090
2.,..leads to a 11% decrease in quantity.
Unit Elastic Demand
- Elasticity equals 1
Quantity
Price
4
$51,A 22%
increase
in price...
Demand
10080
2.,..leads to a 22% decrease in quantity.
Elastic Demand
- Elasticity is greater than 1
Quantity
Price
4
$
51,A 22%increase
in price...
Demand
10050
2.,..leads to a 67% decrease in quantity.
Perfectly Elastic Demand
- Elasticity equals infinity
Quantity
Price
Demand$4
1,At any price
above $4,quantity
demanded is zero.
2,At exactly $4,
consumers will
buy any quantity.
3,At a price below $4,
quantity demanded is infinite.
Elasticity and Total Revenue
Total revenue is the amount paid by
buyers and received by sellers of a good.
Computed as the price of the good
times the quantity sold.
TR = P x Q
$4
Demand
Quantity
P
0
Price
P x Q = $400
(total revenue)
100
Q
Elasticity and Total Revenue
With an inelastic demand
curve,an increase in price
leads to a decrease in
quantity that is
proportionately smaller,
Thus,total revenue increases.
Elasticity and Total Revenue
$3
Quantity0
Price
80
Revenue = $240
Demand$1 Demand
Quantity0
Revenue = $100
100
Price
An increase in price
from $1 to $3...
…leads to an increase
in total revenue
from$100 to $240
Elasticity and Total Revenue,
Inelastic Demand
Elasticity and Total Revenue
With an elastic demand curve,
an increase in the price leads to
a decrease in quantity
demanded that is
proportionately larger,Thus,
total revenue decreases.
Demand
Quantity0
Price
$4
50
Demand
Quantity0
Price
Revenue = $100
$5
20
Revenue = $200
An increase in price
from $4 to $5...
…leads to a decrease
in total revenue
from$200 to $100
Elasticity and Total Revenue,
Elastic Demand
Questions
If you are the curator of the National
Museum,and your financial manager tells
you that you have not enough money to
renew the facilities and advise you to raise
the ticket price to make more money,what
will you do?
If you are a manager of a cinema,then
what will you do in the same situation?
Some Price Elasticities
in the U.S,Economy
Elastic demands
– Purchased meals 2.27
– Metals 1.52
– Furnitures,timber 1.25
– Motor,Vehicles 1.14
– Transportation 1.03
Inelastic demands
– Gas,electricity,water 0.92
– Oil 0.91
– Chemicals 0.89
Some Price Elasticities
in the U.S,Economy
Inelastic demands (contd.)
– Beverages 0.78
– Tobacco 0.61
– Food 0.58
– Housing service 0.55
– Clothing 0.49
– Books,magazines,newspapers 0.34
– Meat 0.2
P r i c e Q ua nt i t y
Tot a l R e v e nue
( P r i c e x
Q ua nt i t y )
P e r c e nt C ha nge
i n P r i c e
P e r c e nt
C ha nge
i n Q ua nt i t y E l a s t i c i t y D e s c r i pt i on
$0 14 $0 200% 15% 0,1 I ne l a s t i c
1 12 12 67 18 0,3 I ne l a s t i c
2 10 20 40 22 0,6 I ne l a s t i c
3 8 24 29 29 1 U ni t e l a s t i c
4 6 24 22 40 1,8 e l a s t i c
5 4 20 18 67 3,7 e l a s t i c
6 2 12 15 200 13 e l a s t i c
7 0 0
Computing the Elasticity of a
Linear Demand Curve
Income Elasticity of Demand
Income elasticity of demand measures
how much the quantity demanded of a
good responds to a change in consumers’
income,
It is computed as the percentage change in
the quantity demanded divided by the
percentage change in income.
Computing Income Elasticity
Income Elasticity
of Demand
Percentage Change
in Quantity Demanded
Percentage Change
in Income
=
Income Elasticity
- Types of Goods -
Normal Goods
Inferior Goods
Higher income raises the quantity
demanded for normal goods but lowers
the quantity demanded for inferior
goods.
Goods consumers regard as necessities
tend to be income inelastic
Examples include food,fuel,clothing,
utilities,and medical services.
Goods consumers regard as luxuries tend
to be income elastic.
Examples include sports cars,furs,and
expensive foods.
Income Elasticity
- Types of Goods -
Application
Whose expenditure weight will rise when
a economy develops and people get rich?
Necessities or Luxuries?
Luxuries!
Example in China
– 80s,TV set,Refrigerator,Washer
– 90s,Air conditioner,Car,PC
– NOW,House,Tourism
1992年去丽江的游客为 20万人,2005年超过 400万。
九寨沟 1990年 15万游客,2006年超过 230万人
Cross-price elasticity of demand
A measure of how much the quantity demanded
of one good responds to a change in the price of
another good.
computed as the percentage change in quantity
demanded of the first good divided by the
percentage change in the price of the second good
Price Elasticity of Supply
Price elasticity of supply is the
percentage change in quantity supplied
resulting from a percent change in price.
It is a measure of how much the
quantity supplied of a good responds to
a change in the price of that good.
Ranges of Elasticity
Perfectly Elastic
ES =?
Relatively Elastic
ES > 1
Unit Elastic
ES = 1
Relatively Inelastic
ES < 1
Perfectly Inelastic
ES = 0
Perfectly Inelastic Supply
- Elasticity equals 0
Quantity
Price
4
$5
Supply
100
2.,..leaves the quantity supplied unchanged.
1,An
increase
in price...
Quantity
Price
4
$51,A 22%
increase
in price...
110100
Supply
2.,..leads to a 10% increase in quantity.
Inelastic Supply
- Elasticity is less than 1
Quantity
Price
4
$51,A 22%
increase
in price...
125100
Supply
2.,..leads to a 22% increase in quantity.
Unit Elastic Supply
- Elasticity equals 1
Quantity
Price
4
$51,A 22%
increase
in price...
200100
Supply
2.,..leads to a 67% increase in quantity.
Elastic Supply
- Elasticity is greater than 1
Quantity
Price
Supply$4
1,At any price
above $4,quantity
supplied is infinite.
2,At exactly $4,
producers will
supply any quantity.
3,At a price below $4,
quantity supplied is zero.
Perfect Elastic Supply
- Elasticity equals infinity
Ability of sellers to change the amount
of the good they produce.
– Beach-front land is inelastic.
– Books,cars,or manufactured goods are
elastic.
Time period,
– Supply is more elastic in the long run.
Determinants of
Elasticity of Supply
Computing the Price
Elasticity of Supply
The price elasticity of supply is computed
as the percentage change in the quantity
supplied divided by the percentage
change in price.E l a s t i c i t y o f S u p p l y =
P e r c e n t a g e C h a n g e i n
Q u a n t i t y S u p p l i e d
P e r c e n t a g e C h a n g e
i n P r i c e
E l a s t i c i t y o f S u p p l y =
P e r c e n t a g e C h a n g e i n
Q u a n t i t y S u p p l i e d
P e r c e n t a g e C h a n g e
i n P r i c e
Application of Elasticity
Can good news for farming be bad news
for farmers?
What happens to wheat farmers and the
market for wheat when university
agronomists discover a new wheat hybrid
that is more productive than existing
varieties?
Application of Elasticity
Examine whether the supply or demand
curve shifts.
Determine the direction of the shift of the
curve.
Use the supply-and-demand diagram to
see how the market equilibrium changes.
3.,..and a proportionately smaller
increase in quantity sold,As a result,
revenue falls from $300 to $220.
An Increase in Supply in the
Market for Wheat
$3
Quantity of Wheat1000
Price of
Wheat 1,When demand is inelastic,an increase in supply...
Demand
S1 S2
2
110
2.,..leads
to a large
fall in
price...
Compute Elasticity
- 0,2 4
0,4
0,0 9 5-
2,0 0 ) / 2( 3,0 0
2,0 0-3,0 0
1 1 0 ) / 2( 1 0 0
1 1 0-1 0 0
=E
D
Demand is inelastic
Policy Implications
Increase in supply of agriculture products
and decrease in quantity of farmers are
inevitable with the progress of agriculture
technology (or institution)
17% of American are farmers in 1948,but
only 2% in 1993,and the production are
doubled.
84% of labor force worked on agriculture in
China in 1952,only 50% in 1999 and the
production are tripled.
Why Did OPEC Fail to Keep the
Price of Oil High?
Supply and Demand can behave
differently in the short run and the long
run
In the short run,both supply and demand
for oil are relatively inelastic
But in the long run,both are elastic
– Production outside of OPEC
– More conservation by consumers
Does Drug Interdiction Increase or
Decrease Drug-Related Crime?
Drug interdiction impacts sellers rather
than buyers.
– Demand is unchanged.
– Equilibrium price rises although quantity falls.
Drug education impacts the buyers rather
than sellers.
– Demand is shifted.
– Equilibrium price and quantity are lowered.
Policies to Reduce the Use of Illegal
Drugs
Price of Drugs
Quantity of Drugs
Price of Drugs
Quantity of Drugs
Drug Interdiction Drug Education
D2D1
D1
S2
S1S1
The demand for illegal
drugs is inelastic.
Interdiction shifts the supply,while education shifts the demand.In each case,the change in pr ce is the same,But in one market the price goes up.
And in the other it goes down.
The changes in quantities (and TR) are
remarkable.
It is amazing
how useful
knowledge of
elasticities can be!
Summary
Price elasticity of demand measures how
much the quantity demanded responds to
changes in the price,
If a demand curve is elastic,total revenue
falls when the price rises,
If it is inelastic,total revenue rises as the
price rises,
Summary
The price elasticity of supply measures
how much the quantity supplied responds
to changes in the price,
In most markets,supply is more elastic in
the long run than in the short run,
The tools of supply and demand can be
applied in many different types of markets.
Exercise #5
Problems and Applications
– #2,#3,#4,#6,#10
