lesson 10 – elasticity of demands - uhmmsosa/math1314/notes/1314day12.pdf · lesson 10 –...

Lesson 10 – Elasticity of Demands

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Lesson 10 - Elasticity of Demand

Suppose you owned a small business and needed to make some decisions about the pricing of your products. It would be helpful to know what effect a small change in price would have on the demand for your product. If a price change will have no real change on demand for the product, it might make good sense to raise the price. However, if a price increase will cause a big drop in demand, then it may not be a good idea to raise prices. There is a measure of the responsiveness of demand for product or service to a change in its price: elasticity of

demand. This is defined as

percentage change in demand

percentage change in price.

To develop this formula, we’ll start by solving our demand function for x, so that we have a function ( )x f p= .

Then we have a demand function in terms of price. If we increase the price by h dollars, then the price is p h+

and the quantity demanded is ( )f p h+ .

The percentage change in demand is ( ) ( )

100( )

f p h f p

f p

+ −⋅ and the percentage change in price is 100h

p⋅ .

If we compute the ratio given above, we have

( ) ( )100

( )

100

f p h f p

f p

h

p

+ −⋅

⋅.

We can simplify this to

( ) ( )

( )

p f p h f p

f p h

+ −

.

For small values of h, ( ) ( )

( )f p h f p

f ph

+ − ′≈ , so we have ( )

( )

p f p

f p

′⋅.

This quantity is almost always negative, and it would be much easier to work with a positive quantity. So the negative of this ratio is the elasticity of demand.

Elasticity of Demand

( )( )

( )

p f pE p

f p

′⋅= − , where p is price and f ( p) is the demand function and is differentiable at x = p .

Revenue responds to elasticity in the following manner: If demand is elastic at p, then

• An increase in unit price will cause revenue to decrease or

• A decrease in unit price will cause revenue to increase


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• An increase/decrease in unit price will cause the revenue to stay about the same. If demand is inelastic at p, then

• An increase in the unit price will cause revenue to increase

• A decrease in unit price will cause revenue to decrease. We have these generalizations about elasticity of demand:

Demand is said to be elastic if ( ) 1E p > .

Demand is said to be unitary if ( ) 1E p = .

Demand is said to be inelastic if ( ) 1E p < .

So, if demand is elastic, then the change in revenue and the change in price will move in opposite directions. If demand is inelastic, then the change in revenue and the change in price will move in the same direction.

Example 1: Find ( )E p for the demand function 2 15 0x p+ − = and determine if demand is elastic, inelastic

or unitary when 4.p =

Example 2: Suppose the demand function for a product is given by 0.02 400p x= − + . This function gives the

unit price in dollars when x units are demanded.

a. Find the elasticity of demand.


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b. Find (100)E and interpret the results.

c. Find (300)E and interpret the results.

d. If the unit price is $100, will raising the price result in an increase in revenues or a decrease in revenues? e. If the unit price is $300, will raising the price result in an increase in revenues or a decrease in revenues?

What else can ( )E p tell you?

Example 3: If 1

( )2

E p = when 250p = , what effect will a 1% increase in price have on revenue?


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Example 4: If 3

( )2

E p = when 250p = , what effect will a 1% increase in price have on revenue?

Question 35: If demand is elastic at p, then a decrease in unit price will cause revenue to increase.

a. True b. False

Lesson 11 – Exponential Functions as Mathematical Models 1

Math 1314

Lesson 11: Exponential Functions as Mathematical Models

Exponential models can be written in two forms: ( ) xf x a e= ⋅ or ( ) x

g x a b= ⋅ . In the first

model, the base of the exponent is the number e, which is approximately 2.71828…. In the

second model, the base of the exponent is a positive number other than 1. In both cases, the

variable is located in the exponent, and that’s why these are called exponential models.

Exponential functions can be either increasing or decreasing.

The function ( ) bxf x a e= ⋅ is an exponential growth function and it’s increasing.

The function ( ) bxf x a e

−= ⋅ is an exponential decay function and it’s decreasing.

In both equations the variable a is called the initial amount and b is called the growth or decay

constant, depending on the type of function.

For a function of the form ( ) xg x a b= ⋅ , the function is an exponential growth function if 1b >

and is an exponential decay function if 0 1b< < .

We can also compute the rate at which an exponential function is increasing or decreasing.

We’ll do this by finding a numerical derivative.

We can use the regression feature to find an exponential equation for data that’s given.

Question 2 is D

Example 1: Identify each function as a growth function or a decay function. Find the initial

value. Calculate (30)f and '(30)f .

a. 0.285( ) 12.8 tf t e

−= b. ( ) 38.6(1.0489)xg x =

Example 2: Suppose the points (0, 10000) and (2, 10940) lie on the graph of a function. Find

the equation of the function in the form ( ) bxf x a e= ⋅ using GeoGebra, assuming that the

function is exponential.

Begin by entering the two given points in the spreadsheet, then make a list.

Command: Answer:


Uninhibited Exponential Growth

Some common exponential applications model uninhibited exponential growth. This means

that there is no “upper limit” on the value of the function. It can simply keep growing and

growing. Problems of this type include population growth problems and growth of investment

assets.

Example 3: A biologist wants to study the growth of a certain strain of bacteria. She starts

with a culture containing 25,000 bacteria. After three hours, the number of bacteria has grown to

63,000. Assume the population grows exponentially and the growth is uninhibited.

a. Find the equation of the function in the form ( ) bxf x a e= ⋅ using GGB.

State the two points given in the problem.

Enter the two points in the spreadsheet, then make a list.

Command: Answer:

b. How many bacteria will be present in the culture 6 hours after she started her study?

Question 26: Given the following function, state the initial value for the function.

�� 54.65�.� ��

a. 0.3541

b. 54.65


Example 4: The sales from company ABC for the years 1998 – 2003 are given below.

Year 1998 1999 2000 2001 2002 2003

profits in millions of dollars 51.4 53.2 55.8 56.1 58.1 59.0

Rescale the data so that x = 0 corresponds to 1998. Begin by making a list.

a. Find an exponential regression model for the data.

Command: Answer:

b. Find the rate at which the company's sales were changing in 2007.

Command: Answer:


Exponential Decay

Example 5: At the beginning of a study, there are 50 grams of a substance present. After 17

days, there are 38.7 grams remaining. Assume the substance decays exponentially.

a. State the two points given in the problem.

Enter the two points in the spreadsheet and make a list.

b. Find an exponential regression model.

Command: Answer:

c. What will be the rate of decay on day 40 of the study?

Command: Answer:

Exponential decay problems frequently involve the half-life of a substance. The half-life of a

substance is the time it takes to reduce the amount of the substance by one-half.

Example 6: A certain drug has a half-life of 4 hours. Suppose you take a dose of 1000

milligrams of the drug.




Command: Answer:

c. How much of it is left in your bloodstream 28 hours later?

Command: Answer:


Example 7: The half-life of Carbon 14 is 5770 years.




Command: Answer:

c. Bones found from an archeological dig were found to have 22% of the amount of Carbon 14

that living bones have. Find the approximate age of the bones.

Command: Answer:

Question 37: Identify the type of exponential function.

( ) 38(0.489)xg x =

a. Growth

b. Decay

c. Neither


Limited Growth Models

Some exponential growth is limited; here’s an example:

A worker on an assembly line performs the same task repeatedly throughout the workday. With

experience, the worker will perform at or near an optimal level. However, when first learning to

do the task, the worker’s productivity will be much lower. During these early experiences, the

worker’s productivity will increase dramatically. Then, once the worker is thoroughly familiar

with the task, there will be little change to his/her productivity.

The function that models this situation will have the form ( ) ktQ t C Ae−= − .

This model is called a learning curve and the graph of the function will look something like

this:

The graph will have a y intercept at C – A and a horizontal asymptote at y = C. Because of the

horizontal asymptote, we know that this function does not model uninhibited growth.

Example 8: Suppose your company’s HR department determines that an employee will be able

to assemble 0.5( ) 50 30 tQ t e−= − products per day, t months after the employee starts working on

the assembly line. Enter the function in GGB.

a. How many units can a new employee assemble as s/he starts the first day at work?

b. How many units should an employee be able to assemble after two months at work?

c. How many units should an experienced worker be able to assemble?

d. At what rate is an employee’s productivity changing 4 months after starting to work?


Logistic Functions

The last growth model that we will consider involves the logistic function. The general form of

the equation is ( )1 kt

AQ t

Be−=+

and the graph looks something like this:

If we looked at this graph up to around x = 2 and didn’t consider the rest of it, we might think

that the data modeled was exponential. Logistic functions typically reach a saturation point – a

point at which the growth slows down and then eventually levels off. The part of this graph to

the right of x = 2 looks more like our learning curve graph from the last example. Logistic

functions have some of the features of both types of models.

Note that the graph has a limiting value at y = 5. In the context of a logistic function, this

asymptote is called the carrying capacity. In general the carrying capacity is A from the

formula above.

Logistic curves are used to model various types of phenomena and other physical situations such

as population management. Suppose a number of animals are introduced into a protected game

reserve, with the expectation that the population will grow. Various factors will work together to

keep the population from growing exponentially (in an uninhibited manner). The natural

resources (food, water, protection) may not exist to support a population that gets larger without

bound. Often such populations grow according to a logistic model.

Example 9: A population study was commissioned to determine the growth rate of the fish

population in a certain area of the Pacific Northwest. The function given below models the

population where t is measured in years and N is measured in millions of tons. Enter the

function in GGB.

0.338

2.4( )

1 2.39 tN t

e−=+

a. What was the initial number of fish in the population?


b. What is the carrying capacity in this population?

c. What is the fish population after 3 years?

d. How fast is the fish population changing after 2 years?

Question 13: Identify the carrying capacity of the following equation.

0.05

31250( )

1 125 tN t

e−=

+

a. 125

b. 31250

c. 0.05

d. 126

e. None of the above

lesson 10 – elasticity of demands - uhmmsosa/math1314/notes/1314day12.pdf · lesson 10 –...

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