Lecture 2

Economic Models, Functions, Logs, Exponents, e

Variables, Constant, Parameters Variables: magnitude can change

Price, profit, revenue…Represented by symbolsCan be ‘frozen’ by setting value

Try to setup models to obtain solutions to variables

Endogenous variable: determined by model Exogenous variable: determined outside model

Variables, Constant, Parameters

Constants, do not change, but can be joined with variables.Called coefficientsEx: 1000q, 1000 is constant, quantity is variable

When constants are not set?Ex: βq, where β stands for the coefficient and q for

quantityNow, β can change! A ‘variable’ constant is called a

parameter.

Equation vs. Identity

An identity, is a definition, like profit, revenue, and cost where π is profit, R is revenue, C is costBehavioral equations

Specify how variables interact

Conditional equations Optimization and Equilibrium conditions are examples

R C

R pq C wl rk

d s

MC MR

Q Q

Real Numbers Rational numbers: (‘ratio’)

Whole numbers: + IntegersFractions

Irrational numbersCan’t be expressed as a fractionNonrepeating, nonterminating decimals

Ex:

Real Numbers combine rational and irrational Non-imaginary, i=2

1

Set Properties and Set Notation Definition: A set is any collection of objects specified

in such a way that we can determine whether a given object is or is not in the collection.

Notation: e A

means “e is an element of A”, or “e belongs to set A”. The notation

e A means “e is not an element of A”.

Null Set

Example. What are the real number solutions of the equation

x2 + 1 = 0? There is no answer to this… you can write this as

{ }, [], or .

Set Builder Notation Sometimes it is convenient to represent sets using set builder

notation. For example, instead of representing the set A (letters in the alphabet) by the roster method, we can use A = {x | x is a letter of the English alphabet}

which means the same as A = {a , b, c, d, e, …, z} In statistics… {x|a} would be read x, given a.Here, we will use the vertical line in set notation Example.

{x | x2 = 9} = {3, -3}This is read as “the set of all x such that the square of x equals 9.” This set consists of the two numbers 3 and -3.

Union of Sets

The union of two sets A and B is the set of all elements formed by combining all the elements of set A and all the elements of set B into one set. It is written A B.

AA BB

In the Venn diagram on the left, the union of A and B is

the entire region shaded.

Intersection of Sets

The intersection of two sets A and B is the set of all elements that are common to both A and B. It is written A B.

AA BB In the Venn diagram on the left, the intersection of A

and B is the shaded region.

The Complement of a SetThe complement of a set A is defined as the set of elements that are contained in U, the universal set, but not contained in set A. The symbolism and notation for the complement of set A are

}{' AxUxA

In the Venn diagram on the left, the rectangle represents the

universe. A is the shaded area outside the set A.

ApplicationA marketing survey of 1,000 commuters found that 600 answered listen to the news, 500 listen to music, and 300

listen to both. Let N = set of commuters in the sample who listen to news and M = set of commuters in the sample who listen to music. Find the number of commuters in the set

The number of elements in a set A is denoted by n(A), so in this case we are looking for

'N M

( ')n N M

SolutionThe study is based on 1000 commuters, so n(U)=1000.

The number of elements in the four sections in the Venn diagram need to add up to 1000.

The red part represents the commuters who listen to both news and music. It has 300 elements.

The set N (news listeners) consists of a green part and a red part. N has 600 elements, the red part has 300, so the

green part must also be 300.

Continue in this fashion.

Solution(continued)

N M

'N Mis the green part,

which contains 300 commuters.

200 people listen to neither news nor

music

U

300 listen to news but

not music.

200 listen

to music

but not news

300 listen to both music and news

Functions The previous graph is the graph of a function. The idea of a

function is this: a correspondence between two sets D and R such that to each element of the first set, D, there corresponds one and only one element of the second set, R.

The first set is called the domain, and the set of corresponding elements in the second set is called the range.For example, the cost of a pizza (C) is related to the size of the pizza. A 10 inch diameter pizza costs $9.00, while a 16 inch diameter pizza costs $12.00.

Function Definition You can visualize a function by the following diagram which

shows a correspondence between two sets: D, the domain of the function, gives the diameter of pizzas, and R, the range of the function gives the cost of the pizza.

10

1216

9.00

12.00

10.00

domain D or x

range R or f(x)

Functions Specified by Equations Consider the equation

2 2y x -2

2

Input: x = -2

Process: square (–2),then subtract 2

Output: result is 2

(-2,2) is an ordered pair of the function.

Vertical Line Test for a Function

If you have the graph of an equation, there is an easy way to determine if it is the graph of an function. It is called the vertical line test which states that:

An equation defines a function if each vertical line in the coordinate system passes through at most one point on the graph of the equation. If any vertical line passes through two or more points on the graph of an equation, then the equation does not define a function.

Vertical Line Test for a Function(continued)

This graph is not the graph of a function because you can draw a vertical line which crosses it twice.

This is the graph of a function because any vertical line crosses only once.

Function Notation The following notation is used to describe functions. The

variable y will now be called f (x).

This is read as “ f of x” and simply means the y coordinate of the function corresponding to a given x value. Our previous equation

can now be expressed as

2 2y x

2( ) 2f x x

Function Evaluation

Consider our function

What does f (-3) mean?

2( ) 2f x x

Function Evaluation

Consider our function

What does f (-3) mean? Replace x with the value –3 and evaluate the expression

The result is 11 . This means that the point (-3,11) is on the graph of the function.

2( ) 2f x x

2( 3) ( 3) 2f

Some Examples

1.

KEEP THIS ‘h’ example in mind for derivatives

( ) 3( ) 2af a

( ) 3( ) 2 18 3 2

1 3

6

6

6h hf h

h

Domain of a Function

Consider

which is not a real number.

Question: for what values of x is the function defined?

( ) 3 2f x x

0

(0) ?

( ) 3( ) 20 2

f

f

Domain of a Function

Answer:

2

3x

is defined only when the radicand (3x-2) is equal to or greater than zero. This implies that

Domain of a Function(continued) Therefore, the domain of our function is the set of

real numbers that are greater than or equal to 2/3.

Example: Find the domain of the function

1( ) 4

2f x x

Domain of a Function(continued) Therefore, the domain of our function is the set of real

numbers that are greater than or equal to 2/3.

Example: Find the domain of the function

Answer:

1( ) 4

2f x x

8 , [8, )x x

Domain of a Function:Another Example Find the domain of

1( )

3 5f x

x

Domain of a Function:Another Example Find the domain of

In this case, the function is defined for all values of x except where the denominator of the fraction is zero. This means all real numbers x except 5/3.

1( )

3 5f x

x

Mathematical ModelingThe price-demand function for a company is given by

where x (variable) represents the number of items and p(x) represents the price of the item. 1000 and -5 are constants. Determine the revenue function and find the revenue generated if 50 items are sold.

( ) 1000 5 , 0 100p x x x

SolutionRevenue = Price ∙ Quantity, so R(x)= p(x) ∙ x = (1000 – 5x) ∙ xWhen 50 items are sold, x = 50, so we will evaluate the revenue function at x = 50:

The domain of the function has already been specified. What is the ‘range’ over this domain?

(50) (1000 5(50)) 50 37,500R 0 100x

Solution

Example 2.4 (5) from Chiang

Production Problem

Either coal (C) or gas (G) can be used to produce steel. If Pc=100 and Pg=500, draw an isocost curve to limit expenditures to $10,000

Production Problem If the price of gas declines by 20%? What

happens to the budget line? What if the price of coal rises by 25%? Expenditures rise 50%?

Production Problem All together?

Break-Even and Profit-Loss Analysis Any manufacturing company has costs C and revenues R. The company will have a loss if R < C, will break even

if R = C, and will have a profit if R > C. Costs include fixed costs such as plant overhead, etc. and

variable costs, which are dependent on the number of items produced. C = a + bx(x is the number of items produced)

a and b are parameters

Break-Even and Profit-Loss Analysis(continued) Price-demand functions, usually determined by financial

departments, play an important role in profit-loss analysis. p = m – nx (x is the number of items than can be sold at $p per item.) [Note: m and n are PARAMETERS]

The revenue function is R = (number of items sold) ∙ (price per item) = xp = x(m - nx)

The profit function is P = R - C = x(m - nx) - (a + bx)

Example of Profit-Loss Analysis

A company manufactures notebook computers. Its marketing research department has determined that the data is modeled by the price-demand function p(x) = 2,000 - 60x, when 1 < x < 25, (x is in thousands).

What is the company’s revenue function and what is its domain?

Answer to Revenue ProblemSince Revenue = Price ∙ Quantity,

2( ) ( ) (2000 60 ) 2000 60R x x p x x x x x

The domain of this function is the same as the domain of the price-demand function, which is 1 ≤ x ≤ 25 (in thousands.)

First, is the demand function, and the plot of it. Below, you can see that x*p(x) is our revenue function…

But, firms don’t maximize revenue, they maximize profits, so we have to consider our cost function

Profit ProblemThe financial department for the company in the preceding problem has established the following cost function for producing and selling x thousand notebook computers:

C(x) = 4,000 + 500x (x is in thousand dollars).

Write a profit function for producing and selling x thousand notebook computers, and indicate the domain of this function.

Answer to Profit Problem

Since Profit = Revenue - Cost, and our revenue function from the preceding problem was R(x) = 2000x - 60x2,

P(x) = R(x) - C(x) = 2000x - 60x2 - (4000 + 500x) = -60x2 + 1500x – 4000.

The domain of this function is the same as the domain of the original price-demand function, 1< x < 25 (in thousands.)

Now, to get this profit function

First, is the cost function, and the plot of it. Below, you can see that the revenue function plotted with our cost function gives us a visual representation of profit…

Where these two lines cross, profit is ZERO, since R=C. If But, firms don’t maximize revenue, they maximize profits, so we have to consider our cost function. We want to maximize the difference between the two functions.

This, is R-C, or our profit function

Below, I have solved for this using

Mathematica’s Maximize function

Types of functions Polynomial functions

2)( xxf

x y

-3

-2

-1

0

1

2

3

Solution of Problem 1

x y

-3 9

-2 4

-1 1

0 0

1 1

2 4

3 9

Polynomials or Quadratics can shift Now, sketch the related

graph given by the equation below and explain, in words, how it is related to the first function you graphed.

2)2()2( xxf

x y

-3

-2

-1

0

1

2

3

Solution of Problem 2

x y

-3 25

-2 16

-1 9

0 4

1 1

2 0

3 1

Cubic functions

x y

-3 -27

-2 -8

-1 -1

0 0

1 1

2 8

3 27-30

-20

-10

0

10

20

30

-4 -3 -2 -1 0 1 2 3 4

Problem 4 Now, graph the

following related function:

55)( 3 xxf

x y

-3

-2

-1

0

1

2

3

Solution to Problem 4

-30

-20

-10

0

10

20

30

40

-4 -3 -2 -1 0 1 2 3 4

x y

-3 -22

-2 -3

-1 4

0 5

1 6

2 13

3 32

Problem 5

Graph

What is the domain of this function?

( )h x x

Solution to Problem 5 The domain is all non-negative real numbers.

Here is the graph:

-6

-4

-2

0

2

4

6

-5 0 5 10 15 20 25 30

Problem 6

Graph:

Explain, in words, how it compares to problem 5.

( )h x x

Solution to Problem 6

-6

-4

-2

0

2

4

6

-5 0 5 10 15 20 25 30

Conclusion:The graph of the function –f (x) is a reflection of the graph of f (x) across the x axis. That is, if the graphs of f (x) and –f (x) were folded along the x axis, the two graphs would coincide.

Absolute Value Function Graph the absolute value function. Be sure to choose x

values that are both positive and negative, as well as zero.

Graph( )a x x

2|1|2)1( xxa

Absolute Value Function(continued)Notice the symmetry of the graph.

||)( xxa

-4

-2

0

2

4

6

8

10

-10 -5 0 5 10

Absolute Value Function(continued)

2|1|2)1( xxa

-3

-2

-1

0

1

2

3

4

5

6

7

8

-10 -5 0 5 10

Shifted left one unit and down

two units.

The purple line uses the absolute value function (Abs) and the second takes the square root of the square… to get the absolute value.

The bottom shifts the axes.

Summary ofGraph Transformations Vertical Translation: y = f (x) + k

k > 0 Shift graph of y = f (x) up k units. k < 0 Shift graph of y = f (x) down |k| units.

Horizontal Translation: y = f (x+h) h > 0 Shift graph of y = f (x) left h units. h < 0 Shift graph of y = f (x) right |h| units.

Reflection: y = -f (x) Reflect the graph of y = f (x) in the x axis.

Vertical Stretch and Shrink: y = Af (x) A > 1: Stretch graph of y = f (x) vertically by multiplying

each ordinate value by A. 0<A<1: Shrink graph of y = f (x) vertically by multiplying

each ordinate value by A.

Piecewise-Defined Functions Earlier we noted that the absolute value of a real number x

can be defined as

Notice that this function is defined by different rules for different parts of its domain. Functions whose definitions involve more than one rule are called piecewise-defined functions.

Graphing one of these functions involves graphing each rule over the appropriate portion of the domain.

0if

0if||

xx

xxx

Example of a Piecewise-Defined Function

Graph the function

2if2

2if22)(

xx

xxxf

Example of a Piecewise-Defined Function

Graph the function

Notice that the point (2,0) is

included but the point (2, -2) is not.

2if2

2if22)(

xx

xxxf

Rational Functions

42

12

xx

xy

Unit Elastic Curve (Rect. Hyperbola)

constant a is a where that so rather or , apqp

aq

x

ay

No matter what the price is, the quantity changes such that expenditures are always equal to a constant (a). Thus, the percentage change in quantity is always equal (and opposite) the percentage change in price!

To sketch a quadratic

You need to find the places it crosses the x-axis, and you need to find where it turns around. When you don’t have Mathematica, or know how to take a derivative this can be tricky.

Trust me, derivatives are easier… and make more sense.

Vertex Form of the Quadratic Function

It is convenient to convert the general form of a quadratic equation

to what is known as the vertex form:

Note, if inside parentheses is x-2, h=2.

2( )f x ax bx c

2( ) ( )f x a x h k

Completing the Square to Find the Vertex of a Quadratic Function

The example below illustrates the procedure: ConsiderComplete the square to find the vertex.

2( ) 3 6 1f x x x

Completing the Square to Find the Vertex of a Quadratic Function

The example below illustrates the procedure: ConsiderComplete the square to find the vertex.

2( ) 3 6 1f x x x

Solution:

Factor the coefficient of x2 out of the first two terms:

f (x) = -3(x2 - 2x) -1

Completing the square (continued)

The vertex is (1, 2) The quadratic function opens down since the

coefficient of the x2 term is -3, which is negative.

Add 1 to complete the square inside the parentheses. Because of the -3 outside the parentheses, we have actually added -3,

so we must add +3 to the outside. The trick is adding and subtracting a number that makes parentheses sum of squares

f (x) = -3(x2 - 2x +1) -1+3

f (x) = -3(x - 1)2 + 2

There has to be an easier way…There is… which will become obvious when we get to start taking derivatives and knowing the quadratic function. This

takes a quadratic function which has many values of f(x), and finds the values where f(x) EQUALS zero (a quadratic

equation). These are also called the ROOTS of the quadratic function.

0 where2

4 22

cbxaxa

acbb

If both roots are real… split the difference and find the values of x and f(x)

Intercepts of a Quadratic Function

Find the x and y intercepts of2( ) 3 6 1f x x x

Intercepts of a Quadratic Function

Find the x and y intercepts of

x intercepts: Set f (x) = 0:

Use the quadratic formula: x = =

2( ) 3 6 1f x x x 20 3 6 1x x

2 4

2

b b ac

a

26 6 4( 3)( 1) 6 240.184,1.816

2( 3) 6

Intercepts of a Quadratic Function(continued) y intercept: Let x = 0. If x = 0, then y = -1, so

(0, -1) is the y intercept.

2( ) 3 6 1f x x x

1)0(6)0(3)0( 2 f

Generalization

If a 0, then the graph of f is a parabola. If a > 0, the graph opens upward. If a < 0, the graph opens downward. Vertex is (h , k)

Axis of symmetry: x = h f (h) = k is the minimum if a > 0, otherwise the maximum Domain = set of all real numbers Range: if a < 0. If a > 0, the range is

y y k

2( ) ( )f x a x h k For

y y k

Solving Quadratic Inequalities

Solve the quadratic inequality -3x2 + 6x -1 > 0

Solving Quadratic Inequalities

Solve the quadratic inequality -3x2 + 6x -1 > 0

Answer:

This inequality holds for those values of x for which the graph of f (x) is at or above the x axis. This

happens for x between the two x intercepts, including the intercepts. Thus, the solution set for the

quadratic inequality is

0.184 < x < 1.816.

Application of Quadratic Functions

A Macon, Georgia, peach orchard farmer now has 20 trees per acre. Each tree produces, on the average, 300 peaches. For each additional tree that the farmer plants, the number of peaches per tree is reduced by 10. How many more trees should the farmer plant to achieve the maximum yield of peaches? What is the maximum yield?

Solution: Yield = (number of peaches per tree) x (number of trees)

Yield = 300 x 20 = 6000 (currently) Plant one more tree: Yield = ( 300 – 1(10)) * ( 20 + 1) =

290 x 21 = 6090 peaches. Plant two more trees: Yield = ( 300 – 2(10)* ( 20 + 2) = 280 x 22 = 6160

Solution (continued) Let x represent the number of additional trees.

Then Yield =( 300 – 10x) (20 + x)= To find the maximum yield, note that the Y (x)

function is a quadratic function opening downward. Hence, the vertex of the function will be the maximum value of the yield. Graph is below, with the y value in thousands.

210 100 6000x x

Solution(continued)

Complete the square to find the vertex of the parabola:

Y (x) =

We have to add 250 on the outside since we multiplied –10 by 25 = -250.

210( 10 25) 6000 250x x

Solution(continued) Y (x) = Thus, the vertex of the quadratic function is (5 ,

6250) . So, the farmer should plant 5 additional trees and obtain a yield of 6250 peaches. We know this yield is the maximum of the quadratic function since the the value of a is -10. The function opens downward, so the vertex must be the maximum.

210( 5) 6250x

Break-Even AnalysisThe financial department of a company that produces

digital cameras has the revenue and cost functions for x million cameras are as follows:

R(x) = x(94.8 - 5x)

C(x) = 156 + 19.7x. Both have domain 1 < x < 15

Break-even points are the production levels at which R(x) = C(x). Find the break-even points algebraically to the

nearest thousand cameras.

Solution to Break-Even ProblemSet R(x) equal to C(x):

x(94.8 - 5x) = 156 + 19.7x

-5x2 + 75.1x - 156 = 0

275.1 75.1 4( 6)( 156)

2( 5)x

x = 2.490 or 12.530

The company breaks even at x = 2.490 and 12.530 million cameras.

Solution to Break-Even Problem(continued)If we graph the cost and revenue functions on a

graphing utility, we obtain the following graphs, showing the

two intersection points:

We can also plot the profit function to see where the

profit is maximized…

Quadratic Regression

A visual inspection of the plot of a data set might indicate that a parabola would be a better model of the data than a straight line. In that case, rather than using linear regression to fit a linear model to the data, we

would use quadratic regression on a graphing calculator to find the function of the form y = ax2 + bx + c that best

fits the data.

Example of Quadratic Regression

An automobile tire manufacturer collected the data in the table relating tire pressure x (in pounds per square

inch) and mileage (in thousands of miles.)

x Mileage

28 45

30 52

32 55

34 51

36 47

Polynomial Functions

A polynomial function is a function that can be written in the form

for n a nonnegative integer, called the degree of the polynomial. The domain of a polynomial function is the

set of all real numbers.

A polynomial of degree 0 is a constant. A polynomial of degree 1 is a linear function. A polynomial of degree 2 is a

quadratic function.

011

1 axaxaxa nn

nn

Shapes of Polynomials A polynomial is called odd if it only contains odd powers

of x It is called even if it only contains even powers of xLet’s look at the shapes of some even and odd polynomials Look for some of the following properties:

Symmetry Number of x axis intercepts Number of local maxima/minima What happens as x goes to +∞ or -∞?

Graph of Odd Polynomial Example 1

xxxxf 45)( 35

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

-3 -2 -1 0 1 2 3

Graph of Odd Polynomial Example 2

3( ) 27f x x x

-60.0

-40.0

-20.0

0.0

20.0

40.0

60.0

-8 -6 -4 -2 0 2 4 6 8

Graph of Even Polynomial Example 1

4 2( ) 6f x x x

-15.0

-10.0

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

-4 -3 -2 -1 0 1 2 3 4

Graph of Even Polynomial Example 2

-5.0

0.0

5.0

10.0

15.0

20.0

25.0

30.0

-4 -3 -2 -1 0 1 2 3 4

13)( 2 xxf

ObservationsOdd Polynomials For an odd polynomial,

the graph is symmetric about the originthe graphs starts negative, ends positive, or vice

versa, depending on whether the leading coefficient is positive or negative

either way, a polynomial of degree n crosses the x axis at least once, at most n times.

ObservationsEven Polynomials For an even polynomial,

the graph is symmetric about the y axisthe graphs starts negative, ends negative, or starts

and ends positive, depending on whether the leading coefficient is positive or negative

either way, a polynomial of degree n crosses the x axis at most n times. It may or may not cross at all.

Characteristics of polynomials Graphs of polynomials are continuous. One can sketch the

graph without lifting up the pencil. Graphs of polynomials have no sharp corners. Graphs of polynomials usually have turning points, which is a

point that separates an increasing portion of the graph from a decreasing portion.

A polynomial of degree n can have at most n linear factors. Therefore, the graph of a polynomial function of positive degree n can intersect the x axis at most n times.

A polynomial of degree n may intersect the x axis fewer than n times.

Rational Functions A rational function is a quotient of two polynomials, P(x) and

Q(x), for all x such that Q(x) is not equal to zero. Example: Let P(x) = x + 5 and Q(x) = x – 2, then

R(x) =

is a rational function whose domain is all real values of x with the exception of 2 (Why?)

5

2

x

x

Vertical Asymptotes of Rational Functions

x values at which the function is undefined represent vertical asymptotes to the graph of the function. A vertical asymptote is a

line of the form x = k which the graph of the function approaches but does not cross. In the figure below, which is the graph of

5

2

x

x

the line x = 2 is a

vertical asymptote.

Horizontal Asymptotes of Rational Functions

A horizontal asymptote of a function is a line of the form y = k which the graph of the

function approaches as x approaches For example, in the

graph of

5

2

x

x

the line y = 1 is a horizontal asymptote.

Generalizations about Asymptotes of Rational Functions

The number of vertical asymptotes of a rational function f (x) = n(x)/d(x) is at most equal to the degree of d(x).

A rational function has at most one horizontal asymptote.

The graph of a rational function approaches the horizontal asymptote (when one exists) both as x increases and decreases without bound.

Exponential functions

The equation

for b>0 defines the exponential function with base b . The domain is the set of all real numbers, while the range is the set of all positive real numbers.

( ) xf x b

Riddle

Here is a problem related to exponential functions: Suppose you received a penny on the first day of

December, two pennies on the second day of December, four pennies on the third day, eight pennies on the fourth day and so on. How many pennies would you receive on December 31 if this pattern continues?

Would you rather take this amount of money or receive a lump sum payment of $10,000,000?

Solution

Day No. pennies

1 1

2 2 2^1

3 4 2^2

4 8 2^3

5 16 ...

6 32

7 64

Complete the table:

Solution(continued) Now, if this pattern continued, how many pennies

would you have on Dec. 31? Your answer should be 230 (two raised to the thirtieth

power). The exponent on two is one less than the day of the month. See the preceding slide.

What is 230? 1,073,741,824 pennies!!! Move the decimal point two

places to the left to find the amount in dollars. You should get: $10,737,418.24

Solution(continued) The obvious answer to the question is to take the

number of pennies on December 31 and not a lump sum payment of $10,000,000(although I would not mind having either amount!)

This example shows how an exponential function grows extremely rapidly. In this case, the exponential function

is used to model this problem. ( ) 2xf x

Graph of

Use a table to graph the exponential function above. Note: x is a real number and can be replaced with numbers such as as well as other irrational numbers. We will use integer values for x in the table:

( ) 2xf x

2

Table of values        

x y

-4 2-4 = 1/24 = 1/16

 -3 2-3 = 1/8

 -2 2-2 = 1/4

 -1 2-1 = 1/2

 0 20 = 1

 1 21 = 2

 2 22 = 4

Graph of y = ( ) 2xf x

Characteristics of the graphs of where b> 1 All graphs will approach the x axis as x

becomes unbounded and negative All graphs will pass through (0,1) (y intercept) There are no x intercepts. Domain is all real numbers Range is all positive real numbers. The graph is always increasing on its domain. All graphs are continuous curves.

( ) xf x b

Graphs of if 0 < b < 1 All graphs will approach the x axis as x gets

large. All graphs will pass through (0,1) (y intercept) There are no x intercepts. Domain is all real numbers Range is all positive real numbers. The graph is always decreasing on its domain. All graphs are continuous curves.

( ) xf x b

Graph of

Using a table of values, you will obtain the following graph. The graphs of and will be reflections of each other about the y-axis, in general.

1( ) 2

2x

xf x

0

2

4

6

8

10

12

-4 -2 0 2 4

graph of y = 2 (̂-x)

approaches the positive x-axis as x gets large

passes through (0,1)

( ) xf x b ( ) xf x b

Base e Exponential Functions Of all the possible bases b we can use for the

exponential function y = bx, probably the most useful one is the exponential function with base e.

The base e is an irrational number, and, like π, cannot be represented exactly by any finite decimal fraction.

However, e can be approximated as closely as we like by evaluating the expression as x gets infinitely large

11

x

x

Exponential Function With Base eThe table to the left illustrates what happens to the expression

as x gets increasingly larger. As we can see from the table, the values approach a number whose approximation is 2.718

x

1 2

10 2.59374246

100 2.704813829

1000 2.716923932

10000 2.718145927

1000000 2.718280469

11

x

x

11

x

x

Leonhard Euler1707-1783 Leonhard Euler first demonstrated that

will approach a fixed constant we now call “e”. So much of our mathematical notation is due to Euler that it will come as

no surprise to find that the notation e for this number is due to him. The claim which has sometimes been made, however, that Euler used the letter e because it was the first letter of his name is ridiculous. It is probably not even the case that the e comes from "exponential", but it may have just be the next vowel after "a" and Euler was already using the notation "a" in his work. Whatever the reason, the notation e made its first appearance in a letter Euler wrote to Goldbach in 1731. http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/e.html#s19

11

x

x

Graph of Graph is similar to the

graphs of

and

It has the same characteristics as these graphs

( ) xf x e

( ) 2xf x

( ) 3xf x

Relative Growth Rates Functions of the form y = cekt, where c and k are constants and

the independent variable t represents time, are often used to model population growth and radioactive decay.

Note that if t = 0, then y = c. So, the constant c represents the initial population (or initial amount.)

The constant k is called the relative growth rate. If the relative growth rate is k = 0.02, then at any time t, the population is growing at a rate of 0.02y persons (2% of the population) per year.

We say that population is growing continuously at relative growth rate k to mean that the population y is given by the model y = cekt.

Growth and Decay Applications:Atmospheric Pressure The atmospheric pressure p

decreases with increasing height. The pressure is related to the number of kilometers h above the sea level by the formula:

Find the pressure at sea level (h = 0)

Find the pressure at a height of 7 kilometers.

0.145( ) 760 hP h e

Solution Find the pressure at sea level (h = 0)

Find the pressure at a height of 7 kilometers

0.145(7)(7) 760 275.43P e

760760)0( 0 eP

Depreciation of a Machine

A machine is initially worth V0 dollars but loses 10% of its value each year. Its value after t years is given by the formula

Find the value after 8 years of a machine whose initial value is $30,000.

0( ) (0.9 )tV t V

Depreciation of a Machine

A machine is initially worth V0 dollars but loses 10% of its value each year. Its value after t years is given by the formula

Find the value after 8 years of a machine whose initial value is $30,000.

Solution:

0( ) (0.9 )tV t V

0( ) (0.9 )tV t V

8(8) 30000(0.9 ) $12,914V

Compound Interest The compound interest formula is

Here, A is the future value of the investment, P is the initial amount (principal), r is the annual interest rate as a decimal, n represents the number of compounding periods per year, and t is the number of years

1nt

rA P

n

Compound Interest Problem

Find the amount to which $1500 will grow if deposited in a bank at 5.75% interest compounded quarterly for 5 years.

Compound Interest Problem Find the amount to which $1500 will grow if deposited in a

bank at 5.75% interest compounded quarterly for 5 years.

Solution: Use the compound interest formula:

Substitute P = 1500, r = 0.0575, n = 4 and t = 5 to obtain =$1995.55

1nt

rA P

n

(4)(5)0.0575

1500 14

A

Logarithmic Functions

In this section, another type of function will be studied called the logarithmic function. There is a close connection between a logarithmic function and an exponential function. We will see that the logarithmic function and exponential functions are inverse functions. We will study the concept of inverse functions as a prerequisite for our study of logarithmic function.

One to One Functions We wish to define an inverse of a function. Before we do so, it is necessary to discuss the topic of one to one functions.

First of all, only certain functions are one to one.

Definition: A function is said to be one to one if distinct inputs of a function correspond to distinct outputs. That is, if

1 2 1 2, ( ) ( )x x f x f x

Graph of One to One Function

This is the graph of a one to one function. Notice that if we choose two different x values, the corresponding y values are different. Here, we see that if x = 0, then y = 1, and if x = 1, then y is about 2.8.Now, choose any other pair of x values. Do you see that the corresponding y values will always be different?

0

1

2

3

4

5

-1 0 1 2

Horizontal Line Test Recall that for an equation to be a function, its graph must pass

the vertical line test. That is, a vertical line that sweeps across the graph of a function from left to right will intersect the graph only once at each x value.

There is a similar geometric test to determine if a function is one to one. It is called the horizontal line test. Any horizontal line drawn through the graph of a one to one function will cross the graph only once. If a horizontal line crosses a graph more than once, then the function that is graphed is not one to one.

Which Functions Are One to One?

-30

-20

-10

0

10

20

30

40

-4 -2 0 2 40

2

4

6

8

10

12

-4 -2 0 2 4

Definition of Inverse Function

Given a one to one function, the inverse function is found by interchanging the x and y values of the original function. That is to say, if an ordered pair (a,b) belongs to the original function then the ordered pair (b,a) belongs to the inverse function.

Note: If a function is not one to one (fails the horizontal line test) then the inverse of such a function does not exist.

Logarithmic Functions The logarithmic function with base two is defined to be the inverse

of the one to one exponential function

Notice that the exponential function

is one to one and therefore has an inverse. 0

1

2

3

4

5

6

7

8

9

-4 -2 0 2 4

graph of y = 2 (̂x)

approaches the negative x-axis as x gets large

passes through (0,1)

2xy

2xy

Inverse of an Exponential Function Start with Now, interchange x and y coordinates:

There are no algebraic techniques that can be used to solve for y, so we simply call this function y the logarithmic function with base 2. The definition of this new function is:

if and only if

2xy

2yx

2log x y 2yx

Graph, Domain, Range of Logarithmic Functions The domain of the logarithmic function y = log2x is the

same as the range of the exponential function y = 2x. Why? The range of the logarithmic function is the same as the

domain of the exponential function (Again, why?) Another fact: If one graphs any one to one function and its

inverse on the same grid, the two graphs will always be symmetric with respect to the line y = x.

Logarithmic-Exponential Conversions

Study the examples below. You should be able to convert a logarithmic into an exponential expression and vice versa. 1.

2.

3.

4.

4log (16) 4 16 2xx x

3125 5 5log 125 3

1

281

181 9 81 9 log 9

2

33 3 33

1 1log ( ) log ( ) log (3 ) 3

27 3

Solving EquationsUsing the definition of a logarithm, you can solve equations involving logarithms. Examples:

3 3 3log (1000) 3 1000 10 10b b b b

56log 5 6 7776x x x

In each of the above, we converted from log form to exponential form and solved the resulting equation.

Properties of Logarithms These are the properties of logarithms. M and N are positive real numbers, b not equal to 1, and p and x are real numbers. (For 4, we need x > 0).

5. log log log

6. log log log

7. log log

8. log log

b b b

b b b

pb b

b b

MN M N

MM N

N

M p M

M N iff M N

log

1.log (1) 0

2.log ( ) 1

3.log

4. b

b

b

xb

x

b

b x

b x

Solving Logarithmic Equations1. Solve for x:

2. Product rule

3. Special product

4. Definition of log

5. x can be +10 only6. Why?

4 4

4

24

3 2

2

2

log ( 6) log ( 6) 3

log ( 6)( 6) 3

log 36 3

4 36

64 36

100

10

10

x x

x x

x

x

x

x

x

x

Another Example1. Solve:

Another Example1. Solve:

2. Quotient rule

3. Simplify (divide out common factor π)

4. rewrite

5 definition of logarithm 6. Property of exponentials

Common Logs and Natural Logs

Common log Natural log

10log logx x ln( ) logex x

2.7181828e If no base is indicated, the logarithm is

assumed to be base 10.

Solving a Logarithmic EquationSolve for x. Obtain the exact

solution of this equation in terms of e (2.71828…)

ln (x + 1) – ln x = 1

Solving a Logarithmic EquationSolve for x. Obtain the exact

solution of this equation in terms of e (2.71828…)

Quotient property of logsDefinition of (natural log)Multiply both sides by x

Collect x terms on left sideFactor out common factor

Solve for x

ln (x + 1) – ln x = 1

1ln 1

x

x

1 1xe

xe

ex = x + 1ex - x = 1

x(e - 1) = 1

1

1x

e

Application How long will it take money to double if compounded monthly at 4 % interest?

Application How long will it take money to double if compounded monthly at 4 % interest? 1. Compound interest formula2. Replace A by 2P (double the amount) 3. Substitute values for r and m 4. Divide both sides by P5. Take ln of both sides 6. Property of logarithms 7. Solve for t and evaluate expression

Solution:

12

12

12

1

0.042 1

12

2 (1.003333...)

ln 2 ln (1.003333...)

ln 2 12 ln(1.00333...)

ln 217.36

12ln(1.00333...)

mt

t

t

t

rA P

m

P P

t

t t