Economic Models, Functions, Logs, Exponents, e

1
Lecture 2

• Economic Models, Functions, Logs, Exponents, e

2
Variables, Constant, Parameters

• Variables magnitude can change
• Price, profit, revenue
• Represented by symbols
• Can be frozen by setting value
• Try to setup models to obtain solutions to
  variables
• Endogenous variable determined by model
• Exogenous variable determined outside model

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Variables, Constant, Parameters

• Constants, do not change, but can be joined with
  variables.
• Called coefficients
• Ex 1000q, 1000 is constant, quantity is variable
• When constants are not set?
• Ex ßq, where ß stands for the coefficient and q
  for quantity
• Now, ß can change! A variable constant is
  called a parameter.

4
Equation vs. Identity

• An identity, is a definition, like profit,
  revenue, and cost
• where p is profit, R is
  revenue, C is cost
• Behavioral equations
• Specify how variables interact
• Conditional equations
• Optimization and Equilibrium conditions are
  examples

5
Real Numbers

• Rational numbers (ratio)
• Whole numbers Integers
• Fractions
• Irrational numbers
• Cant be expressed as a fraction
• Nonrepeating, nonterminating decimals
• Ex
• Real Numbers combine rational and irrational
• Non-imaginary, i

6
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 ? Ameans e is an element of
  A, or e belongs to set A. The notation e ?
  A means e is not an element of A.

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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 ?.

8
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 xa would be read x, given a.
• Here, we will use the vertical line in set
  notation
• Example. x x2 9 3, -3This 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.

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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.

B
A
In the Venn diagram on the left, the union of A
and B is the entire region shaded.
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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.

In the Venn diagram on the left, the intersection
of A and B is the shaded region.
B
A
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The Complement of a Set

• The 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

In the Venn diagram on the left, the rectangle
represents the universe. A? is the shaded area
outside the set A.
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Application
A 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
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Solution
The 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.
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Solution(continued)
U
200 people listen to neither news nor music
is the green part, which contains 300 commuters.
M
N
300 listen to news but not music.
200 listen to music but not news
300 listen to both music and news
15
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.

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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
9.00
12
10.00
16
12.00
domain D or x
range R or f(x)
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Functions Specified by Equations

• Consider the equation

Input x -2
-2
Process square (2),then subtract 2
(-2,2) is an ordered pair of the function.
Output result is 2
2
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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.
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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.
20
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

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Function Evaluation

• Consider our function
• What does f (-3) mean?

22
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.

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Some Examples

• 1.
• KEEP THIS h example in mind for derivatives

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Domain of a Function

• Consider
• which is not a real number.
• Question for what values of x is the function
  defined?

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Domain of a Function

• Answer

is defined only when the radicand (3x-2) is equal
to or greater than zero. This implies that
26
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

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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

28
Domain of a FunctionAnother Example

• Find the domain of

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Domain of a FunctionAnother 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.

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Mathematical Modeling

• The price-demand function for a company is given
  bywhere 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.

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Solution

• Revenue Price Quantity, so
• R(x) p(x) x
  (1000 5x) x
• When 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?

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Solution
33
Example 2.4 (5) from Chiang
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Production Problem

• Either coal (C) or gas (G) can be used to produce
  steel. If Pc100 and Pg500, draw an isocost
  curve to limit expenditures to 10,000

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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?

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Production Problem

• All together?

37
Break-Even and Profit-Loss Analysis

• Any manufacturing company has costs C and
  revenues R.
• The company will have a loss if R lt C, will break
  even if R C, and will have a profit if R gt 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

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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)

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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 lt x lt 25, (x is in
thousands). What is the companys revenue
function and what is its domain?
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Answer to Revenue Problem
Since Revenue Price Quantity,
The domain of this function is the same as the
domain of the price-demand function, which is 1
x 25 (in thousands.)
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First, is the demand function, and the plot of
it. Below, you can see that xp(x) is our revenue
function But, firms dont maximize revenue,
they maximize profits, so we have to consider our
cost function
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Profit Problem
The 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.
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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,
1lt x lt 25 (in thousands.) Now, to get this
profit function
44
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 RC. If But, firms
dont maximize revenue, they maximize profits, so
we have to consider our cost function. We want to
maximize the difference between the two functions.
45
This, is R-C, or our profit function Below, I
have solved for this using Mathematicas Maximize
function
46
Types of functions

• Polynomial functions

x y
-3
-2
-1
0
1
2
3
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Solution of Problem 1
x y
-3 9
-2 4
-1 1
0 0
1 1
2 4
3 9
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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.

x y
-3
-2
-1
0
1
2
3
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Solution of Problem 2
x y
-3 25
-2 16
-1 9
0 4
1 1
2 0
3 1
50
Cubic functions
x y
-3 -27
-2 -8
-1 -1
0 0
1 1
2 8
3 27
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Problem 4

• Now, graph the following related function

x y
-3
-2
-1
0
1
2
3
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Solution to Problem 4
x y
-3 -22
-2 -3
-1 4
0 5
1 6
2 13
3 32
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Problem 5

• Graph
• What is the domain of this function?

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Solution to Problem 5

• The domain is all non-negative real numbers. Here
  is the graph

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Problem 6

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

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Solution to Problem 6
ConclusionThe 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.
57
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

58
Absolute Value Function(continued)

• Notice the symmetry of the graph.

59
Absolute Value Function(continued)
Shifted left one unit and down two units.
60
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.
61
Summary ofGraph Transformations

• Vertical Translation y f (x) k
• k gt 0 Shift graph of y f (x) up k units.
• k lt 0 Shift graph of y f (x) down k units.
• Horizontal Translation y f (xh)
• h gt 0 Shift graph of y f (x) left h units.
• h lt 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 gt 1 Stretch graph of y f (x) vertically
  by multiplying each ordinate value
  by A.
• 0ltAlt1 Shrink graph of y f (x) vertically by
  multiplying each ordinate value by
  A.

62
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.

63
Example of a Piecewise-Defined Function
Graph the function
64
Example of a Piecewise-Defined Function
Graph the function
Notice that the point (2,0) is included but the
point (2, -2) is not.
65
Rational Functions
66
Unit Elastic Curve (Rect. Hyperbola)
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!
67
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 dont have Mathematica, or know
  how to take a derivative this can be tricky.
• Trust me, derivatives are easier and make more
  sense.

68
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, h2.

69
Completing the Square to Find the Vertex of a
Quadratic Function

• The example below illustrates the procedure
• Consider
• Complete the square to find the vertex.

70
Completing the Square to Find the Vertex of a
Quadratic Function

• The example below illustrates the procedure
• Consider
• Complete the square to find the vertex.

• Solution
• Factor the coefficient of x2 out of the first two
  terms
• f (x) -3(x2 - 2x) -1

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Completing the square (continued)

• 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) -13 f (x) -3(x - 1)2 2

• The vertex is (1, 2)
• The quadratic function opens down since the
  coefficient of the x2 term is -3, which is
  negative.

72
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.
If both roots are real split the difference and
find the values of x and f(x)
73
Intercepts of a Quadratic Function

• Find the x and y intercepts of

74
Intercepts of a Quadratic Function

• Find the x and y intercepts of
• x intercepts Set f (x) 0
• Use the quadratic formula
• x

75
Intercepts of a Quadratic Function(continued)

• y intercept Let x 0. If x 0, then y -1, so
  (0, -1) is the y intercept.

76
Generalization
For

• If a ? 0, then the graph of f is a parabola.
• If a gt 0, the graph opens upward.
• If a lt 0, the graph opens downward. Vertex is (h
  , k)
• Axis of symmetry x h
• f (h) k is the minimum if a gt 0, otherwise the
  maximum
• Domain set of all real numbers
• Range if a lt 0. If a gt 0, the
  range is

77
Solving Quadratic Inequalities
Solve the quadratic inequality -3x2 6x -1 gt 0
78
Solving Quadratic Inequalities
Solve the quadratic inequality -3x2 6x -1 gt
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 lt x lt 1.816.
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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?

80

• 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

81
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.

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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.

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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.  

84
Break-Even Analysis
The 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 lt x lt 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.
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Solution to Break-Even Problem
Set R(x) equal to C(x) x(94.8 - 5x) 156
19.7x -5x2 75.1x - 156 0
x 2.490 or 12.530 The company breaks even at x
2.490 and 12.530 million cameras.
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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
87
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.
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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
89
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.
90
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 x
• Lets 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 8 or -8?

91
Graph of Odd Polynomial Example 1
92
Graph of Odd Polynomial Example 2
93
Graph of Even Polynomial Example 1
94
Graph of Even Polynomial Example 2
95
ObservationsOdd Polynomials

• For an odd polynomial,
• the graph is symmetric about the origin
• the 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.

96
ObservationsEven Polynomials

• For an even polynomial,
• the graph is symmetric about the y axis
• the 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.

97
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.

98
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?)

99
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
the line x 2 is a vertical asymptote.
100
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
the line y 1 is a horizontal asymptote.
101
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.

102
Exponential functions

• The equation
• for bgt0 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.

103
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?

104
Solution
Complete the table
Day No. pennies
1 1
2 2 21
3 4 22
4 8 23
5 16 ...
6 32
7 64
105
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

106
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.

107
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

108
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
109
Graph of y
110
Characteristics of the graphs of
where bgt 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.

111
Graphs of if 0 lt b lt 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.

112
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.

113
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 p,
  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

114
Exponential Function With Base e

• The 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
115
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/HistTopi
  cs/e.htmls19

116
Graph of

• Graph is similar to the graphs of
• and
• It has the same characteristics as these
  graphs

117
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.

118
Growth and Decay ApplicationsAtmospheric
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.

119
Solution

• Find the pressure at sea level (h 0)

• Find the pressure at a height of 7 kilometers

120
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.

121
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

122
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

123
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.

124
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

125
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.

126
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

127
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?

128
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.

129
Which Functions Are One to One?
130
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.

131
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.

132
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

133
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.

134
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.

135
Solving Equations

• Using the definition of a logarithm, you can
  solve equations involving logarithms. Examples

In each of the above, we converted from log form
to exponential form and solved the resulting
equation.
136
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 gt 0).

137
Solving Logarithmic Equations

• Solve for x
• Product rule
• Special product
• Definition of log
• x can be 10 only
• Why?

138
Another Example

• 1. Solve

139
Another Example

• 1. Solve
• 2. Quotient rule
• 3. Simplify
• (divide out common factor p)
• 4. rewrite
• 5 definition of logarithm
• 6. Property of exponentials

140
Common Logs and Natural Logs

• Common log

• Natural log

If no base is indicated, the logarithm is assumed
to be base 10.
141
Solving a Logarithmic Equation
Solve for x. Obtain the exact solution of this
equation in terms of e (2.71828)
ln (x 1) ln x 1
142
Solving a Logarithmic Equation
Solve for x. Obtain the exact solution of this
equation in terms of e (2.71828) Quotient
property of logs Definition of (natural
log) Multiply both sides by x Collect x terms on
left side Factor out common factor Solve for x
ln (x 1) ln x 1
ex x 1
ex - x 1
x(e - 1) 1
143
Application

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

144
Application

• Solution

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