Math Review

1
Math Review

• APEC 3001
• Summer 2007

2
Objectives

• Review basic algebraic skills required to
  successfully solve homework, quiz, and exam
  problems in this class.
• Review different ways of describe relationships,
  including
• Functions
• Tables
• Surface Plots
• Contour Plots
• Review the special class of linear functions.
• What is the slope?
• What is the intercept?
• Solutions to systems of linear equations.
• Review the basic differential calculus skills
  required to successfully solve homework, quiz,
  and exam problems.

3
Types of Numbers

• Integers
• ,-3, -2, -1, 0, 1, 2, 3,
• Rational Numbers
• An Integer Divided By An Integer (e.g. ½ 0.5
  -4/3 -1.333)
• All Integers Are Rational Numbers
• Irrational Numbers
• Not Rational (e.g. 1.4142 ?
  3.1415)
• Decimals Are Non-Repeating Non-Terminating
• Any Irrational Number Lies Between Two Rational
  Numbers
• Real
• Includes All Rational Irrational Numbers

4
Properties of Real Numbers(a, b, c are real
numbers)

• Closure a b a ? b Are Real Numbers
• 1 2 3 1 ? 2 2 Are Real
• Commutative a b b a a ? b b ? a
• 1 2 2 1 3 1 ? 2 2 ? 1 2
• Associative (a b) c a (b c) (a ? b)
  ? c a ? (b ? c)
• (1 2) 3 1 (2 3) 6 (1 ? 2) ? 3 1 ?
  (2 ? 3) 6
• Distributive a ? (b c) a ? b a ? c (a
  b) ? c a ? c b ? c
• 1 ? (2 3) 1 ? 2 1 ? 3 5 (1 2) ? 3 1
  ? 3 2 ? 3 9
• Identity a 0 a a ? 1 a
• 1 0 1 1 ? 1 1

5
Properties of Real Numbers Continued(a, b, c
are real numbers)

• Inverse b (-b) 0 b ? 1/b 1
• 2 (-2) 0 2 ? 1/2 1
• Cancellation
• If a b a c, then b c
• If a ? b a ? c, then b c (provided a is not
  0)
• Zero Factor
• a ? 0 0 ? a 0
• If a ? b 0, the a 0, b 0, or a 0 b 0
• Negation
• -1 ? (-a) -(-a) a
• (-a) ? (-b) a ? b
• a ? (-b) (-a) ? b -(a ? b)

6
Exponents(a b are real numbers, m n are
integers)

• Exponent
• an a ? a ? ? a where a is multiplied by
  itself n times
• 24 2 ? 2 ? 2 ? 2 16
• Properties of Exponents
• an ? am anm (e.g. 42 ? 45 425 47
  16,384)
• (an)m an ? m (e.g. (42)5 42 ? 5 410
  1,048,576)
• an ? bn (a ? b)n (e.g. 22 ? 32 (2 ? 3)2
  62 36)
• a-1 1/a (e.g. 2-1 ½ 0.5)

Note that these properties will also hold if m
n are rational numbers, but we can then end up
with complex numbers. In this class, we will
choose rational number for m n, but we will
never use complex numbers.
7
Functions

• What is a function?
• A function describes a definite relationship.
• For example, the relationship between the area
  and radius of a circle is y f(x) where
• y is the area
• x is the radius,
• f(x) ?x2
• Causal Relationship
• The way a function is written is sometimes meant
  to imply a causal relationship.
• For example, y f(x) means x determines y or x
  is the independent variable, while y is the
  dependent variable.
• In this class, the way we write a function will
  not always imply causality.

8
Inverse Functions
Many of the functions we will deal with in this
class will be invertible.
That is, we can write y f(x) or x f -1 (y).
This also implies, we can write y f(f -1(y)).
For example, if y f(x) ?x2, then
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Other Descriptions of Definite Relationships
Functions like y f(x) ?x2 provide a general
and concise way to describe definite
relationships, but they are not the only way .
For example, we could have described the
relationship between the area and radius of a
circle using a Table
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Other Descriptions of Definite Relationships
We could have also described the relationship
between the area and radius of a circle using a
Figure.
y-axis Often Used for Dependent
Variable
x-axis Often Used for Independent Variable
IMPORTANT NOTE WE WILL NOT ALWAYS STICK TO THIS
CONVENTION
11
Could Just As Well Drawn As
12
More Complex Relationships

• What is a function that describes the number of
  words in a book?
• T f(p, n) where
• T is the total number of words in a book
• p is the number of pages in the book
• n is the average number of words per page
• f(p, n) pn
• Many of the functions we will look at in this
  class characterize the relationship between one
  variable and many others.

13
Again, We Can Use Tables
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But, What About Figures?
Surface Plot
Not Very Easy to Draw
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There Has To Be An Easier Way
n 500
Contour Plot
n 400
n 300
n 200
n 100
Each contour shows the relationship between the
total words and the number of pages holding the
words per page constant.
16
Alternatively
p 200
Contour Plot
p 150
p 100
p 50
Each contour shows the relationship between the
total words and the words per page holding the
number of pages constant.
17
Surface Versus Contour

• Surface plots become virtually impossible to draw
  once we are describing a relationship with more
  than three variables.
• Contour plots are easy to draw regardless of how
  many variables you have.
• Plot the relationship between two variables
  holding all the other variables constant.
• Change one of the constant variables and repeat.
• Repeat again as long as you need to in order to
  understand the relationship.

IMPORTANT NOTE AN UNDERSTANDING OF CONTOUR PLOTS
IS ESSENTIAL IN THIS CLASS
18
Linear Functions

• Looks Like y b mx
• b is referred to as the intercept
• The value of y when x 0
• m is referred to as the slope
• How much y changes as x increases by 1
• m ?y/?x where ?y is the change in y (the rise)
  ?x is the change in x (the run)
• m gt 0 implies a positive relationship
• m lt 0 implies a negative relationship
• m 0 implies no relationship
• Inverse is x - (b/m) (1/m)y
• - (b/m) is the intercept for the inverse function
• 1/m is the slope for the inverse function

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Graphical Example
y b mx 10 2.5x
?y 10
?x 4
m ?y/?x 10/4 2.5
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Some More Examples
y 10 2.5x
y 20 0 x
y 30 - 1.5x
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Solving Two Linear Functions

• Suppose we have two linear equations
• y b1 m1x
• y b2 m2x
• Question
• Is there a value for x that makes y the same for
  both functions?
• What if we take two functions from the previous
  slide
• y 10 2.5x
• y 30 - 1.5x
• What is the answer to our question for these two
  functions?

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If y is equal for both equations when x x,
then we know
10 2.5x 30 - 1.5x
10 - 10 2.5x 30 - 10 - 1.5x
2.5x 20 - 1.5x
2.5x 1.5x 20 - 1.5x 1.5x
4x 20
4x / 4 20 / 4
x 5
Is this right?
y 10 2.5 ? 5 22.5
y 30 - 1.5 ? 5 22.5
So, it would seem!
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Does this work in general?
b1 m1x b2 m2x
b1 b1 m1x b2 b1 m2x
m1x b2 b1 m2x
m1x - m2x b2 b1 m2x - m2x
(m1 - m2)x b2 b1
(m1 - m2)x / (m1 - m2) (b2 b1) / (m1 - m2)
If (m1 - m2) ? 0, then
x (b2 b1) / (m1 - m2)
Is this right?
So, it would seem, at least if (m1 - m2) ? 0!
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What is this thing we have found?
What we have found is the point of intersection
between two lines!
Later it will become obvious why this point of
intersection is of interest to us!
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Example
y 10 2.5x
22.5
y 30 - 1.5x
5
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Why doesnt it work with m1 m2?

• If m1 m2 and b1 b2, then the two lines are
  identical and any value of x will give you the
  same value of y.
• If m1 m2 and b1 ? b2, then the two lines are
  different, but have the same slope. They are
  parallel, which means there is no value x that
  will give you the same y. There is no point of
  intersection!

27
Elementary Calculus

• Calculus Has Many Uses
• Compute Areas Volumes
• Compute Velocities
• Describe Functional Relationships
• Find Extremes
• In this class, we will typically be interested in
  
• describing functional relationships
• finding extremes.
• Both these uses can be accomplished using what
  are called derivatives or partial derivatives.

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Derivative of a Function

• Consider the function y f(x)
• We will denote the derivative of a function with
  only one variable using f (x).
• The definition of a derivative is
• What does this definition tell us?
• It tells us how the value of y is changing as x
  changes.

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Graphical Interpretation
f(x) slope as h goes to 0 or slope of line
tangent to f(x) at x.
f(x h) - f(x) 38 32 6
Slope (f(x h) - f(x)) / h 6 / 2
3
h 4 2 2
y f(x) 1015x-2x2
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So why is this useful?

• What can we say about f(x) when f (x) gt 0?
• What can we say about f(x) when f (x) lt 0?
• What can we say about f(x) when f (x) 0?

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When f (x) gt 0, increasing x increases f(x) or y!
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When f (x) lt 0, increasing x decreases f(x) or y!
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When f (x) 0, f(x) or y is at a maximum!
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When f (x) 0, f(x) or y is at a minimum!
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To Summarize

• f (x) gt 0 tells us there is a positive
  relationship between x and y given x.
• f (x) lt 0 tells us there is a negative
  relationship between x and y given x.
• f (x) 0 tells us y is at a maximum or minimum
  given x.
• Note There is a huge difference between being at
  a maximum vs. being at a minimum! It would be
  nice to able to say which is the case.
• What is happening to f (x) when x is increasing
  for a maximum?
• What is happening to f (x) when x is increasing
  for a minimum?

36
For a maximum, f (x) is decreasing as x
increases!
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For a minimum, f (x) is increasing as x
increases!
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But how do we tell what is happening to f (x) as
x increase?

• This is precisely what a derivative tells us!
• Can we take the derivative of a derivative?
• Yes!
• It is called the second derivative, which we will
  denote with f (x).
• The second derivative is defined as
• It is just the derivative of the derivative!
• If f (x) lt 0, it tells us the derivative or
  slope of the tangent is decreasing as x increases
  or that the x that solves f (x) 0 is a
  maximum.
• If f (x) gt 0, it tells us the derivative or
  slope of the tangent is decreasing as x increases
  or that the x that solves f (x) 0 is a
  minimum.

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Useful Properties of Derivatives You Need to Know

• Suppose we have three functions f(x), g(x), and
  h(x)
• If f(x) g(x) h(x), f (x) g(x) h(x).
• If f(x) g(x)h(x), f (x) g(x)h(x)
  g(x)h(x).
• If f(x) h(g(x)), f (x) h (g(x)) g(x).
• If f(x) h(x)/g(x), f (x) (h(x) g(x)-
  h(x)g(x))/g(x)2.
• Suppose a and n are real numbers
• f(x) a, f (x) 0.
• f(x) axn, f (x) naxn - 1

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An Example

• Suppose f(x) 1015x-2x2
• f(x) f1(x) f2(x) f3(x) where f1(x) 10,
  f2(x) 15x, and f3(x) -2x2.
• f (x) f1(x) f2(x) f3(x) 0 15 4x
  15 4x because
• for f1(x) 10, f1(x) 0
• for f2(x) 15x, f2(x) 1 ? 15 ? x1 - 1 1 ?
  15 ? x0 15 and
• for f3(x) -2x2, f3(x) 2 ? (-2) ? x2 - 1 -4
  x1 -4x

41
Another Example

• Suppose f(x) x2x3
• f(x) f1(x)f2(x) where f1(x) x2 and f2(x)
  x3.
• f(x) f1(x) f2(x) f1(x) f2 (x) 2xx3
  x23x2 2x4 3x4 5x4 because
• for f1(x) x2 , f1(x) 2x2 - 1 2x and
• for f2(x) x3, f2(x) 3x3 - 1 3x2
• Notice that f(x) x2x3 x5, such that f (x)
  5x5 - 1 5x4, but things may not always simplify
  so nicely.

42
Yet Another Example

• Suppose f(x) (x2 - 5)3
• f(x) f1(f2(x)) where f1(x) x3 and f2(x) x2
  - 5.
• f (x) f1(f2(x)) f2(x) 2(x2 - 5)2 2x
  4x(x2 - 5)2 because
• for f1(x) x3 , f1(x) 3x3 - 1 3x2 and
• for f2(x) x2 - 5, f2(x) 2x2 - 1 - 0 2x

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One Final Example Before Moving On

• Suppose f(x) (x3 - 5) / x2
• f(x) f1(x) / f2(x) where f1(x) x3 - 5 and
  f2(x) x2
• f (x) (f1(x) f2(x) f1(x) f2(x)) / f2(x)2
  (3x2x2 - (x3 - 5) 2x) / x4
  (3x4 - 2x4 10x) / x4
  because
• for f1(x) x3 - 5, f1(x) 3x3 - 1 - 0 3x2
  and
• for f2(x) x2, f2(x) 2x2 - 1 2x

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What about functions with more than one variable?

• Suppose we have a function y f(x, z)
• The derivative of f(x, z) with respect to x is
  defined as
• The derivative of f(x, z) with respect to z is
  defined as
• These are referred to as partial derivatives
  because we are taking the derivative of the
  function with respect to a single variable, while
  holding the other variable constant.

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What do these partial derivatives mean?

• fx(x,z) tells us the relationship between y and x
  on a contour holding z constant.
• fz(x,z) tells us the relationship between y and z
  on a contour holding x constant.
• Solving fx(x,z) 0 and fz(x,z) 0, we can find
  the x and z that maximizes or minimizes y (there
  is also something called a saddle point that we
  will not deal with in this class).
• Again, we can compute second partial derivatives
  to determine whether we are at a maximum or
  minimum, but explaining how is more than what we
  need to get into at this point.

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An Example

• Suppose y f(x,z) x0.4z0.6
• fx(x,z) 0.4x0.4-1z0.6 0.4x-0.6z0.6
• fz(x,z) x0.40.6z0.6-1 0.6x0.4z-0.4

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Surface Plot for y f(x,z) x0.4z0.6
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Contour Plot for y f(x,z) x0.4z0.6 holding z
constant.
fx(x,z) gives us slopes of tangents to
contours holding z constant.
z 25
z 20
?
z 15
z 10
z 5
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Contour Plot for y f(x,z) x0.4z0.6 holding x
constant.
fz(x,z) gives us slopes of tangents to
contours holding x constant.
x 50
x 40
x 30
?
x 20
x 10
50
Skills for Success

• Manipulate Algebraic Expressions
• Represent Functions Algebraically and Graphically
• Find Solutions to Systems of Equations (Usually
  Two Equations)
• Take Derivatives and Partial Derivative of
  Functions with Exponents