W A T K I N S - J O H N S O N C O M P A N Y Semiconductor Equipment Group

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Chabot Mathematics
9.1aExponential Fcns
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu
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Review

• Any QUESTIONS About
• 8.5 ? Rational InEqualities
• Any QUESTIONS About HomeWork
• 8.5 ? HW-41

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Exponential Function

• A function, f(x), of the form

• is called an EXPONENTIAL function with BASE a.
• The domain of the exponential function is (-8,
  8) i.e., ALL Real Numbers

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Recall Rules of Exponents

• Let a, b, x, and y be real numbers with a gt 0
  and b gt 0. Then

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Evaluate Exponential Functions

• Example ?

• Solution ?

• Example ?

• Solution ?

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Evaluate Exponential Functions

• Example ?

• Solution ?

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Example ? Graph y f(x) 3x

• Graph the exponential fcn

• Make T-Table, Connect Dots

x y
0 1 1 2 2 3 1 3 1/3 9 1/9 27
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Example ? Graph Exponential

• Graph the exponential fcn

• Make T-Table, Connect Dots

x y
0 1 1 2 2 3 1 1/3 3 1/9 9 27

• This fcn is a REFLECTION of y 3x

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Example ? Graph Exponential

• Graph the exponential fcn

• Construct SideWays T-Table

x -3 -2 -1 0 1 2 3
y (1/2)x 8 4 2 1 1/2 1/4 1/8

• Plot Points and Connect Dots with Smooth Curve

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Example ? Graph Exponential

• As x increases in the positive direction, y
  decreases towards 0

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Exponential Fcn Properties

• Let f(x) ax, a gt 0, a ? 1. Then
• The domain of f(x) ax is (-8, 8).
• The range of f(x) ax is (0, 8) thus, the
  entire graph lies above the x-axis.
• For a gt 1 (e.g., 7)
• f is an INcreasing function thus, the graph is
  RISING as we move from left to right
• As x?8, y ax increases indefinitely and VERY
  rapidly

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Exponential Fcn Properties

• Let f(x) ax, a gt 1, a ? 1. Then
• As x?-8, the values of y ax get closer and
  closer to 0.
• For 0 lt a lt 1 (e.g., 1/5)
• f is a DEcreasing function thus, the graph is
  falling as we scan from left to right.
• As x?-8, y ax increases indefinitely and VERY
  rapidly
• As x? 8, the values of y ax get closer and
  closer to 0

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Exponential Fcn Properties

• Let f(x) ax, a gt 0, a ? 1. Then
• Each exponential function f is one-to-one
  i.e., each value of x has exactly ONE target.
  Thus

1. f has an inverse

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Exponential Fcn Properties

• Let f(x) ax, a gt 0, a ? 1. Then
• The graph f(x) ax has no x-intercepts
• In other words, the graph of f(x) ax never
  crosses the x-axis. Put another way, there is no
  value of x that will cause f(x) ax to equal 0
• The x-axis is a horizontal asymptote for every
  exponential function of the form f(x) ax.

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Translate Exponential Graphs
Translation
Equation
Effect on Equation
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Example ? Sketch Graph

• By TranslationMove DOWNy 3x by 3 Units

• Note
• Domain (-8, 8)
• Range (-4, 8)
• Horizontal Asymptote y -4

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Example ? Sketch Graph

• By TranslationMove LEFTy 3x by 1 Unit

• Note
• Domain (-8, 8)
• Range (0, 8)
• Horizontal Asymptote y 0

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Alternative Graph Swap x y

• It will be helpful in later work to be able to
  graph an equation in which the x and y in y ax
  are interchanged

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Example ? Graph x 3y

• Graph the exponential fcn

• Make T-Table, Connect Dots

x y
1 3 1/3 9 1/9 27 0 1 1 2 2 3
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Example ? Apply Exponential

• Example ? Bank Interest compounded annually.
• The amount of money A that a principal P will be
  worth after t years at interest rate i,
  compounded annually, is given by the formula

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Example ? Compound Interest

• Suppose that 60,000 is invested at 5 interest,
  compounded annually
• Find a function for the amount in the account
  after t years
• SOLUTION

60000(1 0.05 )t
60000(1.05)t
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Example ? Compound Interest

• Suppose that 60,000 is invested at 5 interest,
  compounded annually
• Find the amount of money amount in the account at
  t 6.
• SOLUTION

A(6) 60000(1.05)6
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Example ? Bacterial Growth

• A technician to the Great French microbiologist
  Louis Pasteur noticed that a certain culture of
  bacteria in milk doubled every hour.
• Assume that the bacteria count B(t) is modeled by
  the equation

• Where t is time in hours

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Example ? Bacterial Growth

• Given Bacterial Growth Equation

• Find
• the initial number of bacteria,
• the number of bacteria after 10 hours and
• the time when the number of bacteria will be
  32,000.

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Example ? Bacterial Growth

1. INITIALLY time, t, is ZERO ? Sub t 0 into
   Growth Eqn

1. At Ten Hours Sub t 10 into Eqn

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Example ? Bacterial Growth

1. Find t when B(t) 32,000

• Thus 4 hours after the starting time, the number
  of bacteria will be 32k

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WhiteBoard Work

• Problems From 9.1 Exercise Set
• 36, 40, 54
• USAPersonalSavingsRate

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All Done for Today
BacteriaGrowFAST!

• Note 37 C 98.6 F (Body Temperature)

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Chabot Mathematics
Appendix
Bruce Mayer, PE Licensed Electrical Mechanical
EngineerBMayer_at_ChabotCollege.edu

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Irrational Exponents

• By The Properties of Exponents we Can Evaluate
  Bases Raised to Rational-Number Powers Such as

• What about expressions with IRrational exponents
  such as

• To attach meaning to this expression consider
  a rational approximation, r, for the Square Root
  of 2

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Irrational Exponents

• Approximate byITERATION on

1.4 lt r lt 1.5 1.41 lt r lt 1.42 1.414 lt r lt
1.415
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Irrational Exponents

• Thus by Iteration

• Any positive irrational exponent can be
  interpreted in a similar way.
• Negative irrational exponents are then defined
  using reciprocals.