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Mathematics 116 Chapter 5

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# Mathematics 116 Chapter 5 - PowerPoint PPT Presentation

Mathematics 116 Chapter 5. Exponential And Logarithmic Functions. John Quincy Adams. “Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.” Mathematics 116 Exponential Functions and Their Graphs. Def: Relation.

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Mathematics 116Chapter 5
• Exponential
• And
• Logarithmic Functions
John Quincy Adams
• “Patience and perseverance have a magical effect before which difficulties disappear and obstacles vanish.”
• Mathematics 116
• Exponential Functions
• and
• Their Graphs
Def: Relation
• A relation is a set of ordered pairs.
• Designated by:
• Listing
• Graphs
• Tables
• Algebraic equation
• Picture
• Sentence
Def: Function
• A function is a set of ordered pairs in which no two different ordered pairs have the same first component.
• Vertical line test – used to determine whether a graph represents a function.
Defs: domain and range
• Domain: The set of first components of a relation.
• Range: The set of second components of a relation
Objectives
• Determine the domain, range of relations.
• Determine if relation is a function.
Mathematics 116
• Inverse Functions
Objectives:
• Determine the inverse of a function whose ordered pairs are listed.
• Determine if a function is one to one.
Inverse Function
• g is the inverse of f if the domains and ranges are interchanged.
• f = {(1,2),(3,4), (5,6)}
• g= {(2,1), (4,3),(6,5)}
One-to-One Function
• A function f is one-to one if for and and b in its domain, f(a) = f(b) implies a = b.
• Other – each component of the range is unique.
One-to-One function
• Def: A function is a one-to-one function if no two different ordered pairs have the same second coordinate.
Horizontal Line TestA test for one-to one
• If a horizontal line intersects the graph of the function in more than one point, the function is not one-to one
Existence of an Inverse Function
• A function f has an inverse function if and only if f is one to one.
Find an Inverse Function
• 1. Determine if f has an inverse function using horizontal line test.
• 2. Replace f(x) with y
• 3. Interchange x and y
• 4. Solve for y
• 5. Replace y with
Definition of Inverse Function
• Let f and g be two functions such that f(g(x))=x for every x in the domain of g and g(f(x))=x for every x in the domain of.
• g is the inverse function of the function f
Objective
• Recognize and evaluate exponential functions with base b.
Michael Crichton – The Andromeda Strain (1971)
• The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. It this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth.”
Graph
• Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes
Graph
• Determine: Domain, range, function and why, 1-1 function and why, y intercept, x intercept, asymptotes
Exponential functions
• Exponential growth
• Exponential decay
Properties of graphs of exponential functions
• Function and 1 to 1
• y intercept is (0,1) and no x intercept(s)
• Domain is all real numbers
• Range is {y|y>0}
• Graph approaches but does not touch x axis – x axis is asymptote
• Growth or decay determined by base
Calculator Keys
• Second function of divide
• Second function of LN (left side)
Dwight Eisenhower – American President
• “Pessimism never won any battle.”
Property of equivalent exponents
• For b>0 and b not equal to 1
Compound Interest
• A = Amount
• P = Principal
• r = annual interest rate in decimal form
• t= number of years
Continuous Compounding
• A = Amount
• P = Principal
• r = rate in decimal form
• t = number of years
Compound interest problem
• Find the accumulated amount in an account if \$5,000 is deposited at 6% compounded quarterly for 10 years.
Objectives
• Recognize and evaluate exponential functions with base b
• Graph exponential functions
• Recognize, evaluate, and graph exponential functions with base e.
• Use exponential functions to model and solve real-life problems.
Albert Einstein – early 20th century physicist
• “Everything should be made as simple as possible, but not simpler.”
Mathematics 116 – 4.2
• Logarithmic Functions
• and
• Their Graphs
Objectives
• Recognize and evaluate logarithmic function with base b
• Note: this includes base 10 and base e
• Graph logarithmic functions
• By Hand
• By Calculator
Shape of logarithmic graphs
• For b > 1, the graph rises from left to right.
• For 0 < b < 1, the graphs falls from left to right.
Properties of Logarithmic Function
• Domain:{x|x>0}
• Range: all real numbers
• x intercept: (1,0)
• No y intercept
• Approaches y axis as vertical asymptote
• Base determines shape.
Evaluate Logs on calculator
• Common Logs – base of 10
• Natural logs – base of e
**Property of Logarithms
• One to One Property
Objective
• Use logarithmic functions to model and solve real-life problems.
Jim Rohn
• “You must take personal responsibility. You cannot change the circumstances, the seasons, or the wind, but you can change yourself. That is something you have charge of.”
Mathematics 116 – 4.3
• Properties
• of
• Logarithms
Objectives:
• Use properties of logarithms to evaluate or rewrite logarithmic expressions
• Use properties of logarithms to expand logarithmic expressions
• Use properties of logarithms to condense logarithmic expressions.
Albert Einstein
• “The important thing is not to stop questioning.”
Mathematics 116
• Solving
• Exponential
• and
• Logarithmic Equations
Solving Exponential Equations
• 1. *** Rewrite equation so exponential term is isolated.
• 2. Rewrite in logarithmic form
• Use base ln if base is e.
• 3. Solve the equation
• 4. Check the results
• Graphically or algebraically
Solve Logarithmic Equations
• 1. *** Rewrite equation so logarithmic term is isolated. Or use one-one property
• 2. Rewrite in exponential form
• 3. Solve the equation
• 4. Check the results
• Graphically or algebraically
Walt Disney
• “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”
Objectives:
• Solve exponential equations
• Solve logarithmic equations
• Use exponential and logarithmic equations to model and solve real-life problems.
Hans Hofmann – early 20th century teacher and painter
• “The ability to simplify means to eliminate the unnecessary so that the necessary may speak.”
Mathematics 116
• Exponential
• and
• Logarithmic
• Models
Objective
• Recognize the most common types of models involving exponential or logarithmic functions
Models
• Exponential growth
• Exponential decay
• Logarithmic
• Common logs
• Natural logs
Gaussian Model
• “normal curve”
Magnitude of Earthquake
• Uses Richter scale I is intensity which is a measure of the wave energy of an earthquake
Carl Zuckmeyer
• “One-half of life is luck; the other half is discipline – and that’s the important half, for without discipline you wouldn’t know what t do with luck.”
Mathematics 116 – 4.6
• Exploring Data:
• Nonlinear Models
Objectives
• Classify Scatter Plots
• Use scatter plots and a graphing calculator to find models for data and choose a model that best fits a set of data.
• Use a graphing utility to find models to fit data.
• Make predictions from models.
Calculator regression models
• Linear(mx+b) (preferred) and (b+mx)
• Quadratic – 2nd degree
• Cubic – 3rd degree
• Quartic – 4th degree
• Ln (natural logarithmic logarithm)
• Exponential
• Power
• Logistic
• Sin – (trigonometric)
Julie Andrews
• “Perseverance is failing 19 times and succeeding the 20th.”
Walt Disney
• “Disneyland will never be completed. It will continue to grow as long as there is imagination left in the world.”