Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 1 Homework, Page 286 Is the function an exponential function? If so, state the initial value and the base. If not, explain why not. 1. The function is not an exponential function, but rather a power function.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 2 Homework, Page 286 Is the function an exponential function? If so, state the initial value and the base. If not, explain why not. 5. The function is not an exponential function, but rather a power function with a variable base and a variable power.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 3 Homework, Page 286 Compute the exact value of the function for the given x-value without using a calculator. 9.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 4 Homework, Page 286 Determine a formula for the exponential function whose graph is shown. 13.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 5 Homework, Page 286 Describe how to transform the graph of f into the graph of g. Sketch the graphs and support with a grapher. 17. Transform f to g by reflecting the graph of f about the y- axis.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 6 Homework, Page 286 Describe how to transform the graph of f into the graph of g. Sketch the graphs and support with a grapher. 21. Transform f to g by reflecting the graph of f about the y- axis and applying a horizontal stretch of 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 7 Homework, Page 286 (a) Match the given function with its graph. (b) Explain how to make the choice without using a grapher. 25. a. The given function is a basic exponential function and its graph is that of an exponential function with initial value 1.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 8 Homework, Page 286 (a) Match the given function with its graph. (b) Explain how to make the choice without using a grapher. 29. b. The given function is an exponential function that has been reflected about the y-axis and translated two units down.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 3- 9 Homework, Page 286 State whether the function is an exponential growth or exponential decay function and describe its end behavior using limits. 33. The given function is an exponential decay function and its end behavior is given by
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Solve the inequality graphically. 37.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Use a grapher to graph the function. Find the y- intercept and the horizontal asymptotes. 41.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 45.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Graph the function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 49.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Using the data from the table and assuming that growth is exponential, when would the populations of Austin and Columbus be equal? City1990 Pop.2000 Pop. Austin465,622656,562 Columbus632,910711,265
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The populations would be equal about the middle of 2003.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page The number B of bacteria in a Petri dish culture after t hours is given by. (a) What was the initial number of bacteria present? Initially, 100 bacteria were present. (b)How many bacteria were present after 6 hours?
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page Which of the following functions is exponential? A. B. C. D. E.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Graph each function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 65.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Graph each function and analyze it for domain, range, continuity, increasing or decreasing behavior, symmetry, boundedness, extrema, asymptotes, and end behavior. 65.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework, Page 286 Refer to the expression f (a, b, c) =a·b c. If a = 2, b = 3, and c = x, the expression is f (2, 3, x) = 2·3 x, an exponential function. 69. If b = x, state conditions on a and c under which the expression is a quadratic power function. The expression would be a quadratic power function if a ≠ 0 and c = 2.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.2 Exponential and Logistic Modeling
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Constant Percentage Rate and Exponential Functions Exponential Growth and Decay Models Using Regression to Model Population Other Logistic Models … and why Exponential functions model many types of unrestricted growth; logistic functions model restricted growth, including the spread of disease and the spread of rumors.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Constant Percentage Rate Suppose that a population is changing at a constant percentage rate r, where r is the percent rate of change expressed in decimal form. Then the population follows the pattern shown. If r > 0, then P(t) is an exponential growth function. If r < 0, then P(t) is an exponential decay function.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Exponential Population Model
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Bacteria Growth
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Growth with a Logistic Model
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling Atmospheric Pressure
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for 2003.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Example Modeling U.S. Population Using Exponential Regression Slide 3- 34
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Maximum Sustainable Population Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population. These situations are best modeled using a logistic model.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Modeling a Rumor
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Homework Review Section 3.2 Page 296, Exercises: 1 – 57 (EOO), skip 49 Quiz next time
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.3 Logarithmic Functions and Their Graphs
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide What you’ll learn about Inverses of Exponential Functions Common Logarithms – Base 10 Natural Logarithms – Base e Graphs of Logarithmic Functions Measuring Sound Using Decibels … and why Logarithmic functions are used in many applications, including the measurement of the relative intensity of sounds.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Changing Between Logarithmic and Exponential Form
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Inverses of Exponential Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Logarithms Evaluate the logarithmic expression without using a calculator.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide An Exponential Function and Its Inverse
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Common Logarithm – Base 10 Logarithms with base 10 are called common logarithms. The common logarithm log 10 x = log x. The common logarithm is the inverse of the exponential function y = 10 x.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Common Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Solving Simple Logarithmic Equations
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Basic Properties of Natural Logarithms
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Evaluating Natural Logarithms Evaluate the logarithmic expressions:
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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Drawing Logarithmic Graphs Draw the graph of the given function:
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Transforming Logarithmic Graphs
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Decibels
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide Example Computing Decibel Levels Compute the decibel levels of the following Subway train Threshold of pain