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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Exponential and Logarithmic Functions 5 Exponential Functions Logarithmic Functions Compound Interest Differentiation of Exponential Functions Differentiation of Logarithmic Functions Exponential Functions as Mathematical Models

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Exponential Function An exponential function with base b and exponent x is defined by Ex. Domain: All reals Range: y > 0 (0,1) x y x y

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Laws of Exponents LawExample

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Properties of the Exponential Function 1.The domain is. 2. The range is (0, ). 3. It passes through (0, 1). 4. It is continuous everywhere. 5. If b > 1 it is increasing on. If b < 1 it is decreasing on.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Examples Ex.Simplify the expression Ex.Solve the equation

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Logarithms The logarithm of x to the base b is defined by Ex.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Examples Ex. Solve each equation a. b.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Notation: Common Logarithm Natural Logarithm Laws of Logarithms

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Use the laws of logarithms to simplify the expression:

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Logarithmic Function The logarithmic function of x to the base b is defined by Properties: 1. Domain: (0, ) 2.Range: 3. x-intercept: (1, 0) 4. Continuous on (0, ) 5. Increasing on (0, ) if b > 1 Decreasing on (0, ) if b < 1

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Graphs of Logarithmic Functions Ex. (1,0) xx y y (0, 1)

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Ex. Solve Apply ln to both sides.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example A normal child’s systolic blood pressure may be approximated by the function where p(x) is measured in millimeters of mercury, x is measured in pounds, and m and b are constants. Given that m = 19.4 and b = 18, determine the systolic blood pressure of a child who weighs 92 lb.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Compound Interest Formula A = The accumulated amount after mt periods P = Principal r = Nominal interest rate per year m = Number of periods/year t = Number of years

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find the accumulated amount of money after 5 years if $4300 is invested at 6% per year compounded quarterly. = $

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example How long will it take an investment of $10,000 to grow to $15,000 if it earns an interest at the rate of 12% / year compounded quarterly?

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Effective Rate of Interest r eff = Effective rate of interest r = Nominal interest rate/year m = number of conversion periods/year Ex. Find the effective rate of interest corresponding to a nominal rate of 6.5% per year, compounded monthly. It is about 6.7% per year.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Present Value Formula for Compound Interest A = The accumulated amount after mt periods P = Principal r = Nominal interest rate/year m = Number of periods/year t = Number of years

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find the present value of $4800 due in 6 years at an interest rate of 9% per year compounded monthly.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Continuous Compound Interest Formula A = The accumulated amount after t years P = Principal r = Nominal interest rate per year t = Number of years

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Find the accumulated amount of money after 25 years if $7500 is invested at 12% per year compounded continuously.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Differentiation of Exponential Functions Chain Rule for Exponential Functions Derivative of Exponential Function If f (x) is a differentiable function, then

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Examples Find the derivative of Find the relative extrema of – + Relative Min. f (0) = 0 Relative Max. f (-1) = x

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Differentiation of Logarithmic Functions Chain Rule for Exponential Functions Derivative of Exponential Function If f (x) is a differentiable function, then

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Examples Find the derivative of Find an equation of the tangent line to the graph of Slope: Equation:

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Logarithmic Differentiation 1.Take the Natural Logarithm on both sides of the equation and use the properties of logarithms to write as a sum of simpler terms. 2.Differentiate both sides of the equation with respect to x.x. 3.Solve the resulting equation for.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Examples Use logarithmic differentiation to find the derivative of Apply ln Differentiate Properties of ln Solve

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Exponential Growth/Decay Models Q 0 is the initial quantity k is the growth/decay constant A quantity Q whose rate of growth/decay at any time t is directly proportional to the amount present at time t can be modeled by: Growth Decay

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example A certain bacteria culture experiences exponential growth. If the bacteria numbered 20 originally and after 4 hours there were 120, find the number of bacteria present after 6 hours.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Learning Curves C, A, k are positive constants An exponential function may be applied to certain types of learning processes with the model: y = C x y

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example Suppose that the temperature T, in degrees Fahrenheit, of an object after t minutes can be modeled using the following equation: 1. Find the temperature of the object after 5 minutes. 2. Find the time it takes for the temperature of the object to reach 190°.

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Logistic Growth Model A, B, k are positive constants An exponential function may be applied to a logistic growth model: y = A x y

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example The number of people R, in a small school district who have heard a particular rumor after t days can be modeled by: If 10 people know the rumor after 1 day, find the number who heard it after 6 days. …

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. So

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. Example The number of soldiers at Fort MacArthur who contracted influenza after t days during a flu epidemic is approximated by the exponential model: If 40 soldiers contracted the flu by day 7, find how many soldiers contracted the flu by day 15. …

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Copyright © 2006 Brooks/Cole, a division of Thomson Learning, Inc. So

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