Session 6 : 9/221 Exponential and Logarithmic Functions

Session 6 : 9/222 Exponential Functions  Definition: If a is some number greater than 0, and a = 1, then the exponential function with base a is: Examples:

Session 6 : 9/223 Properties of Exponents  If a and b are positive numbers:

Session 6 : 9/224 Graphs of Exponential Functions  Point plotting or graphing tool  If base is raised to positive x, function is an increasing exponential. If base is raised to negative x, function is a decreasing exponential If a>1 and to a (+)x : Increasing Exponential If a<1 and to a (+)x : Decreasing Exponential If a>1 and to a (-)x: Decreasing Exponential If a<1 and to a (-)x: Increasing Exponential

Session 6 : 9/225

6 Sketching an Exponential  Find horizontal asymptote and plot several points How do we find horizontal asymptote?  Take the limit as x approaches infinite (for decreasing exponentials) or negative infinite (for increasing exponentials)

Session 6 : 9/227 INCREASING EXPONENTIAL Asymptote for increasing exponential function Asymptote for decreasing exponential function x + x - 8 8

Session 6 : 9/228 Natural Exponential Functions  In calculus, the most convenient (or natural) base for an exponential function is the irrational number e (will become more obvious once we start trying to differentiate/integrate…) e ≈  Simplest Natural Exponential:

Session 6 : 9/229 Graph of the Natural Exponential Function

Session 6 : 9/2210 Exponential Growth  Exponential functions (particularly natural exponentials) are commonly used to model growth of a quantity or a population  What growth is unrestricted, can be described by a form of the standard exponential function (probably will have multiplying constants, slight changes…):  When growth is restricted, growth may be best described by the logistic growth function: Where a, b, and k are constants defined for a given population under specified conditions.

Session 6 : 9/2211 Comparing Exponential v. Logistic Growth Function y x Exponential Logistic Growth Function

Session 6 : 9/2212 Derivatives of The Natural Exponential Function  From now on, ‘Exponential Function’ will imply an function with base e  Previously, we said that e is the most convenient base to use in calculus. Why? Very simple derivative! Chain rule, where u is a function of x

Session 6 : 9/2213 What does this mean graphically? For the function the slope at any point x is given by the derivative 1 slope =e 1 2 slope = e 2 slope =e 0 =1

Session 6 : 9/2214  Examples:

Session 6 : 9/2215 Logarithmic Functions  Review of ‘log’ If no base specified, log 10

Session 6 : 9/2216 The Natural Log  Natural Log=log e =ln  Definition of the natural log: The natural logarithmic function, denoted by ln(x), is defined as: Why?

Session 6 : 9/2217 Important Properties of Logarithmic Functions Natural log is inverse of exponential Exponential is inverse of natural log

Session 6 : 9/2218 Examples:  Solve the following logarithmic functions for x  Simplify the following:

Session 6 : 9/2219 Examples: Solving Exponential and Logarithmic Equations

Session 6 : 9/2220 Example: Doubling Time  For an account with initial balance P, the function for the account balance (A) after t years (with annual interest rate r compounded continuously) is given by: Find an expression for the time at which the account balance has doubled.

Session 6 : 9/2221 Derivative of logarithmic functions: Where u is a function of x

Session 6 : 9/2222 Examples

Session 6 : 9/2223 Exponential Growth and Decay  Law of exponential growth and decay: If y is a positive quantity whose rate of change with respect to time is proportional to the quantity present at any time t, then y is described by: Where C is the initial value k is the constant of proportionality (often rate constant) If k > 0: Exponential Growth If k < 0: Exponential Decay

Session 6 : 9/2224 Example:  Modeling population growth:  A researcher is trying to develop an equation to describe bacterial growth, and knows that it will follow the fundamental equation for exponential growth. The following data is available: At t=2 hours, there are 1x10 6 cells At t=8 hours, there are 5x10 8 cells  Write an equation for the exponential growth of bacterial cells by the following steps: 1. Find k 2. Find C using the solution for k 3. Write the full model by plugging in C and k values.  Find the time at which the population is double that of the initial population.