Exponential, Logistic, and Logarithmic Functions

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Exponential, Logistic, and Logarithmic Functions

Vocabulary Algebraic Function- A combination of variables and constants involving addition, subtraction, multiplication, division, powers, and rootsExponential Function- A function in the form f(x) = a x b^x Base- the variable b in the exponential functionExponential Growth Function- Growth modeled by f(x) = a x b^x, a>0, b>1 Exponential Decay Function- Decay modeled by f(x) = a x b^x, a>0, b<0, b>1Logistic Growth Function- A model of population growth: f(x) = c/1+a x b^xLimit to Growth- the variable c in hte logistic growth functionLogistic Decay Function- if b>1 or k<0 in the logistic decay functionLogistic Function- f(x) = 1/ 1+e^-xRadioactive Decay- exponential decay function that models the amount of a radioactive substant present in a sampleHalf-Life- The amount of time required for half of a radioactive substance to decayMaximum Sustainable Population- Logarithmic Function with base b- The inverse of the exponential function y=b^x, denoted by y=logbxCommon Logarithms- A logarithm with base 10Natural Logarithms- A logarithm with base e Product Rule- logb(RS) = logbR + logbS Quotient Rule- logb(R/S) = logbR - logbSPower Rule- logbR^c = clogbRRe-Expression- A transformation of a data setRichter Scale-A logarithmic scale used in measuring the intensity of earthquakesOrder of Magnitude- lognNewton's Law of Cooling- T(t) = Tm + (T0 - Tm)e^-ktLinear Regression- A procedure for finding the straight line that is the best fit for the dataNatural Logarithmic Regression- A procedure for fitting a logarithmic curve to a set of data Exponential Regression- A procedure for fitting an exponential function to a set of dataPower Regression- A procedure for fitting a curve y= a x x^b to a set of dataCompound Interest- Interest that becomes part of the investmentCompound Conituously- Interest compounded using the formula A=Pe^rtAnnual Percentage Yield- The rate that would give the same return if interest were computed just once a yearAnnuity- A sequence of equal periodic paymentsOrdinary- An annuity in which deposits are made at the same time interest is postedFuture Value- The net ammount of money returned from an annuityPresent Value- The net amount of your money put into an annuityAnnual Percentage Rate- The annual interest rate

Chapter 3: Exponential, Logistic, and Logarithmic Functions

Q: Suppose a bacteria population starts with 10 bacteria and divides every hour. What is the population seven hours later? A: Use the equation y=a x b^x, where a is the initial value, 10, b is the growth rate, 2 , and x is the amount of time, 7. The population seven hours later y is 1280 bacteria

Useful WebsitesThe Natural Base eExponential Growth and DecayProperties of Logarithms Solving Logarithmic Equations

Newton's Law of Cooling

The Richter Scale

An amount of $2,340.00 is deposited in a bank paying an annual interest rate of 3.1%, compounded continuously. Find the balance after 3 years.Use the continuous compound interest formula, A = Pe^rt, with P = 2340, r = 3.1/100 = 0.031, t = 3. A= 2340e^0.031(3). A ≈ $2,568.06

Real World Applications


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