Exponential Functions & Logarithms: Interest, Growth, Decay, Logarithmic Applications - Pr, Study notes of Pre-Calculus

Download Exponential Functions & Logarithms: Interest, Growth, Decay, Logarithmic Applications - Pr and more Study notes Pre-Calculus in PDF only on Docsity! 241 L21 Applications of Exponential Function and Logarithms; Logistic Models Simple Interest Formula: If a principal of P dollars is invested for a period of t years at a per annum interest rate R, expressed as a decimal, the interest I earned is I P R t= ⋅ ⋅ The interest I is called the simple interest. Compound interest is the interest paid on the principal and previously earned interest. Compound Interest Formula: The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is 1 ntrA P n ⎛ ⎞= ⋅ +⎜ ⎟ ⎝ ⎠ Note: The more frequently the interest rate is compounded (the larger n), the larger is the amount of A. Question: Is it true that A→∞ , as n →∞? 242 Example: Suppose that a principal P =$1.00 is invested at an annual interest rate 1r = (100%) compounded n times per year. (a) Find the future value A after 1t = year. (b) What value does A approach when n →∞? In general, lim 1 nt rt n rP Pe n→∞ ⎛ ⎞+ =⎜ ⎟ ⎝ ⎠ Continuous Compounding: The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is rtA Pe= 245 Exponential Growth and Decay The exponential model is used when the quantity changes with time proportionally to the amount or number present: 0( ) ktA t A e= where 0 (0)A A= is the original amount or number and 0k ≠ is a constant. Uninhibited Growth of Population: 0( ) , 0 ktN t N e k= > . Uninhibited Radioactive Decay: 0( ) , 0 ktA t A e k= < . 246 Example: A sample culture contains 500 bacteria when first measured and 1000 bacteria when measured 72 minutes later. (a) Determine a formula for the number of bacteria ( )N t at any time t hours after the original measurement. (b) What is the number of bacteria at the end of 3 hours? (c) How long does it take for the number to increase to 5000? 247 The half-life is the time it takes for a half of a given amount to decay. Example: Find the half-life of iodine-131 (used in the diagnosis of the thyroid gland) if it decays according to the function 0.0866 0( ) tA t A e−= where t is in days. 250 Richter scale Magnitude of an Earthquake: An earthquake whose seismographic reading measures x millimeters at a distance of 100 km from the epicenter has the magnitude ( )M x , given by 0 ( ) log xM x x ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ where 30 10x −= mm is the reading of a zero-level earthquake at distance of 100 km from its epicenter. Example: Determine the magnitude of an earthquake in Japan in July, 1993 whose seismographic reading measured 63,095.734 mm at 100 km from the epicenter. 251 Logistic Functions Logistic growth model can be used when the value of the dependent variable is limited as the time elapses. Logistic Growth Model: In a logistic growth model, the dependent variable P after time t obeys the equation ( ) 1 bt cP t ae− = + where a, b, and c are constants with 0c > and 0b > Note: If t →∞ , then 0bte− → and ( )P t c→ . The number c is called the carrying capacity. Example: (P. 473, problem #22) The logistic growth model 0.339 0.90( ) 1 3.5 t P t e− = + relates the proportion of new personal computers sold at Best Buy that have Intel’s latest coprocessor t month after it has been introduced. 252 (a) What proportion of new personal computers sold at Best Buy will have Intel’s latest coprocessor when it was first introduced (that is, at 0t = ) (b) Determine the maximum proportion of new personal computers sold at Best Buy that will have Intel’s latest coprocessor (carrying capacity). (c) When will 0.75 (75%) of new personal computers sold at Best Buy have Intel’s latest coprocessor?