University:
High School
Course:
High School Mathematics
Academic year:

2022
Views:

489

Pages:

3
Author:

usarilearlyzs7g

Page 1: The Natural Base e (6.2) Title: 6.2 The Natural Base e A. Vocabulary & Concepts The natural base e (1 + 1/x)^x = e = 2.718 as x approaches ∞ (as x gets bigger) Natural Base Functions y = ae^rx (positive exponent) when a>0 and r>0 => Exponential Growth (negative exponent) when a>0 and r <0 => Exponential Decay Graphs Graph 1: y = e^x labeled points: (0,1), (1, 2.718) Graph 2: y = e^-x labeled points: (0,1), (1, 1/2.718) or (1,.368) Continuously Compounded Interest A = Pe^rt t: time in years r: annual interest rate (as a decimal) P: principal (beginning amount) A: amount Page 2: Examples B. Examples Simplify the Expression a. e^3 * e^6 = e^(3+6) = e^9 b. 16e^5 / 4e^4 = 4e^(5-4) = 4e^1 = 4e c. (3e^-4x)^2 = 3^2 * e^(-4x*2) = 9e^-8x = 9/e^(8x) You Try: d. e^7 * e^4 = e^(7+4) = e^11 e. 24e^8 / 8e^5 = 3e^(8-5) = 3e^3 f. (10e^-3x)^3 = 10^3 * e^(-3x*3) = 1000e^-9x = 1000/e^9x Tell whether the function represents exponential growth or exponential decay. Then graph the function. a. y = 3e^x positive exponent exponential growth table of values (-2, .4), (-1, 1.1) (0, 3), (1, 8.1), (2, 22.2) graph of y = 3e^x diagram included showing x and y values b. y = e^-0.5x * negative exponent * exponential decay table of values (-2, 2.718), (-1, 1.649), (0,1), (1, .606), (2, .368) * graph of y = e^-0.5x c. y = 4e^-x negative exponent exponential decay Table of values (-2, 29.556), (-1, 10.873), (0,4), (1, 1.471), (2, 0.541) graph of y = 4e^-x You deposit $4250 in an account that earns 5% annual interest compounded continuously. What is the balance after 10 years? Continuously Compounded Interest Formula: A = Pe^rt P = 4250 r = 5% = 0.05 t = 10 A = 4250e^(.05 * 10) = 4250e^(.5) = 4250(1.649) A = 7007.07 The balance after 10 years is $7,007.07.