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Understanding Compound and Simple Interest

Learn about compound and simple interest, the effect of compounding period on effective yield, and how to calculate continuously compounded accounts. Also, understand exponential growth and decay models and their applications to real-world scenarios.

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Understanding Compound and Simple Interest

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  1. 5-Minute Check on Activity 5-7 • Match the following interest types: • Compound Earning interest only on the principal • Simple Earning interest on principal and interest • If the effective yield bigger or smaller than the interest rate? • How does the compounding period affect the effective yield? • What is the formula for a continuously compounded account? • How much money would you have at retirement, if a rich uncle deposited $5000 in a stock market fund that earned 10% interest compounded continuously the day that you were born? always bigger more compounding increases the effective yield A = Pert A = Pert = 5000 e0.1(65) = 5000 e6.5 = $3,325,708.17 Click the mouse button or press the Space Bar to display the answers.

  2. Lab 5 - 8 Continuous Growth and Decay Kathmandu, Nepal 11/05/2005

  3. Objectives • Discover the relationship between the equations of exponential functions defined by y = abt and the equations of continuous growth and decay exponential functions defined by y = aekt • Solve problems involving continuous growth and decay models • Graph base e exponential functions using transformations

  4. Vocabulary • None new

  5. Activity The US Census Bureau reported that the US population on April 1, 2000 was 281,421,906. The US population on April 1, 2001 was 284,236,125. Assuming exponential growth, the US population y can be modeled by the equation y = abt, where t is the number of years since April 1, 2000 (when t = 0). What is the initial value, a? What is the annual growth factor, b? a = 281,421,906 b = 284,236,125  281,421,906 = 1.01

  6. Activity cont Assuming exponential growth, the US population y can be modeled by the equation y = abt. What is the annual growth rate? What is the equation for US population as a function of t? Use this to estimate the US population on 1 Apr 2011. r = b – 1 = 1.01 – 1 = 0.01 y(t) = 281,421,906(1.01)t y(11) = 281,421,906(1.01)11 = 281,421,906(1.115668347) ≈ 313,973,513 Estimate as of yesterday: http://www.census.gov/main/www/popclock.html

  7. Activity cont Change the equation y = abt, to a continuous growth form of y = aekt. So bt = ekt and ekt = (ek)t How are b and ek related? Using our calculator, let Y1 = ex and Y2 = 1.01 and find their intersection (solution for b = ek)? Rewrite the US population function in continuous growth format. b = ek or 1.01 = ek k = 0.00995 y = 281,421,906e0.00995t

  8. Continuous Growth Reminder Continuous growth is modeled by the equation: y = aekt where a is the initial amount, k is the constant continuous growth rate and t is time

  9. Continuous Growth Example A bacterial growth in a culture increases by 25% every hour. If 10000 are present when the experiment starts: Determine the constant, k, in continuous growth model Write the equation for the continuous model When will the sample double? b = 1 + .25 = 1.25 b = ek 1.25 = ek via graph k = 0.2231 T = A0ekt = 10000e0.2231t 20000 = 10000e0.2231t t ≈ 3.11 hours

  10. Continuous Decay Example Tylenol (acetaminophen) is metabolized in your body and eliminated at a rate of 24% per hour. You take two Tylenol tablets (1000 milligrams) at 1200 noon. What is the initial value? Determine the decay factor, b. Find the constant continuous decay rate, k. Write the continuous decay function 1000 milligrams b = 1 - .24 = 0.76 b = ek 0.76 = ek via graph k = -0.27444 T = A0ekt = 1000e-0.27444t

  11. y x Graph of ex function y = ex Domain: all real numbers Range: y > 0 Increasing or Decreasing: always increasing (positive slopes) y-intercept = 1; no x-intercept y = 0, x-axis, is a horizontal asymptote

  12. ex Transformations Compared to y = ex, describe the graphic relationship between its graph and the following graphs: • y = - ex • y = ex+2 • y = ex + 2 • y = 2ex • y = e-x • y = 1 – 2ex Outside: Reflection across x-axis Inside: Shift left 2 units Outside: Shift up 2 units Outside: Vertical stretch by 2 Inside: Reflection across y-axis Outside: Vertical stretch by 2; reflected across x-axis and shifted up by 1

  13. Summary and Homework • Summary • Quantities that increase or decrease continuously at a constant rate can be modeled by y = aekt. • Increasing: k > 0 k is continuous rate of increase • Decreasing: k < 0 |k| is continuous rate of decrease • The initial quantity at t=0, a, may be written in other forms such as y0, P0, etc • Remember the general shapes of the graphs • Homework • page 604-09; problems 2, 3, 8

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