Notetaking with Vocabulary

Notetaking with Vocabulary Core Concepts The Natural Base e It is defined as follows: The equation (1 + 1/x)^x approaches e ≈ 2.71828182846 as x approaches infinity. Examples: x = 1: (1 + 1/1)^1 = (2)^1 = 2 x = 2: (1 + 1/2)^2 = (1.5)^2 = 2.25 x = 5: (1 + 1/5)^5 = (1.2)^5 = 2.488 x = 50: (1 + 1/50)^50 = (1.02)^50 = 2.691 Treat it like any other variable 3e + 5e = 8e e^3 * e^5 = e^(3+5) = e^8 (e^3)^5 = e^(3*5) = e^15 Natural Base Functions A function of the form y = ae^rx is called a natural base exponential function. When a > 0 and r > 0, the function is an exponential growth function.   Graph: An upward-sloping curve that approaches the x-axis asymptotically. Example: y = e^x Asymptote: y = 0 (x-axis) When a > 0 and r < 0, the function is an exponential decay function. Graph: A downward-sloping curve that approaches the x-axis asymptotically. Example: y = e^(-x) Asymptote: y = 0 (x-axis) Continuously Compounded Interest When interest is compounded continuously, the amount A in an account after t years is given by the formula: A = Pe^(rt) where P is the principal and r is the annual interest rate expressed as a decimal.   "No more 'n' since it is continuously compounded" "rate is now an exponent" 6.2 Extra Practice Section 1: Simplifying Expressions with "e" 1. e^-9 * e^12 Add exponents = e^3 2. 25x^2 / 35e^7 Reduce by 5 = 5x^2 / 7e^7 3. (2e^-3x)^5 * 2e^(x+1) Distribute the exponent = 2^5 * e^(-15x) * 2e^(x+1) Combine exponents = 32e^(-14x+1) 4. √(16e^(24x)) √(16e^(24x)) = 4e^(24x/4) = 4e^(6x) Section 2: Identifying and Graphing Exponential Growth/Decay Functions 5. y = 2e^(-x) Decay Graph: (-3, 40.17) (-2, 14.78) (-1, 5.44) (0, 2) (1, 0.74) (2, 0.27) (3, 0.1) 6. y = 0.75e^(4x) Growth Graph: (-3, 0.000041) (-2, 0.0005) (-1, 0.0061) (0, 0.75) (1, 10.1) (2, 135.9) (3, 1825.1) 7. y = 5e^(0.25x) Growth Graph: (-4, 1.84) (-3, 2.23) (-2, 2.72) (-1, 3.35) (0, 5) (1, 6.18) (2, 7.53) (3, 9.12) (4, 11.08) (5, 13.54) (6, 16.48) (7, 20.09) (8, 24.53) 8. y = 0.8e^(-3x) Decay Graph: (-3, 322.74) (-2, 16.27) (-1, 0.81) (0, 0.8) (1, 0.036) (2, 0.0006) Section 1: Graphing Exponential Functions 9. y = e^(-x) Graph: x: -3, -2, -1, 0, 1, 2, 3 y: 20.09, 7.39, 2.72, 1, 0.37, 0.14, 0.05 Domain: All reals Range: y > 0 10. y = 2e^(x+3) Graph: x: -6, -5, -4, -3, -2, -1, 0 y: 0.09, 0.27, 0.82, 2, 5.44, 14.78, 40.17 Domain: All reals Range: y > 0 11. y = -5e^x Graph: x: -3, -2, -1, 0, 1, 2, 3 y: -0.18, -0.68, -2.28, -5, -13.59, -36.79, -100.43 Domain: All reals Range: y < 0 Section 2: Word Problems with Exponential Functions 12. Population Decline Problem: The population of Evans City is currently 48,500 and is declining at a rate of 2.5% each year. Predict the population of Evans City after 25 years. Equation: P(t) = P₀ * e^(rt) Solution: P(t) = 48500 * e^(-0.025 * 25) P(t) = 48500 * e^(-0.625) P(t) ≈ 12560 Answer: The population of the town after 25 years will be approximately 12,560 people. 13. College Tuition Savings Problem: Your parents will need $25,000 in 10 years to pay for your brother's college tuition. They can invest in an account with an interest rate of 9.8% 1 that compounds continuously. How much should your parents invest today in order to have your brother's full tuition available in 10 years? 2   Equation: A = Pe^(rt) Solution: 25000 = P * e^(0.098 * 10) 25000 = P * e^0.98 P = 25000 / e^0.98 P ≈ 938.28 Answer: Your parents will need to invest a minimum of $938.28 to have $25,000 for your brother's college tuition in 10 years.