4.3 – Use Functions involving e

1. 4.3 – Use Functions involving e GPS: MM3A2e, MM3A2f, MM3A2d

2. GPS • MM3A2e – Investigate and explain characteristics of exponential and logarithmic functions including domain and range, asymptotes, zeros, intercepts, intervals of increase and decrease, and rate of change. • MM3A2f – Graph functions as transformations of f(x) = ax, f(x) = logax, f(x) = ex, f(x) = ln x. • MM3A2g - Explore real phenomena related to exponential and logarithmic functions including half-life and doubling time. • MM3A3d –Solve a variety of types of equations by appropriate means, choosing among mental calculation, pencil and paper, or appropriate technology.

3. Vocabulary • The number denoted by the letter e is called the natural base e. • It is an irrational number defined as follows: As n approaches +∞, (1 + )napproaches e≈2.718281828. • A function of the form f(x)=aerxis called a natural base exponential function. • Continuously Compounded Interest – When interest is compounded continuously, the amount A in an account after t years is given by the formula A=Pert, where P is the principal and r is the annual interest rate expressed as a decimal.

4. Remember Exponent Properties?

5. Simplify natural base expressions • Simplify the expression • e6 * e4 = e6+4= e10 • (-2ex)3 = (-2)3(ex)3 = -8e3x • 1

6. Graph natural base functions • Graph • Steps: • Make a table • Plot your points • Asymptote? • Domain & Range

7. Graph natural base functions • Graph • Steps: • Make a table • Plot your points • Asymptote? • Domain & Range

8. Graph y=abx-h +k for 0 < b < 1 • Graph • Steps: • Make a table • Plot your points • Asymptote? • Domain & Range

9. Model continuously compounded interest • You deposit $500 in an account that pays 3% annual interest compounded continuously. What is the balance after 2 years? • Use the formula for continuously compounded interest. • A = Pert

10. Continuously Compounded Interest

11. Guided Practice Page • Try page 141: 1-8