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Section 3.1 Vocabulary

Section 3.1 Vocabulary. Algebraic functions. Include: Polynomial functions, and rational functions. Exponential Function. The exponential function f with base a is denoted by f(x) = a x Where a > 0, a ≠ 1, and x is any real number. The natural base. e ≈ 2.718281828….

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Section 3.1 Vocabulary

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  1. Section 3.1 Vocabulary

  2. Algebraic functions • Include: Polynomial functions, and rational functions.

  3. Exponential Function • The exponential function f with base a is denoted by • f(x) = ax Where a > 0, a ≠ 1, and x is any real number

  4. The natural base • e ≈ 2.718281828…..

  5. Natural Exponential Function f(x) = ex

  6. Continuous compounding Interest • After t years, the balance A in an account with principal P and annual interest rate r is given by the following formula: A = Pert

  7. Interest (n) compoundings per year • After t years, the balance A in an account with principal P and annual interest rate r is given by the following formula A = P(1+ (r/n) ) nt

  8. Section 3.2 Vocabulary

  9. Logarithmic Function with base a • For x > 0, a > 0, and a ≠ 1, y = log ax if and only if x = ay The function given by f(x) = log ax Is called the logarithmic function with base a

  10. Common Logarithmic Function • The logarithmic function with base 10 is called the common logarithmic function. • Note: on most calculators this is denoted by the log button.

  11. The Natural Logarithmic Function • For x > 0 y = ln (x) if and only if x = ey The function given by F(x) = loge(x) = ln(x) Is called the natural logarithmic function.

  12. Properties of Natural Logarithms • ln(1) = 0 because e0 = 1 • ln(e) = 1 because e1 = e • ln(ex) = x and e lnx = x • If ln(x) = ln(y), then x = y

  13. Section 3.3 Vocabulary

  14. Change of Base Formula • Log ax = log(x) / log(a) • Log ax = ln(x) / ln(a)

  15. Properties of Logs • Product Property: loga(uv) = loga(u) + loga(v) Quotient Property: 2. loga(u/v) = loga(u)- loga(v) 3. Power Property: loga(un) = nloga(u)

  16. Section 3.4 Vocabulary

  17. Section 3.5 Vocabulary

  18. Mathematical Models 1. Exponential growth Model: Y = aebx , b > 0 2. Exponential Decay model: Y = ae-bx , b > 0 3. Gaussian model : Y = ae-(x-b)^2/c 4. Logistic growth model Y = 1/ (1 + be-rx) 5. Logistic models: Y = a + b ln(x)

  19. Logistic Curve • S- shaped graph, given by the function: Y = a /(1 + be-rx)

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