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State “exponential growth” or “exponential decay” (no calculator needed)

State “exponential growth” or “exponential decay” (no calculator needed). a.) y = e 2x b.) y = e –2x c.) y = 2 –x d.) y = 0.6 –x. k < 0, exponential decay. k > 0, exponential growth. 0<b<1 so decay, but reflect over y-axis, so growth. b>1 so growth,

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State “exponential growth” or “exponential decay” (no calculator needed)

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  1. State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e2x b.) y = e–2x c.) y = 2–x d.) y = 0.6–x k < 0, exponential decay k > 0, exponential growth 0<b<1 so decay, but reflect over y-axis, so growth b>1 so growth, but reflect over y-axis, so decay

  2. =  lim f(x) x   lim f(x) x  -  = 0 Characteristics of a Basic Exponential Function: Domain: Range: Continuity: Symmetry: Boundedness: Extrema: ( - ,  ) ( 0,  ) continuous none Asymptotes: End Behavior: b = 0 y = 0 none

  3. Question Use properties of logarithms to rewrite the expression as a single logarithm. log x + log y 1/5 log z log x + log 5 2 ln x + 3 ln y ln y – ln 3 4 log y – log z ln x – ln y 4 log (xy) – 3 log (yz) 1/3 log x 3 ln (x3y) + 2 ln (yz2)

  4. Change of Base Formula for Logarithms

  5. “the” exponential function the “natural base” 2.718281828459 (irrational, like ) Leonhard Euler(1707 – 1783) f(x) = a • e kx for an appropriately chosen real number, k, so ek = b exponential growth function exponential decay function

  6. State “exponential growth” or “exponential decay” (no calculator needed) a.) y = e2x b.) y = e–2x c.) y = 2–x d.) y = 0.6–x k < 0, exponential decay k > 0, exponential growth 0>b>1 so decay, but reflect over y-axis, so growth b>1 so growth, but reflect over y-axis, so decay

  7. Rewrite with e; approximate k to the nearest tenth. a.) y = 2x b.) y = 0.3x e? = 2 e? = 0.3 y = e0.7x y = e–1.2x

  8. = 1 lim f(x) x   lim f(x) x  -  = 0 Characteristics of a Basic Logistic Function: Domain: Range: Continuity: Symmetry: Boundedness: Extrema: ( - ,  ) ( 0, 1 ) continuous about ½, but not odd or even Asymptotes: End Behavior: B = 0, b = 0 y = 0, 1 none

  9. Based on exponential growth models, will Mexico’s population surpass that of the U.S. and if so, when? Based on logistic growth models, will Mexico’s population surpass that of the U.S. and if so, when? What are the maximum sustainable populations for the two countries? Which model – exponential or logistic – is more valid in this case? Justify your choice.

  10. Logarithmic Functions inverse of the exponential function logbn = p bp = n logbn = p iff bp = n find the power = 5 2? = 32 = 0 3? = 1 = ½ 4? = 2 = 1 5? = 5 = ½ 2? = 2

  11. Basic Properties of Logarithms (where n > 0, b > 0 but ≠ 1, and p is any real number) Example log51 = 0 logb1 = 0 because b0 = 1 logbb = 1 because b1 = b log22 = 1 logbbp = p because bp = bp log443 = 3 blogbn = n because logbn = logbn 6log611 = 11

  12. With a Calculator: log 32.6 = log 710 = log 0.59 = log (–4) = Evaluating Common Log Expressions Without a Calculator: 2 1.5132176 log 100 = –0.22914… 1/7 undefined 8 10 log 8 =

  13. Solving Simple Equations with Common Logs and Exponents Solve: log x = – 1.6 10 x = 3.7 x = 10 –1.6 x = log 3.7 x ≈ 0.03 x ≈ 0.57

  14. With a Calculator: ln 3e = ln 31.3 ln 0.39 ln (–3) Evaluating Natural Log Expressions Without a Calculator: 1/3 ≈ 3.443 7 ≈ – 0.9416 log e7 = 5 = undefined e ln 5 =

  15. Solving Simple Equations with Natural Logs and Exponents Solve: ex = 6.18 ln x = 3.45 x = ln 6.18 x = e 3.45 x ≈ 1.82 x ≈ 31.50

  16. Logarithmic Functions ≈ 0.91 ln x ≈ – 0.91 ln x • reflect over the x-axis • vertical shrink by 0.91 • vertical shrink by 0.91

  17. Graph the function and state its domain and range: f(x) = log4x 0.721 ln x Vertical shrink by 0.721 f(x) = log5x 0.621 ln x Vertical shrink by 0.621 f(x) = log7(x – 2) 0.514 ln (x – 2) Vertical shrink by 0.514, shift right 2 0.091 ln (–(x – 2) f(x) = log3(2 – x) Vertical shrink by 0.091 Reflect across y-axis Shift right 2

  18. Logarithmic Functions one-to-one functions u = v 2x = 25 x = 5 log22x = log27 x = log27 isolate the exponential expression take the logarithm of both sides and solve

  19. Newton’s Law of Cooling An object that has been heated will cool to the temperature of the medium in which it is placed (such as the surrounding air or water). The temperature, T, of the object at time, t, can be modeled by: where Tm = temp. of surrounding medium T0 = initial temp. of the object Example: A hard-boiled egg at temp. 96 C is placed in 16 C water to cool. Four (4) minutes later the temp. of the egg is 45 C. Use Newton’s Law of Cooling to determine when the egg will be 20 C.

  20. Compound Interest Interest Compounded Annually A = P (1 + r)t A = Amount P = Principal r = Rate t = Time Interest Compounded k Times Per Year A = P (1 + r/k)kt k = Compoundings Per Year Interest Compounded Continuously A = Pert

  21. Annual Percentage Yield Compounded Continuously APY = (1 + r/k)k – 1 APY = er – 1 Annual Percentage Yield

  22. Annuities Future Value of an Annuity R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments # 11 (p. 324) $14,755.51

  23. Present Value of an Annuity R = Value of Payments i = r/k = interest rate per compounding n = kt = number of payments Annuities For loans, the bank uses a similar formula

  24. Annuities If you loan money to buy a truck for $27,500, what are the monthly pay-ments if the annual percentage rate (APR) on the loan is 3.9% for 5 years?

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