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UNIT A

UNIT A. PreCalculus Review. Unit Objectives. 1. Review characteristics of fundamental functions (R) 2. Review/Extend application of function models (R/E) 3. Introduce new function concepts pertinent to Calculus (N). A8 - Logarithmic Functions. Calculus - Santowski. Lesson Objectives.

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UNIT A

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  1. UNIT A PreCalculus Review

  2. Unit Objectives • 1. Review characteristics of fundamental functions (R) • 2. Review/Extend application of function models (R/E) • 3. Introduce new function concepts pertinent to Calculus (N)

  3. A8 - Logarithmic Functions Calculus - Santowski

  4. Lesson Objectives • 1. Simplify and solve logarithmic expressions • 2. Sketch and graph logarithmic fcns to find graphic features • 3. Explore logarithmic functions in the context of calculus related ideas (limits, continuity, in/decreases and its concavity) • 4. Logarithmic models in biology (populations), business (profit, cost, revenue)

  5. Fast Five • 1. Solve ln(x + 2) = 3 • 2. Sketch f(x) = -ln(x + 2) • 3. State the domain of f(x) = (ln(x - 1))0.5 • 4. Evaluate log232 + log31/81+ log0.2516 • 5. Sketch the inverse of f(x) = 3 - log2x • 6. Find the domain of log3(9 - x2) • 7. Evaluate log3324 - log34 • 8. Expand using LoL • 9. Evaluate log79 + log35 using GDC • 10. Solve 3x = 11

  6. Explore • Using your calculator for confirmation, and remembering thatlogarithms are exponents, explain why it is predictable that: • (a)log 64 is three times log 4; • (b)log 12 is the sum of log 3 and log 4; • (c)log 0.02 and log 50 differ only in sign.

  7. (A) Logarithmic Fcns & Algebra • (1) Find the inverse of f(t) = 67.38(1.026)t • (2) Solve e3-2x = 4 • (3) Solve log6x + log6(x - 5) = 2 • (3) Express as a single log • (4) Solve log2x + log4x + log8x = 11 • (5) Simplify

  8. (B) Logarithmic Fcns &Graphs • Be able to identify asymptotes, intercepts, end behaviour, domain, range for y = logax • Ex. Given the function y = log2(x - 1) - 2, determine the following: • - domain and range • - asymptotes • - intercepts • - end behaviour • - sketch and then state intervals of increase/decrease as well as concavities

  9. (B) Logarithmic Fcns & Graphs • Ex 3. Given the functions f(x) = logx and g(x) = x1/3, • (a) when is f(x) > g(x) • (b) when is 100f(x) < g(x) • (c) Which function increases faster, 100f(x) or g(x)?

  10. (C) Logarithmic Fcns & Calculus Concepts • Now we will apply the concepts of limits, continuities, rates of change, intervals of increase/decreasing & concavity to exponential function • Ex 1. Graph • From the graph, determine: domain, range, max and/or min, where f(x) is increasing, decreasing, concave up/down, asymptotes

  11. (C) Logarithmic Fcns & Calculus Concepts • Ex 2. Evaluate the following limits numerically or algebraically. Interpret the meaning of the limit value. Then verify your limits and interpretations graphically.

  12. (C) Logarithmic Fcns & Calculus Concepts • Ex 3. Given the function : • (i) find the intervals of increase/decrease of f(x) • (ii) is the rate of change at x = 2 equal to/more/less than the rate of change equal to/greater/less than the rate at x = 1? • (iii) find intervals of x in which the rate of change of the function is increasing. Explain why you are sure of your answer. • (iv) where is the rate of change of f(x) equal to 0? Explain how you know that?

  13. (C) Logarithmic Fcns & Calculus Concepts • Ex 4. Given the function , find the average rate of change of f(x) between: • (a) 1 and 1.5 • (b) 1.4 and 1.5 • (c) 1.499 and 1.5 • (d) predict the rate of change of the fcn at x = 1.5 • (e) evaluate limx1.5 f(x). • (f) Explain what is happening in the function at x = 1.5 • (g) evaluate f(1.5) • (h) is the function continuous at x = 1.5? • (i) is the function continuous at x = 0?

  14. (D) Applications of Logarithmic Functions • The population of Kenya was 19.5 million in 1984 and was 32.0 million in 2004. Assuming the population increases exponentially, find a formula for the population of Kenya as a function of time. Then using logs, find the doubling time of Kenya’s population

  15. (E) Internet Links • Logarithm Rules Lesson from Purple Math • College Algebra Tutorial on Logarithmic Properties from West Texas AM • You can try some on-line word problems from U of Sask EMR problems and worked solutions • More work sheets from EdHelper's Applications of Logarithms: Worksheets and Word Problems

  16. (F) Homework • Text pages 113-117 • (1) Evaluate logs, Q19,20,23 • (1) properties of logs, Q31,33,35 • (2) solving eqns, Q39,41,51,53,60 • (3) apps, Q71,77,81

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