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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). A7 - Exponential 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. A7 - Exponential Functions Calculus - Santowski

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

  5. Fast Five • 1 Solve 2-x+2 = 0.125 • 2. Sketch a graph of y = (0.5)x + 3 • 3. Solve 4x2 - 4x - 15 = 0 • 4. Evaluate limx∞ (3-x) • 5. Solve 3x+2 - 3x = 216 • 6. Solve log4(1/256) = x • 7. Evaluate limx3 ln(x - 3) • 8. Solve 22x + 2x - 6 = 0 • 9. State the exact solution for 2x-1 = 5 (2 possible answers) • 10. Is f(x) = -e-x an increasing or decreasing function?

  6. Explore • Given 100.301= 2 and 100.477= 3, solve without a calculator: • (a)10x= 6; • (b)10x= 8; • (c)10x= 2/3; • (d)10x= 1

  7. Explore • A function is defined as follows: • (i) Evaluate limx-2 if a = 1 • (ii) Evaluate limx3 if b = 1 • (iii) find values for a and b such f(x) is continuous at both x = -2 and x = 3

  8. (A) Exponentials & Algebra • (1) Factor e2x - ex • (2) Factor and solve xex - 2x = 0 algebraically. Give exact and approximate solutions (CF) • (3) Factor 22x - x2 (DOS) • (4) Express 32x - 5 in the form of a3bx (EL) • (5) Solve 3e2x - 7ex + 4 = 0 algebraically. Give exact and approximate solutions (F) • (6) Solve 4x + 5(2x) - 12 = 0 algebraically. Give exact and approximate solutions (F)

  9. (B) Exponentials & Their Graphs • Be able to identify asymptotes, intercepts, end behaviour, domain, range for y = ax • Ex. Given the function y = 2 + 3-x, determine the following: • - domain and range • - asymptotes • - intercepts • - end behaviour • - sketch and then state intervals of increase/decrease as well as concavities

  10. (B) Exponentials & Their Graphs • Be able to identify asymptotes, intercepts, end behaviour, domain, range for y = ax • Ex 1. Given the function y = 2 + 5(1 - ex+1), determine the following: • - domain and range • - asymptotes • - intercepts • - end behaviour • - sketch and then state intervals of increase/decrease as well as concavities

  11. (B) Exponentials & Their Graphs • Ex 2. Given the graphs of f(x) = x5 and g(x) = 5x, plot the graphs and determine when f(x) > g(x). Which function rises faster? • Ex 3. Given the points (1,6) and (3,24): • (i) determine the exponential fcn y = Cax that passes through these points • (ii) determine the linear fcn y = mx + b that passes through these points • (iii) determine the quadratic fcn y = ax2 + bx + c that passes through these points

  12. (C) Exponentials & 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

  13. (C) Exponentials & 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.

  14. (C) Exponentials & Calculus Concepts • Ex 3. Given the function f(x) = x2e-x: • (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?

  15. (C) Exponentials & Calculus Concepts • Ex 4. Given the function f(x) = x2e-x, 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 x2e-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?

  16. (D) Applications of Exponential Functions • The population of a small town appears to be increasing exponentially. In 1980, the population was 35,000 and in 1990, the population was 57,000. • (a) Determine an algebraic model for the town’s population • (b) Predict the population in 1995. Given the fact that the town population was actually 74,024, is our model accurate? • (c) When will the population be 100,000? • (d) Find the average growth rate between 1985 and 1992 • (e) Find the growth rate on New Years day, 1992 • (f) Find on what day the growth rate was 6%

  17. (E) Internet Links • Exponential functions from WTAMU • Exponential functions from AnalyzeMath • Solving Exponential Equations from PurpleMath

  18. (F) Homework • From our textbook, p99-103 • (1) for work with graphs, Q3-11 • (2) for work with solving eqns, Q15,1619,10,21,22 • (3) for applications, Q35,40 (see pg95-6) • (4) for calculus related work, see HO (scanned copy on website)

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