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Understanding Exponential Functions in Precalculus

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This comprehensive review focuses on the characteristics and applications of exponential functions, essential for calculus. You'll learn to simplify and solve exponential expressions, sketch graphs to identify key features, and explore their behavior in calculus, including concepts like limits and continuity. The unit covers the real-world application of exponential models in biology and business, enhancing your understanding of population growth, profit, cost, and revenue dynamics. Engage with examples and exercises to solidify your comprehension of exponential functions and their relevance in mathematical analysis.

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Understanding Exponential Functions in Precalculus

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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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