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### UNIT A

### A7 - Exponential Functions

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)

Calculus - Santowski

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)

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 limx3 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?

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

Explore

- A function is defined as follows:
- (i) Evaluate limx-2 if a = 1
- (ii) Evaluate limx3 if b = 1
- (iii) find values for a and b such f(x) is continuous at both x = -2 and x = 3

(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)

(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

(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

(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

(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

(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.

(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?

(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 limx1.5 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?

(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%

(E) Internet Links

- Exponential functions from WTAMU
- Exponential functions from AnalyzeMath
- Solving Exponential Equations from PurpleMath

(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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