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# UNIT A - PowerPoint PPT Presentation

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

PreCalculus Review

• 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

• 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)

• 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?

• 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

• 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

• (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)

• 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

• 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

• 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

• 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

• Ex 2. Evaluate the following limits numerically or algebraically. Interpret the meaning of the limit value. Then verify your limits and interpretations graphically.

• 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?

• 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?

• 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%

• Exponential functions from WTAMU

• Exponential functions from AnalyzeMath

• Solving Exponential Equations from PurpleMath

• 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)