a. Graph each scenario. How many days will it take to infect our whole class in each scenario?

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1 Math 111 Section.3 Notes Zombie Tag! A Zombie is loose in our classroom! How long until we are all infected? Eample 1. Fill in the table for each scenario. Scenario 1: The initial zombie infects one new person in our class per da. Newl infected zombies cannot infect others. Das # of People Infected Da 0 1 Da 1 Da Da 3 Da Da 5 Da 6 Da 7 Da 8 Scenario : The initial zombie and each infected person infect one new person per da. Das # of People Infected Da 0 1 Da 1 Da Da 3 Da Da 5 Da 6 Da 7 Da 8 a. Graph each scenario. How man das will it take to infect our whole class in each scenario? Cara Lee Page 1

2 b. Write an equation for each scenario: Scenario 1: Scenario : c. How man people would be infected on da 30? Scenario 1: Scenario : d. On which da would the zombie outbreak infect one million people? Scenario 1: Scenario : Cara Lee Page

3 Section.3: Eponential Functions An eponential function is of the form where C is the initial value a is the growth factor and a > 0 f() = C a Consequentl, an eponential function is a function that increases or decreases at a constant percent rate. Let s review percent increase and decrease as we work through these eamples. Eample. You start a new job with an initial salar of $36,000 per ear. Each ear thereafter, ou receive a 3% raise. Let S(t) be our salar t ears after ou start our new job. (a) Write the formula for S(t). (b) What will our salar be after 10 ears? (c) When will our salar reach $50,000? (Use our graphing calculator to solve this). Eample 3. A compost pile has 7 cubic feet of waste and decas at a rate of 10% per month. Let Q(t) be the volume of compost (in cubic feet) t months since deca began. Write the formula for this decreasing eponential function. Instructor: A.E.Car Page 3 of 10

4 Section.3: Eponential Functions Eample. Graph of = in Figure 3. Use this to graph the various transformations listed. Figure 3. = Figure. = + 1 Figure 5. = 3 Figure 6. = Figure 7. = Figure 8. = 1 Instructor: A.E.Car Page of 10

5 Section.3: Eponential Functions Eample 5. Solve the following equations. List our solution set. (a) 5 = 5 6 (d) 1 = (b) 5 = 1 16 (e) 3 1 = 3 (c) 5 +8 = 15 (f) 9 7 = 3 1 Instructor: A.E.Car Page 5 of 10

6 Section.3: Eponential Functions What s e? The number e is a number that occurs in nature, and is a frequent base for eponential and logarithmic epressions. It is defined b: ( e = lim ) n n n It can also be epressed b the following: e = 1 0! + 1 1! + 1! + 1 3! + 1! + 1 5! + This number is irrational and is approimated b The graph of the function given b = e looks a lot like the graphs of the functions given b = and = 3, as shown in Figure 9. In calculus, ou will stud that the special propert of e is that the slope of the tangent line at zero is eactl 1, as shown in Figure 10. = 0 = = e = 3 Figure 9 5 = 0 = e = + 1 Figure Eample 6. Solve the following equation. e 3 = e Instructor: A.E.Car Page 6 of 10

7 Section.3: Eponential Functions Eample 7. In 1990, the population of Oregon was.8 million people. In 010, the population of Oregon was 3.83 million people. Let P (t) be the population of Oregon in millions, where t is the number of ears after 000. This can be modeled b P (t) = 3.98e 0.015t. (a) According to this model, what will the population be in 00? (b) According to this model, when will the population reach million people? Use our graphing calculator to solve this. Instructor: A.E.Car Page 7 of 10

8 Section.3: Eponential Functions Eample 8. Find an algebraic rule (or formula) for an eponential function f that passes through the points ( 1, 8) and (, 1). Also find the algebraic rule (or formula) for a linear function g that passes through the points ( 1, 8) and (, 1). Figure Instructor: A.E.Car Page 8 of 10

9 Section.3: Eponential Functions Eample 9. Find an algebraic rule (or formula) for an eponential function f that passes through the points (, 3 ) and (, 1). Eample 10. Find an algebraic rule (or formula) for an eponential function f that passes through the points (1, 8) and (3, 18). Instructor: A.E.Car Page 9 of 10

10 Section.3: Eponential Functions Eample 11. After caffeine is consumed, it leaves the bod at a fairl fied rate. A person consumes 00 mg of caffeine at 8:00am. Four hours later, about 100 milligrams of caffeine are remaining in their bloodstream. Let Q(t) be the number of milligrams of caffeine in the bod t hours after consumption. (a) Write the formula for the function modeling this eponential deca. (b) How much caffeine will still be in the bod at 8:00pm? Instructor: A.E.Car Page 10 of 10