Download - Aim: The Power of e Course: Math Literacy Aim: How does the exponential model fit into our lives? Do Now: f(x) = 1000(0.5) x/30 Find f(x) when x = 80 f(80)

Aim: The Power of e Course: Math Literacy

Aim: How does the exponential model fit into our lives?

Do Now:

f(x) = 1000(0.5)x/30

Find f(x) when x = 80

f(80) = 1000(0.5)80/30

f(80) 157 kilograms

habitation not

possible


6

4

2

-2

-5 5

r x = 1+1

x x

Where’d e Come From?

Graph x

f xx

11

6

4

2

-2

-5 5

y 2.7183

y 2.7183 is asymptotic to f(x).

x = 1 x = 100 x = 10000



y = a • bxExponential function

or e

x

f xx

11

Leonard Euler



Leonard Euler

e

1 1 11 1! 2! 3!e K

11 1

11

12

11

11

14

11

11

6

e

K


4

3

2

1

-1

-2 2

t x = ex

Natural Exponential Function

f(x) = ex

f(x) = (2.71828. . .)x

Evaluate e2

e-1

e0.48

= 7.389056099

= 0.3678794412

= 1.616074402

10

8

6

4

2

-2

-5 5 10

q1 x = 2e0.24x

Graph f(x) = 2e0.24x Graph f(x) = (1/2)e-0.58x

10

8

6

4

2

-2

-5 5 10

r1 x = 1

2e-0.58x


Model Problem

At a constant temperature, the atmospheric pressure p in pascals is given by the form p = 101e -0.001h , where h is the altitude in meters. Find p at an altitude of 300 m.

p = 101e -0.001h h = 300

p = 101e -0.001(300)

p = 101e -0.3

p = 101(0.7408182207)

p = 74.823


Model Problem 1

In a report entitled Resources and Man, The US National Academy of Sciences concluded that a world population of 10 billion “is close to (if not above) the maximum that an intensely managed world might hope to support with some degree of comfort and individual choice.” At the time the report was issued in 1969, the world population was approximately 3.6 billion, with a growth rate of 2% per year. The function models world population, f(x), in billions, x years after 1969. Use the function to find world population in the year 2020. Is there cause for alarm?

0.02( ) 3.6 xf x e

0.02( ) 3.6 xf x e

x = 51: 2020 – 1969

9.98billion 0.02 51(51) 3.6f e


Model Problem 2

World population in 2000 was approximately 6 billion, but the growth rate was no longer 2%. It had slowed to 1.3%. Using this current growth rate, exponential functions now predict a world population of 7,8 billion in the year 2020. Experts think the population may stabilize at 10 billion after 2200 if the deceleration of growth rate continues. The function models world population, f(x), in billions, x years after 2000 subject to a growth rate of 1.3% annually. Find the world population in 2050.

0.013( ) 6 xf x e

0.013( ) 6 xf x e

x = 50: 2050 – 2000

11.49billion 0.013 50(50) 6f e

World population passed 7 billion in Oct 2011


Model Problem 3

In college, we study large volumes of information – information that, unfortunately, we do not often retain for very long. The function describes the percentage a particular person remembers x weeks after learning the information.

a) Substitute 0 for x and, without using a calculator, find the percentage of information remembered at the time it is first learned.

b) Substitute 1 for x and find the percentage of information that is remembered after 1 week.

c) Find the percentage of information that is remembered after 4 weeks.

d) Find the percentage of information that is remembered after 52 weeks.

0.5( ) 80 20xf x e

100%

68.5%

30.8%

20%


Model Problem 4

Sociologists have found that information spreads among a population at an exponential rate. Suppose that the function y = 525(1 – e-0.038t) models the number people in a town of 525 people who have heard news within t hours of its distribution.

a. How many people will have heard about the opening of a new grocery store within 24 hours of the announcement?

b. Graph the function on a graphing calculator. When will 90% of the people have heard about the grocery store opening?


Model Problem 5

The probability P that a person has responded to an advertisement can be modeled by the exponential equation P = 1 – e-0.047t where t is the number of days since the advertisement began to appear in the media.

a. What are the probabilities that a person has responded after 5 days, 20 days, and 90 days?

b. Find when the probability that an individual has responded to the advertisement is 75%.

c. If you were planning a marketing campaign, how would you use this model to plan the introduction of new advertisements?


Model Problem

How long will it take for prices in the economy to double at a 10% annual inflation rate?

22 is what percent of 143?

A shirt sold for $35.99. The markup on the shirt was 65% over cost. What was the cost?


Model Problem

Find the simple interest. Principal is $650, rate = 6.25% annually, time in 3 years

Evaluate: 35(0.45)x when x = 24

Write as a percent. 11/36