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Lecture 1:

Topic: Introduction

Reading

H&P, Ch 4; Ch 1.1

(For introductory material, see Ch 0, pp. 10-16.)

Reading for this lecture

H&P, Ch 4, Sect 4.1; Ch 1.1 (modeling).

Homework for Tutorial in Week 2

Ex 4.1, pp. 173, problems 1, 5, 15.2

Function

A function describes the relationship betweentwo or more variables. Eg:

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Variable

Something whose magnitude can change.

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Types of Functions

- Constant Function

Eg: Y =a

- Linear Functions

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Types of Functions

- A Polynomial Function

- Quadratic Function

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Types of Functions

- Exponential Functions: The exponent is avariable.

- Logarithmic Functions

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Mathematical Models

Growth of Populations(people, animals, plants, organisms etc)

What will the population be after 20 periods?

Answer:8

More Mathematical Models

Growth of Populations

What will the population be after 20 periods?

Answer:

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Mathematical Modeling

The general model of population growth is:

y =

where y is population size

after x periods,

b is number of offspring

per female per period,

x is the number of periods

from the beginning.

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f(x) = or y =

where b > 0, b 1, x is any real number. b is called the base x is called the exponent f(x) is called the exponential function

Application

b is

x is

Exponential Functions

Definition (p.163)

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Example f(x) = 3x

x -3 -2 -1 0 1 2 3

f(x)

Exponential Functions

Example Growth (b > 1)

x

f(x)

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An investment increasesby 3% every year.After 10 years $1 of theinitial investment is worth$(1 + 0.03)10 = $1.3439.

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Example f(x) = (1/3)x

x -3 -2 -1 0 1 2 3

f(x)

Exponential Functions

Example Decay (0 < b < 1)

x

f(x)

1

An asset depreciates by3% every year.After 5 years every $1 ofthe assets value is worth$(1 0.03)5 = $0.8587.

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Rules for Exponents (p. 163)

1. axay = ax+y

2. ax/ay = ax-y

3. (ax)y = axy

4. (ab)x = axbx

5. (a/b)x = ax/bx

6. a1 = a

7. a0 = 1

8. a-x = 1/ax14

Investment

Ex 4.1, p. 173, problem 29.

A $6,500 certificate of deposit is purchased

for $6,500 and is held for six years. If the

certificate earns 4% compounded quarterly,

what is it worth at the end of six years?

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Investment

Ex 4.1, p. 173, problem 29.

Initial deposit =

Time held =

Interest rate =

Value at the end of 6 years =

=

=

At the end of 6 years the certificate of deposit

is worth16

Bacteria Growth

Ex 4.1, p. 173, problem 31.

Bacteria are growing in a culture, and their number

is increasing at a rate of 5% an hour. Initially, 400

bacteria are present.

(a) Determine an equation that gives the number,

N, of bacteria present after t hours.

(b) How many bacteria are present after one hour?

(c) After four hours?

Give your answers to (b) and (c) to the nearest

integer.

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Bacteria GrowthEx 4.1, p. 173, problem 31.

Let Nt be the population of bacteria after t hours

Initial population, N0 =

Rate of increase, r =

(a) Nt =

(b) N1 =

(c) N4 =

(to the nearest integer) are present

after 4 hours18

Population

Ex 4.1, p. 174, problem 35.

Because of an economic downturn, the

population of a certain urban area declines at a

rate of 1.5% per year. Initially the population is

350,000. To the nearest person, what is the

population after three years?

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Population

Ex 4.1, p. 174, problem 35.Initial population =

Rate of population decline =

Population after 3 years =

After 3 years the population of the area will be