MAT 213Brief Calculus

Section 1.1

Models, Functions and Graphs

Mathematical Models• The process of translating a real world problem

situation into a usable mathematical equation is called mathematical modeling

• For example, in business it is important to know how many units to produce to maximize profit– Thus if we can model our profits as a function of the

number of units produced, we can use calculus to determine how many products will maximize our profit

A function is a rule that assigns exactly one output to every input

• In mathematics, a function is often used to represent the dependence of one quantity upon another

• We therefore define the input as the independent variable, and the resulting output as the dependent variable

• Note: the output does not have to depend on the input in order to have a function

Definitions

Independent Variable Its values are the elements of the DOMAIN Plotted on the horizontal axis Its values are known when collecting data

Dependent Variable Its values are the elements of the RANGE Plotted on the vertical axis The quantity measured for a specific value of the independent

variable.

Is it a function???

Input Output

-2

-1

0

1

2

9

7

5

3

1

Is it a function???

Input Output

5

5

5

5

5

-2

-1

0

1

2

Is it a function???

Input Output

-2

-1

0

1

2

5

5

5

5

5

What about these?

A = {(0,4), (7,4), (5,3), (1,0)}

B = {(0,1), (1,1), (1,0)}

C = {(1,1)}

Is it a function???

5 10-5-10

5

10

-5

-10

Is it a function???

5 10-5-10

5

10

-5

-10

We use the notation f(x) to denote a function.

It is read "f of x," meaning the value of the function f evaluated at point x

Actually, any combination of letters can be used in function notation

Example: If we were writing a function that described the area of a square in terms of

the length of a side, we may choose A(s) to mean the area A when the side is length s.

The parentheses DO NOT mean multiplication!!!The parentheses DO NOT mean multiplication!!!

Function Notation

Examples

Find the function values.

Do not worry about simplifying right now

h(x) = x2 + 2x - 4a. h(4)b. h(-3x)c. h(a – 1)

d. h(x+1) – 3h(x)

g(x)

1. g(-2) = ?

2. g(-1) = ?

3. Find the values of x that make g(x) = 0.

Rule of Four

• Functions can be represented in 4 ways1. Numerical data such as a table

2. Graphically

3. In words

4. By an equation

We will encounter all 4 of these representations during the semester

In Business

Fixed costs (overhead)

Variable costs

Total Cost = Fixed costs + Variable costs

Average cost = -----------------------------

Profit = Revenue – Total Cost

When does a company break even?

ProducedUnitsofNumber

CostTotal

Break-Even Point

$ Revenue

Total Cost

10 20 30

Number of Units (in millions)

How many units would this business need to sell in order to break even?

Break-Even Point

$ Profit

5 10 15

How many units would this business need to sell in order to break even?# of units

(in thousands)

Combinations of Functions• Now if we have a revenue function, R(x), and a cost

function, C(x), we saw that we can create a profit function, P(x)

• We would get P(x) = R(x) - C(x)• Thus we have combined two functions via subtraction to

get another function• We can also add, multiply or divide two functions

Composition of FunctionsNotation

Take the functions f(x) and g(x)

f(g(x)) = (f◦g)(x)

To evaluate f(g(x)), always work from the inside out. First find g(x) then plug that result into f.

For all x in the domain of f such that f(x) is in the domain of g

Composition of FunctionsExample

Let f(x) = 5x + 1 and g(x) = x2

Evaluate the following:

(f◦g)(x) (f◦f)(x) g(f(x))

f(g(-2))

In groups let’s try the following from the book

• 1, 13, 25, 27, 35, 53