mat 213 brief calculus section 1.1 models, functions and graphs
TRANSCRIPT
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MAT 213Brief Calculus
Section 1.1
Models, Functions and Graphs
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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
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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
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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.
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Is it a function???
Input Output
-2
-1
0
1
2
9
7
5
3
1
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Is it a function???
Input Output
5
5
5
5
5
-2
-1
0
1
2
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Is it a function???
Input Output
-2
-1
0
1
2
5
5
5
5
5
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What about these?
A = {(0,4), (7,4), (5,3), (1,0)}
B = {(0,1), (1,1), (1,0)}
C = {(1,1)}
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Is it a function???
5 10-5-10
5
10
-5
-10
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Is it a function???
5 10-5-10
5
10
-5
-10
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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
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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)
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g(x)
1. g(-2) = ?
2. g(-1) = ?
3. Find the values of x that make g(x) = 0.
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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
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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
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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?
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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)
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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
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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
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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))
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In groups let’s try the following from the book
• 1, 13, 25, 27, 35, 53