1.1 Mathematical Modeling
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Transcript of 1.1 Mathematical Modeling
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Chapter 1
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Mathematical Model
A mathematical model is a graphical, verbal, numerical, or symbolic representation of a problem situation.
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Example- Page 17
15. Teacher Salary Comparison
Over 60% of men not in the teaching profession earn a higher salary than men who are teachers. The table shows how much more money the average college-educated male non-teacher makes as compared to the average male teacher.
Year Percent MoreEarned by Non-Teachers asCompared toTeachers
1940 -3.6%
1950 2.1%
1960 19.7%
1970 33.1%
1980 36.1%
1990 37.5%
2000 60.4%Source: www.nea.orgFor example, in 1990 male non-teachersmade 37.5% more than male teachers on average.
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Example- Page 17
a. Describe the trend observed in these data.
b. Why was there such a big jump in the percentage of non-teachers who earn a higher salary than teachers from 1990 to 2000?
c. What does the –3.6% in 1940 indicate about salaries of male teachers?
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Example- Page 19
18. Super Bowl Ticket Prices
The table shows the price of a Super Bowl ticket for selected Super Bowls.
Super Bowl Ticket
FaceValue
I (1) $10
V (5) $15
X (10) $20
XV (15) $40
XX (20) $75
XXV (25) $150
XXX (30) $300
XXXV (35) $325
XL (40) $600
(Source: www.superbowl.com.)
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a. Describe the trend seen in these data. Are prices increasing or decreasing over time?
b. Compare the difference in ticket face value from one year to the next. What patterns do you notice?
c. Predict the face value of a ticket for the 60th Super Bowl.
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1.2
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Function Notation
y = f(x)
Input (independent variable)
Output (dependent variable)
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Functions
A relation is a function if each input value has exactly one output value
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Determining Functions
Determine if the relation is a function (3, 2), (4, 2), (5, 2)
(1, 2), (-1, 3), (1, 7)
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Vertical Line Test
Use the vertical line test to determine whether each graph represents a function
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1.3
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Is it a function?
Just verify that each input value corresponds to a single output value.
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Slope
Slope = rate of change
or = change in y change in x
= ∆y
or ∆x
or = Rise Run
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Slope- given points
For points slope =
A positive slope goes up as you read the graph from left to right. (A negative slope goes down as you read the graph from left to right.)
As the absolute value of the slope gets larger, the steepness of the incline increases.
2 1
2 1
y y
x x
1 1 2 2( , ), ( , )x y x y
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Examples
Calculate the slope between each pair of points (7, 2) and (4, -3)
(8, 2) and (-8, 6)
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Average Rate of Change
= Change in function value Change in x value
= f(b) – f(a)b – a
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Average Rate of Change
Page 75 # 4- Baby Girl’s Average Weight
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Average Rate of Change
Page 75 #6- Divorce Rate
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Average Rate of Change
Page 79 #16- Exxon Mobile
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Interpreting Data
Page 81 #20-U.S. Home Sales
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Functions of several variables Sometimes functions depend on more than
one variable. A pay as you go service is as follows:
Phone bill = airtime + text messages + picture messaging
M = .25a + .05t + .25p, where M is measured in dollars. M = M(a, t, p) airtime = 25 cents per minute text messages = 5 cents each picture messaging = 25 cents each
Explain what M(300,15,8) means and find its value.
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Borrowing money
If you borrow P dollars at a monthly interest rate of r (as a decimal) and want to pay off the loan in t months, then the monthly payment M(P, r, t) in dollars is calculated using
P = amount borrowedr = monthly interest rate, as
a decimalt = number of monthsM = monthly payment
Mr
r
t
t
Pr( )
( )
1
1 1
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Examples
What does M(6000, .035, 60) mean?
Find the monthly payment.
Suppose we borrow $5600 with a monthly interest rate of .825% and want to pay the loan off in 4 years.
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Example
Page 81 #21-22
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Solving Equations
Temperature Problems Page 83 #33-36
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Example
Page 86 #47-48
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1.4
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Two-Variable Data
Ordered Pair
Domain
Range
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Domain and Range Domain- the input values
In an (x, y) situation, these are the “x” values. The domain is the set of values that are
plugged into a function. When it is limited to values that make sense in
the real world it is called the practical domain. Range- the output values
In an (x, y) situation, these are the “y” values. The range is the set of solutions we get from
plugging the domain values into a function. When it is limited to values that make sense in
the real world it is called the practical domain.
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State the Domain and the Range (2, 3), (-3, 7), (8,3)
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Gathering Data from graphs Page 102 #1-6
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Example
Page 103 #16 Cell Phone Subscribers
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Examples
Page 104-105 #18-22