Checklist and Assignment
Notes
Checklist:
Identify Variables
Graph Mathematical Models
Assignment:
1. p 519 Quick Quiz
2. p 520 - 521 Exercises 2, 3, 6, 7, 9, 11, 14, 15, 17, 21,
23
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Key Words
Notes
Mathematical Model
Function
Independent Variable
Dependent Variable
Domain
Range
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Function Notation - Variables
Notes
Functions describe how a dependent varialbe changes
with respect to independent variables.
An ordered pair summarizes two variables, with the
independent variable first:
(independent variable, dependent variable) = (x, y )
A dependent variable is function of the independent
variable. If we label them as y and x, we write the
function as
y = f (x)
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Example 1 - Writing Functions
Notes
Write the two variables as an ordered pair and write down
the function with the notation y = f (x).
Example: Over the course of a day, the temperature
varies. You notice that the mornings are cold, the
afternoons are warm, and the evenings are cold again.
Ans: The temperature depends on the time, so we say
the temperature changes with respect to time.
Temp = dependent variable, time = independent variable.
The pair is (time, Temp). If time = t and temp = T , then
T = f (t)
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Example 1 - Writing Functions
Notes
a. You are riding in a hot-air balloon. As the balloon
rises, the surronding atmospheric pressure descreases
(causing you ears to pop.)
b. You’re on a barge headed south down the Mississippi
River. You notice that the width of the river changes as
you travel southward with the current.
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Representing Functions
Notes
You can represent a function with:
1. ...a data table.
2. ...a picture or graph.
3. ...an equation.
9A mainly uses graphs to represent functions.
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Domain and Range
Notes
Recall that sets are collection of objects.
The domain of a function is the set of values that make
sense and are of interest for the independent variable.
The range of a function is the set of values for the
dependent variable that correspond to the values of
domain.
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Models
Notes
Plot the data table on a graph:
Time
6:00 a.m.
7:00 a.m.
8:00 a.m.
9:00 a.m.
10:00 a.m.
11:00 a.m.
12:00 p.m.
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Temp
50◦ F
52◦ F
55◦ F
58◦ F
61◦ F
65◦ F
70◦ F
Time
1:00 p.m.
2:00 p.m.
3:00 p.m.
4:00 p.m.
5:00 p.m.
6:00 p.m.
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Temp
73◦ F
73◦ F
70◦ F
68◦ F
65◦ F
61◦ F
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Models
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Notes
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Models - Graphs
Notes
1. Write the independent and dependent variables of the
function.
2. Scale and label the axes using the domain and range.
3. Use the data to make a graph.
4. To make any predictions, evaluate the data, model,
and the assumptions for building the model.
Zoom in on the region of interest.
Fill in the gaps between data points if that is
appropriate.
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Example 2 - Pressure-Altitude Function
Notes
In a hot-air balloon, you notice the pressure is different at
different altitudes. Pressure is measured in inches of
mercruy. Graph a function using the data below.
Altitude (ft) Pressure (in. mercury)
0
30
5,000
25
10,000
22
20,000
16
30,000
10
Use the graph to predict the pressure at an altitude of
15,000 feet. Is the predition valid?
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Example 3 - Hours of Daylight
Notes
The number of hours of daylight varies with the seasons.
Use the data from cities at 40◦ latitude to model the
change in the number of daylight hours with time.
During summer soltice (June 21), the number of
daylight hours is the greatest at 14 hours.
During winter soltice (Dec 21), the number of daylight
hours is the smallest at 10 hours.
During the equinoxes (Mar 21, Sept 21) the are about
12 hours of daylight.
What time of the year does the daylight change most
gradually? Most quickly? Is the model valid?
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