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Math 140
February 10, 2017
Mathematical modeling
Modeling Process
Car wash problem
• How was this a modeling problem?
Linear functions
• A linear function is one that represents constant
change
• Its graph is a straight line
Is it linear?
x
0
100
300
600
y
50
100
150
200
x
0
10
20
30
y
20
40
50
55
x
0
5
10
15
y
10
20
30
40
Equation of horizontal
line
• y=constant
Equation of vertical line
• x=constant
Equation of slanted line
• Two forms:
o y=mx+b
o y-y0=m(x-x0)
• What do these letters mean?
o m is the slope
o b is the intercept
o (x0,y0) is a point on the line
Slope
• Note that every line is uniquely determined by two
points on it
• The slope of a line is one of the most important
concepts of calculus
• Its practical meaning is that it is a rate of change
• Slope =
∆𝑦
∆𝑥
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑦
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑥
=
𝑦2 −𝑦1
𝑥2 −𝑥1
=
𝑦1 −𝑦2
𝑥1 −𝑥2
• Linear functions have a constant slope. In fact, that
is how you recognize a linear function.
Meaning of slope and
intercept
• The cost C in dollars of renting a car through a
rental agency and driving it d days is
C = 30 + 25d
• What do 30 and 25 mean?
• Another agency charges C = 30d. Where would
you rent the car and why?
• Note: these are not actual prices
Questions
• What is the slope of a horizontal line?
• What is the slope of a vertical line?
Practice
• Write the equation for a line that
o Has slope -4 and intercept 7
o Passes through points (-1,5) and (2,-1)
o Passes through points (3,5) and (6,5)
Parallel and
perpendicular lines
• Two lines are parallel if they have the same slopes
• Two lines are perpendicular if the product of their
slopes is -1.
Practice
• Are the following pairs of lines, perpendicular,
parallel, or neither?
o y = 5x-7, y=5x+8
o y=4x+3, y=13-0.25x
o y=4x+7, y=0.25x - 2
Earning power
Below are data from the book on the median incomes for men
and women working full-time.
Based on these data, answer the following:
1. If incomes for men and women grew linearly from 1970 to
today, what should the median incomes for women and men
have been in 2015?
2. When will women catch up with men?
Earning power, cont.
• Women’s median income in 2015 was $40,742 and
men’s median income was $51,212. How did this
compare with your predictions?
• What do you know about the income gap?