Padmanabhan Seshaiyer

1
Introductory Calculus with Business Applications
Math 108 (Section 003)
Padmanabhan Seshaiyer Mathematical
Sciences George Mason University Email
pseshaiy_at_gmu.edu Lecture 4 (September 28, 2009)
2
1.3 Linear Functions
Padmanabhan Seshaiyer George Mason University

• A manufacturers total cost consists of fixed
  overhead of 200 plus production costs of 50 per
  unit. Express the total cost as a function of the
  number of units produced and draw the graph.

C(x) 50 x 100
3
Linear Depreciation
Padmanabhan Seshaiyer George Mason University

• A moving company buys a 40-foot van with a useful
  lifetime of 5 years. After x months of use, the
  value y of the van is estimated by the linear
  equation y25,000 400x
• Sketch the graph of this linear function.
• What is the value of the van after 5 years?
• What economic interpretation can be given to the
  y-intercept of the graph?

4
Slope of a straight line
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y 2 x1
Y 2 x 1
5
Geometric definition of slope of the line
Padmanabhan Seshaiyer George Mason University

• Find the slope of the line joining the points
  (2, 5) and (2, 5).

(x2,y2)
(x1,y1)
6
Slope of a horizontal line
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• y 3
• y (0) x 3

y3
7
Steepness property of positive slope
Padmanabhan Seshaiyer George Mason University
y 2 x1
8
Steepness property of negative slope
Padmanabhan Seshaiyer George Mason University
9
Steepness property of zero slope
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y3
10
Implications of steepness property
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y x 1
y 3 x 1
y ½ x 1
11
Implications of steepness property
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• Use the steepness property to draw the graph of
• y ½ x ½
• y -½ x 2
• Start at y-intercept
• Apply Steepness property to find a new point
• Connect the y-intercept and the new point

(1, 1)
(1, -2.5)
12
Graph of an equation
Padmanabhan Seshaiyer George Mason University

• Sketch the graph of the equation 8x 4y4
• X-intercept (0.5, 0)
• Y-intercept (0, 1)

8X 4Y4
13
Slope-intercept form of a linear equation
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• y m x b
• Find the slope-intercept form of the equation
• 8 x 4 y 4
• 8 x 8 x 4 y 4 8 x
• 4 y 8 x 4
• y 2x 1

m-2
b1
14
Point-Slope Form of the equation of a line
Padmanabhan Seshaiyer George Mason University

• The equation of the straight line passing through
  the point (x1, y1) and having slope m is given
  by
• y y1 m (x x1)
• Find the equation of a line that passes through
  the point (2, 3) and has slope ½.
• Find the equation of a line that passes through
  the points (2, 3) and (2, 4).

15
General Form of the equation of a line
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• The general form of the equation of a straight
  line is given by
• A x B y C 0
• If A 0, then the line is horizontal.
• If B 0, then the line is vertical.
• If B 0, then the line has slope A/B and
  y-intercept C/B

16
Perpendicular Property of slope
Padmanabhan Seshaiyer George Mason University

• When two lines are perpendicular, their slopes
  are negative reciprocals of one another. That is,
  if a line has slope m1, then the line
  perpendicular to it will have the slope

17
Perpendicular Line
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• Find the equation of the line perpendicular to
  the graph of y 6x 3 and passing through the
  point (0, 2). Also, sketch the graph of the two
  lines.

18
Parallel Property of Slope
Padmanabhan Seshaiyer George Mason University

• Parallel Lines have the same slope. That is, if a
  line has slope m1, then the line parallel to it
  will have the slope
• m2 m1

19
Parallel Line
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• Find the equation of the line parallel to the
  graph of y 3x 4 and passing through the point
  (0, 8). Also, sketch the graph of the two lines.

20
College Tuition
Padmanabhan Seshaiyer George Mason University

• The average college tuition and fees at four-year
  public colleges increased from 2035 in 1990 to
  4128 in 2004. Assuming that average tuition and
  fees increased linearly with respect to time,
  find the equation that relates the average
  tuition and fees, y, to the number of years after
  1990, x. What were the average tuition and fees
  in 2000?

21
Collinear Points ?
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• Show that the points (1, 3), (2, 4) and (3, 2)
  are not on the same line.

There must be a Smarter Way!
22
Announcements
Padmanabhan Seshaiyer George Mason University

• Suggested HW Problems
• Section 1.3 1 47 (odd problems)
• Quiz III
• October 5, 2009
• Section 1.3