lecture 7.1 to 7.2 bt
TRANSCRIPT
Today’s Agenda
Attendance / Announcements
Questions from 6.1 / 6.2
Sections 7.1 / 7.2
E.C. Quiz Today
Exam Schedule
Exam 5 (Ch 6.1, 7)
Wed 4/1
Exam 6 (Ch 10)
Monday 4/27
Final Exam (Cumulative)
Monday 5/4
Graphing Systems of Inequalities
1. Graph boundary lines
2. Check inequality signs
(Dashed or Solid?)
3. Shade accordingly
5. Test Points
4𝑥 + 𝑦 ≥ 92𝑥 + 3𝑦 ≤ 7
4. Identify Intersecting
regions (“Feasible”)
Find “Corner Points” of the Feasible
Region3𝑥 + 2𝑦 ≤ 6−2𝑥 + 4𝑦 ≤ 8
𝑥 + 𝑦 ≥ 1𝑥 ≥ 0𝑦 ≥ 0
𝑦 ≤ −32
𝑥+3
𝑦 ≤ 12
𝑥 + 2
𝑦 ≥ −𝑥 + 1𝑥 ≥ 0𝑦 ≥ 0
Finding Feasible RegionsFind the system
whose feasible region
is a triangle with
vertices (2,4),
(-4,0), and (2,-1)
2
46
832
x
yx
yx
Linear Programming
Businesses use linear
programming to find out how to
maximize profit or minimize
costs. Most have constraints on
what they can use or buy.
Linear Programming
The Objective Function is
what we need to maximize or
minimize. For us, this will be a
function of 2 variables, f(x, y)
The general idea… (pg 398)Find max/min values of the objective
function, subject to the constraints.
yxyxf 52),(
0,0
1
842
623
yx
yx
yx
yx
Objective Function Constraints
The general idea… (pg 398)
The Feasible Region makes up the possible inputs to the Objective Function
yxyxf 52),(
Corner Point Thm (pg 400)
If a feasible region is bounded, then the objective function has both a maximum and minimum value, with each occurring at one or more corner points.
Find the minimum and maximum
value of the function f(x, y) = 3x - 2y.
We are given the constraints:
• y ≥ 2
• 1 ≤ x ≤5
• y ≤ x + 3
• The vertices (corners) of the
feasible region are:
(1, 2) (1, 4) (5, 2) (5, 8)
• Plug these points into the
function f(x, y) = 3x - 2y
Note: plug in BOTH x, and y values.
Evaluate the function at each vertex
to find min/max values
f(x, y) = 3x - 2y
• f(1, 2) = 3(1) - 2(2) = 3 - 4 = -1
• f(1, 4) = 3(1) - 2(4) = 3 - 8 = -5
• f(5, 2) = 3(5) - 2(2) = 15 - 4 = 11
• f(5, 8) = 3(5) - 2(8) = 15 - 16 = -1
Find the minimum and maximum value
of the function f(x, y) = 4x + 3y
With the constraints:
52
24
1
2
xy
xy
xy
f(x, y) = 4x + 3y
• f(0, 2) = 4(0) + 3(2) = 6
• f(4, 3) = 4(4) + 3(3) = 25
• f( , - ) = 4( ) + 3(- ) = -1 = 7
3
1
3
1
3
7
3
28
3
25
3
Evaluate the function at each vertex
to find min/max values