Download Fall 2013 Slides for Math Camp – Day 1

Math Camp: Day 1
School of Public Policy
George Mason University
August 19, 2013
6:00 to 8:00 pm
http://www.youtube.com/watch?v=YVp46B8DHok
Amit Patel, PhD
[email protected]
718-866-5757
Teaching Associates
Lokesh Dani
Lisardo A. Bolanos Fletes
[email protected]
[email protected]
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Course Outline
Monday 8/19
Wednesday 8/21
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Pre-test
Percentage, Rate of Change
Review of concepts, notation
Algebra Review, Functions
Exponents
Solving linear systems
Coordinate geometry
Wrap-up, assignments
Review assignment
Probability
Linear functions (cont.)
Non-linear functions
Derivatives (cont.)
Intro to Optimization
Post-test, course assessment
Course Website: https://sites.google.com/site/sppmathcamp/
(Lokesh Dani)
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Pre-Test
15 Minutes
3
Concepts and Notation
• Variable: a symbol for a number that can change
(often use x and y)
• Constant: a number that does not change
• Coefficient: a number used to multiply a variable
Variable Constants
Coefficient
4x – 7 = 5
4
Concepts and Notation: Subscripts
• Subscripts: letters written below and usually
to the right of variables to distinguish different
elements
xi – value of x for a particular element i (where i is
defined as going from 1 to I)
i
1
2
3
4
5
6
7
8
9
10
x
23
40
35
28
38
31
29
27
50
41
value of x4 = 28
I = 10
yt – value of y for a particular time period
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Concepts and Notation: Summation
• Summation symbol Σ: sum up the expression
that follows for all values for elements i to I
i
1
2
3
4
5
6
7
8
9
10
x
23
40
35
28
38
31
29
27
50
41
10
Σ xi = 23 + 40 + 35 + 28 + 38 + 31 + 29 + 27 + 50 + 41 = 342
i=1
Can also be written as Σ xi
i
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Using the Summation Symbol
Exercise
Solve
Σ 2xi =
5
i=1
where
i
1
2
3
4
5
x
0
2
4
6
8
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Percentage and Rate of Change
Percentage: part or fraction of a total
23 out of 50 states = 46%
(= 0.46 = 46/100 = 23/50)
117 out of 500 students = 23.4%
(= 0.234 = 234/1000 = 117/500)
Percent change ∆/rate of change ∆: measure of
relative change between old and new value
x1 = 231
x2 = 253
Percent ∆: (253 – 231)/231 = 0.095 = 9.5%
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Percent Changes
Exercise:
Jurisdiction
2010
Population
Pop
Change
2000 - 2010
Virginia
8,001,024
922,509
Culpeper
County
46,689
12,427
Pop Percent
Change
2000-2010
Avg. Annual Pop
Growth Rate
Source: US Census Bureau 2000 and 2010 Census SF1
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Percent Changes
Exercise:
What’s the better deal?
Original item price: $125
20% off original
price, then 25% off
markdown price
40% off original price
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Order of Operations
PEMDAS: order that must be adhered to when
solving expressions
P
Parentheses
Parentheses
E
Exponents
Exponents
M
Multiplication
Multiplication
D
Division
Division
A
Addition
Addition
S
Subtraction
Subtraction
3 * (4-2)2 + 8 / 2 – 1
3 * 22 + 8 / 2 – 1 =
3*4+8/2–1=
12 + 8 / 2 – 1 =
12 + 4 – 1 =
16 – 1 = 15
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Order of operations
Exercises:
5 * (10 – 8)2 – 10 =
5 + 10 – 3 * 3 / 9 =
12
Adding and Subtracting Fractions
To add or subtract, find the lowest common
denominator then add or subtract across the
top
Exercise
=
13
Multiplying & Dividing by Fractions
To multiply, multiply across the top and bottom
(then simplify)
Exercise:
=
To divide, flip the fraction you are dividing by
and multiply across the bottom and top
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Operations with Negative Numbers
• Adding:
• 4 + (-2) = 4 – 2 = 2
• Subtracting:
• 4 – (-2) = 4 + 2 = 6
• Multiplying:
• 4 * (-2) = - 8
• -2 * -2 = 4
• Dividing:
• 4 / (-2) = - 2
• -4 /( -2) = 2
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Exponents: Rules
• zx = z*z*z*z … (x times)
e.g. 43 = 4 * 4 * 4 = 64
• zx *zy =z(x+y)
e.g. 42*43
= 4(2+3)
Must be the same base number
• zx /zy =z(x-y)
e.g. 54/53
= 5(4-3)
• (zx)y = z(x*y)
e.g. (32)3 = 36 = 729
Must be the same base number
Exponents: Rules
• z-x = 1/zx
e.g. 3-2
= 1/32 = 1/9
• z½ = √z e.g. 4½ = √4 = ± 2
• z1 = z
e.g. 71 = 7
• z0 = 1
e.g. 80 = 1, 2190 = 1
Exponents: Exercises
24 * 22 =
73 * 33
=
4
2
7 *3
(54)2 =
140 =
Algebraic Functions
• Function: relates one quantity or input with
another quantity or output
• Input: independent variable, exogenous variable, x
or t
• Output: dependent variable, endogenous variable,
y
So, y equals some function of x (or t)  y = f(x)
f is often used as the notation for a function but it
is not a must  g(x), v(t)
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Algebraic Functions
• Linear functions:
slope-intercept form f(x) = mx + b
f(x) = 3x + 7 (or y = 3x + 7)
ℓ2
ℓ1
Degree = 1 (no squared+ x)
• Nonlinear functions: includes
squared x, cubed x, etc.
Degree > 1
f(x) = x2
g(x) = -7x3 + 5
f1
f2
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Algebra Review
Solve for x
Combine x terms on one side,
constants on the other side
by adding and subtracting
on both sides of the
equation
Divide by the coefficient (on
both sides) to isolate x
2x – 4 = 8 – x
+x
+x
3x – 4 = 8
+4
+4
3x = 12
3
3
x=4
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Algebra Review
Exercises
Solve for x
7x – 12 = 3x + 8
½x–1=⅓x+5
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Linear Functions
f(x) = ⅔x – 4
Evaluate the following when
x=6
Or what is the value of the
function f(6)?
Exercise
Given v(t) = 2t + 4
What is the value of the
function v(10)?
f(6) = ⅔(6) – 4
= 12/3 – 4
=4–4=0
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Linear Functions: Word Problem
Write a linear function f(x) for the following
scenario:
In the School of Public Policy, the tuition for
each course credit is $693 (in-state.)
What is the cost for a 3 credit course?
f(x) = 693x
*slope = 693, y-intercept = 0
f(3) = 693(3) = 2,079
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Solving Linear Systems: 2 Variables
Two Variables  Two Equations
10x – 5y = 20
y = 3x – 1
x+y=2
y = 2x
Re-arrange equation 2
Substitute y = 2x into
equation 1
y=2–x
2x = 3x – 1
Substitute into equation 1
1=x
10x – 5(2 – x) = 20
10x – 10 +5x = 20
15x – 10 = 20
15x = 30
x = 2, y = 0 (b/c y = 2 – x)
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Solving Linear Systems: 2 Variables
Exercises
Solve for x and y
3x – y = 2
2x + 2y = 12
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Solving Linear Systems: 3 Variables
Solve for x, y and z
3x – y + z = 2
2x + 2y – z = 12
x+y–z=6
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Coordinate Geometry
(2,4)
(1, 2)
Plotting points:
(1, 2) (2, 4)
Calculating slope:
m = y 2 – y1
x2 – x1
= (4 – 2)
(2 – 1)
=2
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Coordinate Geometry
What is the slope
of Line A?
(3,1)
(-2,-2)
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Coordinate Geometry
Sketch the following
linear function.
y = 2/3x – 1
run = 3
rise = 2
Slope = 2/3
y-intercept = -1
Plug-and-chug method:
x
y
0
-1
1
-1/3
2
1/3
3
1
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Coordinate Geometry
Sketch the following
linear function.
4x – 2y = 4
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Coordinate Geometry
Undefined
slope
Positive slope
Zero slope
Negative slope
32
Calculating Area
Area of a Rectangle = Base * Height
Height
Area of a Triangle = ½ * Base * Height
Base
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Area under the Linear Function
What is the area under the linear
function?
Height
Area of a triangle = ½ * base * height
Base
Area = ½ * 1 * 4
=2
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Area Calculation Exercise
Calculate the area of the shaded
region.
(Assume each square is 1 unit.)
Area of rectangle + Area of triangle
3*2
6
6.5
+ ½*1*1
+ 0.5
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Thank You
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