Math Camp: Day 1. School of Public Policy George Mason University August 19, :00 to 8:00 pm.

Transcription

1 Math Camp: Day 1 School of Public Policy George Mason University August 19, :00 to 8:00 pm Amit Patel, PhD apatelh@gmu.edu Teaching Associates Lokesh Dani Lisardo A. Bolanos Fletes ldani@masonlive.gmu.edu lbolano2@masonlive.gmu.edu 1

2 Course Outline Monday 8/19 Pre-test Percentage, Rate of Change Review of concepts, notation Algebra Review, Functions Exponents Solving linear systems Coordinate geometry Wrap-up, assignments Wednesday 8/21 Review assignment Probability Linear functions (cont.) Non-linear functions Derivatives (cont.) Intro to Optimization Post-test, course assessment Course Website: (Lokesh Dani) 2

3 Pre-Test 15 Minutes 3

4 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 Coefficient Variable Constants 4x 7 = 5 4

5 Concepts and Notation: Subscripts Subscripts: letters written below and usually to the right of variables to distinguish different elements x i value of x for a particular element i (where i is defined as going from 1 to I) i x value of x 4 = 28 I = 10 y t value of y for a particular time period 5

6 Concepts and Notation: Summation Summation symbol Σ: sum up the expression that follows for all values for elements i to I i x Σ x i = = 342 i = 1 Can also be written as Σ x i i 6

7 Using the Summation Symbol Exercise Solve 5 Σ 2x i = i = 1 where i x

8 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% (= = 234/1000 = 117/500) Percent change /rate of change : measure of relative change between old and new value x 1 = 231 x 2 = 253 Percent : ( )/231 = = 9.5% 8

9 Percent Changes Exercise: Jurisdiction 2010 Population Pop Change Pop Percent Change Avg. Annual Pop Growth Rate Virginia 8,001, ,509 Culpeper County 46,689 12,427 Source: US Census Bureau 2000 and 2010 Census SF1 9

10 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 10

11 Order of Operations PEMDAS: order that must be adhered to when solving expressions P E M D A S Parentheses Exponents Multiplication Division Addition Subtraction Parentheses 3 * (4-2) / 2 1 Exponents 3 * / 2 1 = Multiplication 3 * / 2 1 = Division / 2 1 = Addition = Subtraction 16 1 = 15 11

12 Order of operations Exercises: 5 * (10 8) 2 10 = * 3 / 9 = 12

13 Adding and Subtracting Fractions To add or subtract, find the lowest common denominator then add or subtract across the top Exercise = 13

14 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 14

15 Operations with Negative Numbers Adding: 4 + (-2) = 4 2 = 2 Subtracting: 4 (-2) = = 6 Multiplying: 4 * (-2) = * -2 = 4 Dividing: 4 / (-2) = /( -2) = 2 15

16 Exponents: Rules z x = z*z*z*z (x times) e.g. 4 3 = 4 * 4 * 4 = 64 z x *z y =z (x+y) e.g. 4 2 *4 3 = 4 (2+3) z x /z y =z (x-y) e.g. 5 4 /5 3 = 5 (4-3) Must be the same base number Must be the same base number (z x ) y = z (x*y) e.g. (3 2 ) 3 = 3 6 = 729

17 Exponents: Rules z -x = 1/z x e.g. 3-2 = 1/3 2 = 1/9 z ½ = z e.g. 4 ½ = 4 = ± 2 z 1 = z e.g. 7 1 = 7 z 0 = 1 e.g. 8 0 = 1, = 1

18 Exponents: Exercises 2 4 * 2 2 = (5 4 ) 2 = 7 3 * *3 2 = 14 0 =

19 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) 19

20 Algebraic Functions Linear functions: slope-intercept form f(x) = mx + b f(x) = 3x + 7 (or y = 3x + 7) Degree = 1 (no squared+ x) Nonlinear functions: includes squared x, cubed x, etc. l 1 l 2 Degree > 1 f 1 f(x) = x 2 g(x) = -7x f 2 20

21 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 = x = x = 4 21

22 Algebra Review Exercises Solve for x 7x 12 = 3x + 8 ½ x 1 = ⅓ x

23 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 23

24 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 24

25 Solving Linear Systems: 2 Variables Two Variables Two Equations y = 3x 1 y = 2x Substitute y = 2x into equation 1 2x = 3x 1 1 = x 10x 5y = 20 x + y = 2 Re-arrange equation 2 y = 2 x Substitute into equation 1 10x 5(2 x) = 20 10x 10 +5x = 20 15x 10 = 20 15x = 30 x = 2, y = 0 (b/c y = 2 x) 25

26 Solving Linear Systems: 2 Variables Exercises Solve for x and y 3x y = 2 2x + 2y = 12 26

27 Solving Linear Systems: 3 Variables Solve for x, y and z 3x y + z = 2 2x + 2y z = 12 x + y z = 6 27

28 Coordinate Geometry (1, 2) (2,4) Plotting points: (1, 2) (2, 4) Calculating slope: m = y 2 y 1 x 2 x 1 = (4 2) (2 1) = 2 28

29 Coordinate Geometry What is the slope of Line A? (3,1) (-2,-2) 29

30 Coordinate Geometry rise = 2 run = 3 Sketch the following linear function. y = 2/3x 1 Slope = 2/3 y-intercept = -1 Plug-and-chug method: x y /3 2 1/

31 Coordinate Geometry Sketch the following linear function. 4x 2y = 4 31

32 Coordinate Geometry Undefined slope Positive slope Zero slope Negative slope 32

33 Height Calculating Area Area of a Rectangle = Base * Height Area of a Triangle = ½ * Base * Height Base 33

34 Height Area under the Linear Function What is the area under the linear function? Area of a triangle = ½ * base * height Base Area = ½ * 1 * 4 = 2 34

35 Area Calculation Exercise Calculate the area of the shaded region. (Assume each square is 1 unit.) Area of rectangle + Area of triangle 3 * 2 + ½ * 1 *

36 Thank You 36