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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] 1 Course Outline Monday 8/19 Wednesday 8/21 • • • • • • • • • • • • • • • 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) 2 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 5 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 6 Using the Summation Symbol Exercise Solve Σ 2xi = 5 i=1 where i 1 2 3 4 5 x 0 2 4 6 8 7 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% 8 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 9 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 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 11 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 14 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 15 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) 19 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 20 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 21 Algebra Review Exercises Solve for x 7x – 12 = 3x + 8 ½x–1=⅓x+5 22 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 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 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) 25 Solving Linear Systems: 2 Variables Exercises Solve for x and y 3x – y = 2 2x + 2y = 12 26 Solving Linear Systems: 3 Variables Solve for x, y and z 3x – y + z = 2 2x + 2y – z = 12 x+y–z=6 27 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 28 Coordinate Geometry What is the slope of Line A? (3,1) (-2,-2) 29 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 30 Coordinate Geometry Sketch the following linear function. 4x – 2y = 4 31 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 33 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 34 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 35 Thank You 36