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Math for Econ 1; Week 1

by: Sara Mendez

Math for Econ 1; Week 1 Math-UA-211-011

Marketplace > New York University > Economcs > Math-UA-211-011 > Math for Econ 1 Week 1
Sara Mendez
GPA 3.6
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About this Document

Functions and finding the domain
Mathematics for Economics I
Trushant S. Majmudar
Class Notes
Math, Economics, Math for Economics




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This 5 page Class Notes was uploaded by Sara Mendez on Sunday January 31, 2016. The Class Notes belongs to Math-UA-211-011 at New York University taught by Trushant S. Majmudar in Spring 2016. Since its upload, it has received 333 views. For similar materials see Mathematics for Economics I in Economcs at New York University.

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Date Created: 01/31/16
Lecture 1: Functions & Finding the Domain  Basic Functions: 1. Linear  x ; 2x – 1  2. Integer Powers  x^2 ; X^3 ; X^4 3. Polynomials  2x^2 – 1 ; x^3 + x ; x^5 – 3x^4 +1  4. Root    x  5. Exponential  (1.03)^x ; 2^x; e^x  6. Logarithmic  Domain: all permissible values depending on x­variable ­the denominator of a fraction cannot equal 0 ­you cannot have a negative inside a radical  Ex 1: Find the domain of f(x) = x / 3x­1  ­ Set the denominator equal to 0 3x – 1 = 0 3x = 1 x = 1/3 ­ x cannot equal 1/3 ­ the domain of this equation is all real numbers except for 1/3 ­ Domain: (­INFINITY, 1/3) U (1/3, INFINITY) Ex 2: Find the domain of f(x) =  x  +  4 – x  ­ You cannot have a negative inside a radical  x ≥ 0 4 – x ≥ 0 ­x ≥ ­4 x ≤ 4 ­ Domain [0,4] “Piece­wise Functions”    ­ Use the equation x^2 if trying to find f(x) when x is less than 2 ­ Use the equation 10 – x to find f(x) when x is greater than 2 and less than or  equal  to 6  ­ When x = 2, f(x) = 6 Linear Functions  y = f(x) = mx + b ­ Two points entirely determine a line   ­ m = slope = rise/run = (y – y1) / (x – x2)  ­ b = y­intercept; y­value when x = 0 Ex 3: Linear Models: the % of people in the U.S below poverty, t years after 2000 Years after 2000 0 1 2 3 Poverty % 11.3 11.7 12.1 12.5 a) Find a formula for p, assuming it is a linear function of t  ­ Pick any two points to find the slope  m = (12.1 – 11.7) / (2 – 1) = .4 ­ b = the y­value when x = 0 which is 11.3 ­ f(t) = .4t + 11.3 b) Use this formula to predict the % of poverty in 2006 ­ To predict poverty % in 2006, plug in t = 6 in the formula ­ f(6) = .4(6) + 11.3 = 13.7%  c) Reality: in 2006, 12.3% of the U.S lived in poverty. Is the model accurate?  ­ No, the linear model is not accurate.  ­ To make it accurate you’d need to modify your function by making a piece­wise  function  ­ This function works for the years 2000­2003, but it is inaccurate for 2006 Ex 4: if the price for gasoline is p = $3.50 per gallon, the demand is q = 40 gallons. If p = $4.00 per gallon, the demand goes down to q =35 gallons. a) Find an equation for the demand q as a function of p (assume it’s linear) ­ q = mp + b m = (35 – 40) / (4 – 3.5) = ­10 ­ to find b, plug in (4, 35) to the equation  35 = ­10(4) + b b = 75 ­ q = ­10p + 75 Note: we usually graph supply and demand and put q on the horizontal axis and p  on the vertical axis. If we wanted to graph this equation, we’d have to inverse it.  b) At what price would consumers refuse to buy at all?  ­ When consumers refuse to buy, demand = 0 0 = ­10p + 75 ­75 = ­10p p = 7.5  ­ Consumers would refuse to buy when the price is $7.50 per gallon  Supply & Demand Supply: the quantity of a good that suppliers are willing to make.  ­ The higher the price, the more suppliers are willing to make  ­ Supply curve relates quantity suppliers are willing to make to sale price ­ p or below  suppliers won’t produce at all     Demand: quantity of a good consumers are willing to buy.  ­ p = price is too high, consumers won’t buy ­ q = the good is free, consumers won’t buy  Supply & Demand Equilibrium  ­ the point where supply and demand for a good are equal  ­ we assume the market works its way toward the equilibrium point  Ex 5: Find equilibrium point if  Quantity supplied = 3p – 50  Quantity demanded = 100 – 2p  Note: the slope of supply is always positive; the slope of demand is always negative ­ Set equations equal to each other  3p – 50 = 100 – 2p  5p = 150 p = 30 ­ Now that we know the equilibrium price is p=30, plug that into one of the  equations to find the equilibrium quantity  q = 3p – 50 q = 3(30) – 50 q = 40 ­ The equilibrium point is (40,30)  Taxes on Demanders (Linear demand curve) ­ If a tax of $t is imposed on consumers, the demand curve shifts down by t  ­ P is replaced by p + t, since consumers pay more  Taxes on Suppliers (linear supply curve)  ­ If a tax of $t is imposed on consumers, the supply curve shifts up by t  ­ p is replaced by p – 6, since suppliers get paid less for their sales   


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