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Chapter 1 Notes: Sections 1.1-1.6

by: Ellie Karayan

Chapter 1 Notes: Sections 1.1-1.6 MATH 109

Marketplace > Chapman University > Mathmatics > MATH 109 > Chapter 1 Notes Sections 1 1 1 6
Ellie Karayan
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These are the notes for the first half of Chapter 1. This covers everything from the basic functions to the natural logarithm. If you have any questions, I am always available. Feel free to email m...
Calculus with Applications in Business and Social Science
Jill Dunham
Class Notes
Math, Calculus, functions, math109, Exponential, Exponential Functions, exponents, natural, log, rate, Of, change, Linear, Tips, concavity, concave, relative, Economics, cost, Revenue, profit, break, Even, e, ln, Marginal, supply, demand, Equilibrium, tax, sales




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This 9 page Class Notes was uploaded by Ellie Karayan on Tuesday August 30, 2016. The Class Notes belongs to MATH 109 at Chapman University taught by Jill Dunham in Fall 2016. Since its upload, it has received 5 views. For similar materials see Calculus with Applications in Business and Social Science in Mathmatics at Chapman University.


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Date Created: 08/30/16
Chapter 1: Sections 1-6 1.1 What is a Function? A function is a rule that takes certain numbers as inputs and assigns to each a definite output number. Domain: the set of all input numbers Range: the set of all output numbers Independent Variable: input Dependent Variable: output Interval Notation: Closed Intervals: a ⦤ t ⦤b and is written [a,b] Open Intervals: a < t < b and is written (a,b) A mathematical model is a mathematical description of a real situation. Increasing and Decreasing Functions: A function is increasing if the values of f(x) increase as x increases A function is decreasing if the values of f(x) decrease as x increases 1.2 Linear Functions A linear function has a graph that has a straight line. Slope= rise/run = ∆y/∆x= f(x2)-f(1 )2x 1x A linear function has the form y=b+mx If m is positive, f(x) is increasing. If m is negative, f(x) is decreasing. If m is 0 we have a straight line. 1.3 Average Rate of Change and Relative Change Visualizing Rate of Change The change in a function( ∆y) is represented by a vertical distance. The average rate of change (blue line) is represented by the slope of a line. Concavity Concavity is a way of classifying a graph. Concave Up: Concave Down: Relative Change In general, when a quantity P changes from P to P0 1 Relative change in P= change in P/ P = P0- P 1 P 0 0 You can remember it like a cup. If is could hold water, it is concave up. If it cannot, it is concave down. 1.4 Applications of Functions to Economics The Cost Function The cost function, C(q), gives the total cost of producing a quantity, q, of some good. This can be broken down into: Fixed costs: incurred even if nothing is produced Variable costs: depend on how many units are produced The variable cost for one additional unit is called marginal cost for a linear cost function, this is the rate of change/slope. If C(q) is a linear cost function: -fixed costs are represented by the vertical intercept -marginal cost is represented by the slope The Revenue Function The revenue function, R(q), gives the total revenue received by a firm from selling a quantity, q, of some good. If the good sells for a price of p per unit and quantity sold is q, Revenue= price (quantity) so R=pq If price does not depend on quantity sold, so p is a constant, the graph of revenue as a function of q is a line through the origin, with slope equal to the price p. The Profit Function In this case we use π to represent profit. Profit= Revenue - Cost so π= R-C The break-even point for a company is where π=0 and R=C The Marginal Cost, Marginal Revenue and Marginal Profit Marginal revenue: rate of change, or slope, of linear revenue functions Marginal profit: rate of change, or slope, of linear profit functions Supply and Demand Curves Supply curve: relates quantity, q, of the item that manufactures are willing to make per unit time to the price, p, for which the item can be sold. Demand curve: relates the quantity, q, of an item demanded by consumers per unit time to the price, p, of the item. Equilibrium Price and Quantity The point where the two lines intersect is called the equilibrium point. It is assumed that the market naturally settles to this point (the green dot) The Effect of Taxes on Equilibrium Specific tax: a fixed amount per unit regardless of selling price; usually imposed on the producer Examples of this include: gasoline, alcohol and cigarettes Sales tax: a fixed percentage of the selling price A budget constraint is a specific allocation of how money is able to be spent. The best example of this is in the case of guns and butter: A government can choose to allocate its money to national defense to protect its citizens or it can use the money to make the life of its citizens better through investments in the lives of their citizens. We use the term "marginal" because we are looking for how the cost, revenue or profit change "at the margin"/ by the addition of one or more unit. Supply: q(p)= mp+ b b- represents the maximum possible supplied m- how supply changes as price changes (positive) Demand: q(p)= mp+ b b- represents the maximum possible demand m- how demand changes as price changes (negative) 1.5 Exponential Functions x Exponential functions in the form f(x)= ka are often used in natural and social sciences. Population Growth Exponential Growth: as t increases, P increases P=P 0 t P0= the initial population a = growth factor This is used when we have a constant percent increase. Elimination of a Drug from the Body Exponential Decay: as t increases, the function gets closer to zero Q= f(t)= Q a0 t Q 0 the initial quantity a = decay factor The General Exponential Function Exponential functions are often described in terms of percent growth or decay rates. t P=P a0 P0= the initial quantity a = growth/ decay factor a= 1+r r= a decimal representation of the percent rate of change Exponential Growth: a>1 r will be positive Exponential Decay: 0< a < 1 r will be negative Comparison Between Linear and Exponential Functions Linear functions: change at a constant or absolute rate Ex. 2 inches per day Exponential functions: changes at a constant percent or relative rate Ex. 3.4% per year Exponential decay functions have horizontal asymptotes. These are lines that the graph of the function approaches as x approaches infinity. 1.6 The Natural Logarithm The natural logarithm of x, ln x, is the power of e needed to get x Ln x= 1 means e = x Properties of the Natural Logarithm 1 Ln (AB) = ln A + ln B 2 Ln (A/B)= ln A- ln B P 3 Ln (A ) = p ln A 4 Ln e = x 5 eln= x 0 1 In addition, ln 1= 0 because e = 1 and ln e = 1 because e = e F(x) = ln x The x- intercept is 1 since ln 1= 0 Exponential Functions with Base e P=P a0 t x For any positive value of a, we can write a= e where k= ln a So the exponential formula can be written as P=P a0= P (e0) = P e 0 kt k is the continuous rate of growth or decay Exponential Growth: a > 1: k is positive x > 1: ln x = positive Exponential Decay: 0< a < 1: k is negative 0 < x < 1: ln x = negative


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