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If a, b, c, and d are constants such that lim xl0 ax 2 1

Calculus | 8th Edition | ISBN: 9781285740621 | Authors: James Stewart ISBN: 9781285740621 127

Solution for problem 7 Chapter 4

Calculus | 8th Edition

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Calculus | 8th Edition | ISBN: 9781285740621 | Authors: James Stewart

Calculus | 8th Edition

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Problem 7

If a, b, c, and d are constants such that lim xl0 ax 2 1 sin bx 1 sin cx 1 sin dx 3x 2 1 5x 4 1 7x 6 8 find the value of the sum a 1 b 1 c 1 d.

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Econ 2010 Notes 3 Supply -The amount of a good that a firm could and would sell at all possible prices, given the environment. Q: What determines how much is supplied to the market -Free markets determine how much is supplied. -The price! (Important for supply and demand) Supply Schedule -A table that shows the relationship between the price of a good and the quantity supplied. Law of Supply -As the price goes up, quantity supplied goes up -Opposite of the law of demand. -So it is positively related -As a seller, you want to make as much profit as possible. Q: What is the slope of the Supply Curve -The curve will have a positive slope and increase. Market Supply vs. Individual Supply -The quantity supplied in the market is the sum of the quantities supplied by all sellers at each price. -Just add to get the total quantity supplied! -They are also positively related. Determinants of Supply: 1. Price of the good 2. Prices of Inputs 3. Technology 4. Number of Suppliers 5. Expectations of future prices Price decreases  Quantity supplied decreases -Just like demand you MOVE ALONG THE SUPPLY CURVE Q: What is the ultimate goal in doing business -To make a profit! -Profit = total revenue-total cost -Price x quantity A change in supply -Opposite of a change in demand An increase in supply =The entire SC shifts to the right (out) A decrease in supply =The entire SC shifts left (in) Q: What determines cost of production -if a firm can produce the good cheaper then it will want to produce more of the good at all possible prices. Input prices -The cost of production. -What is used in order to create the output! -Wages cost of machines, rent, raw materials, and workers. Ex: Illegal immigrants are willing to work for low wages. This means less cost of production and the supply will increase. The supply curve will shift to the right because we are supplying more! Technology -Better technology makes firm more efficient = more output for some amount of input. Q: What other factors that effect supply Number of suppliers -If there are more firms in a industry, then more will be supplied at any price Expectation -Future prices -If you are a seller, you expect the price will go up in the future. Do you sell now or later You sell later! You will make maximize your profit! -Opposite of demand expectations Ex: Suppose there is an increase in the price of steel. We would expect the supply curve for steel beams to____ -Shift to the left -Price of inputs goes up -Cost of production goes up -Supply goes down -Therefore shifts to the left Equilibrium Price -Where price has reached the level where quantity supplied equals quantity demanded. -On the chart, the demand line and the supply line cross at this point. -A perfect market! Disequilibrium -Shortage (=excess demand) Qs < Qd -Surplus (=excess supply) Qs > Qd Three Steps to Analyze Changes in Equilibrium 1. Decide whether event shifts supply curve, or the demand curve, or both, 2. Decide in which direction curve shifts (left or right) 3. Use supply-demand diagram to see how the shift changes price and quantity. What happens when both supply and demand shifts at the same time -Supply increases, Demand increases  Price , Quantity rises -Supply decreases, Demand deceases  Price , Quantity falls -Supply increases, Demand decreases  Price falls, Quantity -Supply decreases, Demand increases  Price rises, Quantity Chapter 5 Elasticity and its Applications Elasticity -A numerical measure of responsiveness of quantity demanded or quantity supplied to on of the determinants. -Always has to do with the responsiveness between 2 variables. Price Elasticity of Demand - Measures how much quantity demanded responds to a change in price. - The size of this number determines customers’ sensitivity towards price. - Smaller numbers means people will not react as much to price changes. - Larger numbers means people will react more to price changes. Price elasticity demanded = % change in quantity demanded / % change in price Ex: -Price rises by 10%. -Quantity falls by 15%. - -15%/10% = (WILL ALWAYS BE A NEGATIVE NUMBER) the absolute value of -1.5 = 1.5 Ex: -Price falls by 10%. -Income increases by 5%. -Quantity increases by 20%. -Price elasticity cannot by calculated because there is a changing income variable. Calculating Percentage Changes Standard method (do not use because gives different answers depending on which variable you start with) -End value – start value / start value x 100 Midpoint Method -End value – start value / midpoint x 100 -The midpoint is the number halfway between the start and the end values, the average of those values. -Does not matter where you start or where you end! Change in quantity demanded / quantity average (midpoint) = (Q2 – Q1) / ((Q2 + Q1) /2) x100 Change in price / price average (midpoint) = (P2-P1) / ((P2 + P1) / 2) x100 Q: -When the price of a good is %5, the quantity demanded is 100 units per month; when the price is $7, the quantity demanded is 80 units per month. Using the midpoint formula, the price elasticity of demand is about… A: -Elasticity demanded = % change in quantity / % change in price -(100 – 80 / 90) / (7 – 5 / 6) = 22.2 / 33.3 = 0.67 The Variety of Demand Curves -The flatter the curve, the more elastic it is! -The price of elasticity of demand is closely related to the slope of the demand curve Demand Elasticity Elastic Demand Ed > 1 Unit elastic demand Ed = 1 Inelastic demand Ed < 1 Perfectly elastic demand (ex: $1 for 4 quarters) Ed = infinite Perfectly inelastic demand (ex: medicine, medical treatment) Ed = 0 Determinants of Price Elasticity 1. Availability of close substitutes (more substitutes > less substitutes) -Ex: breakfast cereal vs. sunscreen -Price goes up  which good is more elastic (Which good does quantity demanded drop) -Breakfast cereal has many close substitutes, so buyers can easily switch if the price rises. It is more elastic and people respond more to price change. -Sunscreen does not have close substitutes, and therefore is inelastic and people respond less towards price changes. 2. Necessities vs. luxuries (necessities < luxuries) -Ex: insulin vs. Caribbean cruises -Price goes up  which is more elastic -Insulin is a necessity. A rise in its price would cause little or no decrease in demand. It is inelastic and people react less to price change. -Caribbean Cruises are a luxury and you don’t really need it to survive. If it goes up you are likely not to go. It is elastic and people react to price change. 3. Definition of the market (narrower good > broader good) -Ex: blue jeans vs. clothing -Price goes up  which is elastic -For a narrowly defined good, such as blue jeans, there are many substitutes. This is elastic and people react more to price change. -For a broader good such as clothing, there are not many substitutes (nude). This is inelastic and people react less to price. 4. Time horizon (long time > short time) -Ex: Gasoline in the short run vs. gasoline in the long run -Price goes up  which is more elastic -There’s not many people that can do a short run, other than ride the bus or carpool. Fewer substitutes. People react less to price because they don’t have enough time to do so and therefore it is inelastic. In a long run, people can buy smaller cars or live closer to where they work. They have more substitutes. The long run person react more to price change and is elastic. 5. Cost relative to income (expensive good > cheap goods) -Ex: pencil vs. vacation trip -Price goes up  which is more elastic -Pencils are a small portion of a persons’ income and people don’t care. It is inelastic and people react less to price change. -Vacation trips are a large portion of an income. People react to price change and it is elastic. Price Elasticity of Demand and Total Revenue Total Revenue -Total Revenue = Price * Quantity -A price increase has two effects on revenue Price Effect -Higher price means more revenue on each unit you sell. Quantity Effect -The price increases, you make more revenue, but you sell fewer units due to the Law of Demand. Q: Which of these effects is bigger A: It depends on the price elasticity of demand. -If demand is elastic, than elasticity of demand > 1 -This means the total revenue falls. (Quantity Effect > Price Effect) -If demand is inelastic, than elasticity of demand < 1 -This means that the total revenue will rise. (Price Effect > Quantity Effect) Cross-Price Elasticity of Demand -Measures the response of demand for one good to changes in the price of another. =% change in quantity for good X / % change in price for good Y -For substitutes, cross-price elasticity >0 Ex: An increase in the price of beef causes an increase in demand for chicken. -For compliments, cross-price elasticity < 0 (negative) Ex: an increase in the price of computers causes a decrease in the demand for software. Price Elasticity of Supply -Measures how much quantity supplied responds to a change in price. =% change in quantity supplied / % change in price -It measures sellers’ price-sensitivity -In a graph, the steeper the curve, the more elastic the supply Chapter 7 Consumers, Producers Efficiency -How society can best allocate its scarce resources Allocation of resources refers to: -How much of each good is produced -Which producers produce it -Which consumers consume it Welfare Economics -Studies how the allocation of resources affects economic well- being. Marginal benefit -Measures how much a buyer values a good -The benefit a person receives from consuming one more unit of a good. -The max amount that a person is willing to pay for an additional unit of a good. -The willingness to pay Ex: -You’re willing to pay $5 for one slice of pizza because you are super hungry. -You’re still hungry, but not as much, so you’re willing to pay $3 for the second slice. -For the third slice, you’re even less full and less desperate for food, my willingness to pay is $1 The principle of decreasing marginal benefit -MB decreases as you consume more of a good. -Demand Curve = Willingness to Pay Curve = Marginal Benefit Curve -The demand curve looks like a staircase. (one step per buyer) Consumer Surplus -The amount a buyer is willing to pay minus the amount the buyer actually pays. -Consumer Surplus = Willingness to Pay – Price = Marginal benefit – Price Ex: -Price is $260 -Ryan’s willingness to buy is $300 -300 – 260 = 40 -Consumer Surplus = $40

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Chapter 4, Problem 7 is Solved
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Textbook: Calculus
Edition: 8
Author: James Stewart
ISBN: 9781285740621

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If a, b, c, and d are constants such that lim xl0 ax 2 1