Presentation on theme: "Business Calculus Other Bases & Elasticity of Demand."— Presentation transcript:
1Business CalculusOther Bases & Elasticity of Demand
23.5 The Exponential Function Know your facts forKnow the graphs: A horizontal asymptote on one side at y = 0.Through the point (0,1)Domain: (-∞, ∞) Range: (0, ∞)a > < a < 1
32. Evaluate exponential functions by calculator. 3. Differentiate :4. Differentiate exponential functions using the sum/difference,coefficient, product, quotient, or chain rule.
4Logarithmic Function Know your facts for Know the graph: A vertical asymptote on one side of the x axisat x = 0.Through the point (1,0).Domain: (0, ∞) Range: (-∞, ∞)a > < a < 1
52. Evaluate logarithmic functions by calculator. 3. Change of Base formula:4. Differentiate :5. Differentiate exponential functions using the sum/difference,coefficient, product, quotient, or chain rule.
63.6 Elasticity of DemandGiven a demand function D(x) (quantity, also called q) where xis the price of an item, we want to determine how a smallpercentage increase in price affects the demand for the item.Percent change in price: (given as a decimal)Percent change in quantity: (given as a decimal)E(x) is the negative ratio of percent change in quantity topercent change in price.
7For example: if a small increase in price causes a larger decrease in demand, this could result in a decrease in revenue.(Bad for business)In this case, E(x) would be a number > 1, andthe demand is called elastic.On the other hand, a small increase in price may cause a smallerdecrease in demand, resulting in an increase in revenue. Here,E(x) would be a number < 1, and the demand is called inelastic.If a percent change in price results in an equal percent changein demand, E(x) = 1, and demand is unit elastic.
8Calculus & Elasticity and Revenue Using calculus, and taking the limit as ∆x → 0, we can show thatWe will use this formula for elasticity to find the elasticity ofdemand for a given demand function D(x), and tocalculate elasticity at various prices.
9Revenue is related to elasticity since R(x) = price * demand. If demand is inelastic, a small increase in price will result in anincrease in revenue.If demand is elastic, a small increase in price will result in andecrease in revenue.If demand is unit elastic, the current price represents maximumrevenue.