Presentation on theme: "Business Calculus Other Bases & Elasticity of Demand."— Presentation transcript:
Business Calculus Other Bases & Elasticity of Demand
3.5 The Exponential Function Know your facts for 1.Know the graphs: A horizontal asymptote on one side at y = 0. Through the point (0,1) Domain: (-, ) Range: (0, ) a > 1 0 < a < 1
2. Evaluate exponential functions by calculator. 3. Differentiate : 4. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule.
Logarithmic Function Know your facts for 1.Know the graph: A vertical asymptote on one side of the x axis at x = 0. Through the point (1,0). Domain: (0, ) Range: (-, ) a > 1 0 < a < 1
3. Change of Base formula: 4. Differentiate : 5. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule. 2. Evaluate logarithmic functions by calculator.
Given a demand function D(x) (quantity, also called q) where x is the price of an item, we want to determine how a small percentage 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 to percent change in price. 3.6 Elasticity of Demand
For 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, and the demand is called elastic. On the other hand, a small increase in price may cause a smaller decrease 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 change in demand, E(x) = 1, and demand is unit elastic.
Using calculus, and taking the limit as x 0, we can show that We will use this formula for elasticity to find the elasticity of demand for a given demand function D(x), and to calculate elasticity at various prices. Calculus & Elasticity and Revenue
Revenue is related to elasticity since R(x) = price * demand. If demand is inelastic, a small increase in price will result in an increase in revenue. If demand is elastic, a small increase in price will result in an decrease in revenue. If demand is unit elastic, the current price represents maximum revenue.