#### Transcript Business Calculus - Front Range Community College

Business Calculus Other Bases & Elasticity of Demand 3.5 The Exponential Function Know your facts for f ( x) a x 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 ya x : dy a x (lna) dx 4. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule. Logarithmic Function Know your facts for f ( x) loga x 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 2. Evaluate logarithmic functions by calculator. 3. Change of Base formula: 4. Differentiate lnb loga b ln a y loga: x dy 1 dx x ln a 5. Differentiate exponential functions using the sum/difference, coefficient, product, quotient, or chain rule. 3.6 Elasticity of Demand 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. x Percent change in price: x Percent change in quantity: (given as a decimal) q q (given as a decimal) E(x) is the negative ratio of percent change in quantity to percent change in price. q q E ( x) x x 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) q q E ( x) x x 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. Calculus & Elasticity and Revenue Using calculus, and taking the limit as ∆x → 0, we can show that x D ( x) E ( x) D( x) 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. 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.