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Chapter 11 Review Important Terms, Symbols, Concepts 11.1. The Constant e and Continuous Compound Interest The number e is defined as either one of the limits n 1 e lim 1 s 1 e lim 1 s n n s0 If the number of compounding periods in one year is increased without limit, we obtain the compound interest formula A = Pert, where P = principal, r = annual interest rate compounded continuously, t = time in years, and A = amount at time t. Barnett/Ziegler/Byleen Business Calculus 11e 1 Chapter 11 Review 11.2. Derivatives of Exponential and Logarithmic Functions For b > 0, b 1 d x d x e e x b b x ln b dx dx d 1 ln x d logb x 1 1 ( ) dx x dx ln b x The change of base formulas allow conversion from base e to any base b > 0, b 1: bx = ex ln b, logb x = ln x/ln b. Barnett/Ziegler/Byleen Business Calculus 11e 2 Chapter 11 Review 11.3. Derivatives of Products and Quotients Product Rule: If f (x) = F(x) S(x), then dS dF f ' ( x) F S dx dx Quotient Rule: If f (x) = T (x) / B(x), then B ( x) T ' ( x) T ( x) B ' ( x) f ' ( x) [ B ( x)] 2 11.4. Chain Rule If m(x) = f [g(x)], then m’(x) = f ’[g(x)] g’(x) Barnett/Ziegler/Byleen Business Calculus 11e 3 Chapter 11 Review 11.4. Chain Rule (continued) A special case of the chain rule is the general power rule: d f x n n f x n 1 f ' ( x) dx Other special cases of the chain rule are the following general derivative rules: d 1 d f ( x) ln [ f ( x)] f ' ( x) e e f ( x ) f ' ( x) dx f ( x) dx Barnett/Ziegler/Byleen Business Calculus 11e 4 Chapter 11 Review 11.5. Implicit Differentiation If y = y(x) is a function defined by an equation of the form F(x, y) = 0, we can use implicit differentiation to find y’ in terms of x, y. 11.6. Related Rates If x and y represent quantities that are changing with respect to time and are related by an equation of the form F(x, y) = 0, then implicit differentiation produces an equation that relates x, y, dy/dt and dx/dt. Problems of this type are called related rates problems. Barnett/Ziegler/Byleen Business Calculus 11e 5 Chapter 11 Review 11.7. Elasticity of Demand The relative rate of change, or the logarithmic derivative, of a function f (x) is f ’(x) / f (x), and the percentage rate of change is 100 (f ’(x) / f (x). If price and demand are related by x = f (p), then the elasticity of demand is given by p f ' ( p) relative rate of change of demand E ( p) f ( p) relative rate of change of price Demand is inelastic if 0 < E(p) < 1. (Demand is not sensitive to changes in price). Demand is elastic if E(p) > 1. (Demand is sensitive to changes in price). Barnett/Ziegler/Byleen Business Calculus 11e 6

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posted: | 8/20/2012 |

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