Chapter 3 The IntegralBusiness Calculus214Graphically, the amount of extra money that ended up in consumers’ pockets is the area between the demand curve and the horizontal line at p*. This is the difference in price, summed up over all the consumers who spent less than they expected to –a definite integral.Notice that since the area under the horizontal lineis a rectangle, we can simplify the area integral:()()()()*****0*0*0*0qpdqqddqpdqqfdqpqdqqqq−=−=−∫∫∫∫The amount of extra money that ended up in producers’ pockets is the area between the supply curve and the horizontal line at p*. This is the difference in price, summed up over all the producers who received more than they expected to.Similar to consumer surplus, this integral can be simplified:()()()()∫∫∫∫−=−=−*0*0*0*0****qqqqdqqsqpdqqsdqpdqqspConsumer and Producer SurplusGiven a demand function p = d(q) and a supply function p = s(q), and the equilibrium point (q*, p*)The consumer surplus= ()***0qpdqqdq−∫The producer surplus= ()∫−*0**qdqqsqpThe sum of the consumer surplus and producer surplus is the total gains from trad

The definite integralof a positive function f(x) over an interval [a, b] is the area between f, the x-axis, x = a and x = b. The definite integralof a positive function f(x) from a to b is the area under the curve between a and b.If f(t) representsa positive rate (in y-units per t-units), then the definite integralof f from a to b is the total y-units that accumulate between t = a and t = b. Notation for the Definite Integral:The definite integral of f from a to b is writtenbadxxfThe symbol is called an integral sign; it’s an elongated letter S, standing for sum. (The is actuallythe Σ from the Riemann sum, written in Roman letters instead of Greek letters.)The dx on the end must be included; you can think of and dx as left and right parentheses. The dx tells what the variable is –in this example, the variable is x. (The dx is actually thexfrom the Riemann sum, written in Roman letters instead of Greek letters.) The function f is called theintegrand.The a and b are called the limits of integration.Verb forms:Weintegrate, or find the definite integralof a function. This process is called integration.Formal Algebraic Definition:niixnbaxxfdxxf10lim. (*)Practical Definition:The definite integral can be approximated with a Riemann sum (dividing the area into rectangles where the height of each rectangle comes from the function, computing the area of each rectangle, and adding them up). The more rectangles you use, the narrower the rectangles are, the better y

Chapter1 ReviewBusiness Calculus14Solution:Because the doubling time is constant, we know the bacteria are growing exponentially. a.This part is easy to figure out without writing a formula, by just counting up. If they double every 20 minutes, then there are 6 million at 12:20, there are 12 million at 12:40, and there are 24 million at 1:00.b.This part is not so easy –1:30 isn’t a whole number of 20-minute chunks after noon. So we will build the formula. Our units will be millions of bacteria and hours. The initial amount, the principal, 0Ais the 3 million bacteria we started with at noon. Our population is doubling every twenty minutes, so it’s being multiplied by 2 every 1/3 hour. Over one hour, then, it will be multiplied by32.

Chapter 2 The DerivativeBusiness Calculus982.4 Exercises1. Fill in the table with the appropriate units for f '(x).units for xunits for f(x)units for f '(x)hoursmilespeopleautomobilesdollarspancakesdaystroutsecondsmiles per secondsecondsgallonsstudy hourstest

Chapter 2 The DerivativeBusiness Calculus96The Profit (P)for q items is TR(q) –TC(q), the difference between total revenue and total costsThe average profit for q items is P/q. The marginal profit at q items is P(q + 1) –P(q), or ()qP′Graphical Interpretations of the Basic Business Math TermsIllustration/Example:Here are the graphs of TR and TC for producing and selling a certain item. The horizontalaxis is the number of items, in thousands. The vertical axis is the number of dollars, also in thousands. First, notice how to find the fixed cost and variable cost from the graph here. FC is the y-intercept of the TC graph. (FC = TC(0).) The graph of TVC would have the same shape as the graph of TC, shifted down. (TVC = TC –FC.) We already know that we can find average rates of change by finding slopes of secant lines. AC, AR, MC, and MR are all rates of change, and we can find them with slopes, too.AC(q) is the slope of a diagonal line, from (0, 0) to (q, TC(q)).AR(q) is the slope of the line from (0, 0) to (q, TR(q)).The Marginal Revenue (MR)at q items is the cost of producing the nextitem, MR(q) = TR(q + 1) –TR(q). Just as with marginal cost, we will use both this definition and the derivative definition MR(q) = TR’(q).Your pro

Chapter2 The DerivativeBusiness Calculus17Solution:8.1104170018.1

OK, enough of that. Here are the basic rules, all in one place.Derivative Rules:Building BlocksIn what follows, fand gare differentiable functions of x.(a)Constant Multiple Rule:'kfkfdxd(b)Sum (or Difference) Rule:''gfgfdxd(or ''gfgfdxd)(c) Power Rule:1nnnxxdxdSpecial cases: 0kdxd(because 0kxk)1xdxd(because 1xx)(d) Exponential Functions:xxeedxdxxaaadxdln(e)Natural Logarithm:xxdxd1lnThe sum, difference, and constant multiple rule combined with the power rule allow us to easilyfind the derivative of any polynomial.Example:Find the derivative of 1003

Domain and RangeDomain:The set of possible input values to a functionRange:The set of possible output values of a function

Square root:2()fx x x= =Cube root:3()fx x