G.1 Differentiation and Integration Formulas - …

1 appendix Differentiation and Integration FormulasG1 Use Differentiation and Integration tables to supplement Differentiation and Integration FormulasForms Involving Involving 1u a bu du 1a ln ua bu Cn 1, 2, 3 a2 n 1 a bu n 1 C, u2 a bu n du 1b3 1 n 3 a bu n 3 2a n 2 a bu n 2 u2 a bu 3 du 1b3 2aa bu a22 a bu 2 ln a bu C u2 a bu 2 du 1b3 bu a2a bu 2a ln a bu C u2a bu du 1b3 bu2 2a bu a2 ln a bu Cn 1, 2 u a bu n du 1b2 1 n 2 a bu n 2 a n 1 a bu n 1 C, u a bu 2 du 1b2 aa bu ln a bu C ua bu du 1b2 bu a ln a bu Ca bu 1u du ln u Cn 1 un du un 1n 1 C,unddx csc u csc u cot u u ddx sec u sec u tan u u ddx cot u csc2 u u ddx tan u sec2 u u ddx cos u sin u u ddx sin u cos u u ddx eu euu ddx ln u u uddx x 1ddx un nun 1u ddx c 0ddx uv vu uv v2ddx uv uv vu ddx u v u v ddx cu cu Differentiation and Integration 12/27/11 1.

2 47 PM Page G1 Integration Formulas (continued) Involving Involving Involving 1u u2 a2 du 1a ln a u2 a2u C 1 u2 a2 du ln u u2 a2 C u2 a2u2 du u2 a2u ln u u2 a2 C u2 a2u du u2 a2 a ln a u2 a2u C u2 u2 a2 du 18 u 2u2 a2 u2 a2 a4 ln u u2 a2 C u2 a2 du 12 u u2 a2 a2 ln u u2 a2 C u2 a2, a>0n 1 1 u2 a2 n du 12a2 n 1 u u2 a2 n 1 2n 3 1 u2 a2 n 1 du , 1u2 a2 du 1a2 u2 du 12a ln u au a Cu2 a2, a>0 un a bu du 2 2n 1 b un a bu na un 1 a bu du u a bu du 2 2a bu 3b2 a bu Cn 1 a buun du 1a n 1 a bu 3 2un 1 2n 5 b2 a buun 1 du , a buu du 2 a bu a 1u a bu dun 1 1un a bu du 1a n 1 a buun 1 2n 3 b2 1un 1 a bu du ,a>0 1u a bu du 1 a ln a bu a a bu a C, un a bu du 2b 2n 3 un a bu 3 2 na un 1 a bu du a bu 1u2 a bu 2 du 1a2 a 2buu a bu 2ba ln ua bu C 1u2 a bu du 1a 1u ba ln ua bu C 1u a bu 2 du 1a 1a bu 1a ln ua bu CG2 appendix G 12/27/11 1.

3 47 PM Page Involving Involving Involving Involving sin or cos un sin u du un cos u n un 1 cos u du u cos u du cos u u sin u C u sin u du sin u u cos u C cosn u du cosn 1 u sin un n 1n cosn 2 u du sinn u du sinn 1 u cos un n 1n sinn 2 u du cos2 u du 12 u sin u cos u C sin2 u du 12 u sin u cos u C cos u du sin u C sin u du cos u Cuu ln u n du u ln u n n ln u n 1 du ln u 2 du u 2 2 ln u ln u 2 Cn 1 un ln u du un 1 n 1 2 1 n 1 ln u C, u ln u du u24 1 2 ln u C ln u du u 1 ln u Cln u 11 enu du u 1n ln 1 enu C 11 eu du u ln 1 eu C uneu du uneu n un 1eu du ueu du u 1 eu C eu du eu Ceu 1 a2 u2 3 2 du ua2 a2 u2 C 1u2 a2 u2 du a2 u2a2u C 1u a2 u2 du 1a ln a a2 u2u C a2 u2u du a2 u2 a ln a a2 u2u C a2 u2, a>0 1 u2 a2 3 2 du ua2 u2 a2 C 1u2 u2 a2 du u2 a2a2u C u2 u2 a2 du 12 u u2 a2 a2 ln u u2 a2 CAppendix Differentiation and Integration 12/27/11 1.

4 47 PM Page G3 Integration Formulas (continued) Involving tan , cot , sec , or csc 11 csc u du u tan u sec u C 11 sec u du u cot u csc u C 11 cot u du 12 u ln sin u cos u C 11 tan u du 12 u ln cos u sin u Cn 1 cscn u du cscn 2 u cot un 1 n 2n 1 cscn 2 u du,n 1 secn u du secn 2 u tan un 1 n 2n 1 secn 2 u du,n 1 cotn u du cotn 1 un 1 cotn 2 u du,n 1 tann u du tann 1 un 1 tann 2 u du, csc2 u du cot u C sec2 u du tan u C cot2 u du u cot u C tan2 u du u tan u C csc u du ln csc u cot u C sec u du ln sec u tan u C cot u du ln sin u C tan u du ln cos u Cuuuu 1sin u cos u du ln tan u C 11 cos u du cot u csc u C 11 sin u du tan u sec u C un cos u du un sin u n un 1 sin u duG4 appendix G 12/27/11 1.

5 47 PM Page G4 appendix Formulas from Business and FinanceG5 Summary of business and finance formulasFormulas from BusinessBasic Termsnumber of units produced (or sold)price per unittotal revenue from selling unitstotal cost of producing unitsaverage cost per unittotal profit from selling unitsBasic EquationsTypical Graphs of Supply and Demand CurvesSupply curves increase as price increases and demand curves decrease as price increases. The equilibrium point occurs when the supply and demand curves Function:price required to sell unitsWhen the demand is inelastic. When the demand is Graphs of Revenue, Cost, and Profit FunctionsRevenue FunctionCost FunctionProfit FunctionThe low prices required toThe total cost to produceThe break-even point occurssell more units eventually units includes the fixed when result in a decreasing profit x Negative of fixed cost Break-even point P x Fixed cost C Inelastic demand x ElasticdemandR >1, <1, p xdp dx price elasticity of demandxp f x x Demand Equilibrium point(x0, p0)Supply x0p0 Equilibrium quantity Equilibrium price p P R CC CxR xpx P C x C x R p x Formulas from Business and 12/27/11 1.

6 47 PM Page G5 Formulas from Business (continued)Marginalsmarginal revenuethe extrarevenue from selling one additional unitmarginal costthe extracost of producing one additional unitmarginal profitthe extraprofit from selling one additional unitFormulas from FinanceBasic Termsamount of depositinterest ratenumber of times interest is compounded per yearnumber of yearsbalance after yearsCompound Interest when interest is compounded times per when interest is compounded continuously:Effective Rate of InterestPresent Value of a Future InvestmentBalance of an Increasing Annuity After Deposits of per Year for YearsInitial Deposit for a Decreasing Annuity with Withdrawals of per Year for YearsMonthly Installment for a Loan of Dollars over Years at % InterestAmount of an Annuityis the continuous income function in dollars per year and is the term of the annuity in t erT T0 c t e rt dtM P r 121 11 r 12 12t rtPMP W nr 1 11 r n nt tWnA P 1 rn nt 1 1 nr tPnP A 1 rn ntreff 1 rn n 1A PertA P 1 rn ntnt A t n r P dPdx dCdx dRdx G6 appendix G Formulas1 unit Extra revenue for one unit Marginal revenue Revenue 12/27/11 1:47 PM Page G6