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^— ^M^B CALCULUS WITH GEOMETRY ANALYTIC E Simmons :orge 1 ALGEBRA EXPONENTS LOGARITHMS ama" = flm+", >)n = anbn logu(Ar) = log,,.v + log,, log,, 1 =0, log,**- 1 am (\ab\)n " ab"" log, - = log, x - log, y akm.x=Xt log^ (a*) -= X (a"1)" = amn, Qm/n ^ vflm ==(Va) log, x> = y lo^x QUADRATIC FORMULA FACTORING The roots ofthe quadratic equation a2 -b2 = (a- b\a + b) = are a3 - b> = (a - b\a2 + ab + b2 ) -b ±ylb2 -4ac a* + b* = (a + b)(a2 - ab + b2 ) la BINOMIAL THEOREM (a + bf = a2 + lab + b2 (a + bf = a3 + 3a2b + lab2 + b3 , Ifn is a positive integer. — (a + p)" = a" + na" '& +-1-- - a" 2b2 + • • • + — - r^r j- : a" kbk + + 6' GEOMETRIC PROGRESSIONS AND SERIES -A"+1 v + .XT2 + • • • + A" = 1 -x V + A"2 + -A, |A|<1 1 1 DETERMINANTS ab\\ ab\2 = a>bU2-aAU, ab,x ab22 aby3 = a, bc22 b<->l <*2 bx b3 + a3 bCix bc22 GEOMETRY iy VI) v y Rectangle Triangle Circle Circular Sector = = = = Area hb Area \hb Area nr2 Length ofarc rd = Circumference == lnr Area irs ider Cone Sphere Volume = nr2h Volume = \nr2h Volume = \nr* area = Inrh Lateral area = nrs Area = Anr2 — — TRIGONOMETRY fYlB = sin y = sin tan0 cos B — x cos /LL. = cos tan cot sin (-1,0) = X = 1 cot sec y cos sec = j_ esc = - 1 sin esc = j_ cot 1 tan0 sin (--0) = —sin sin2 + cos2 = 1 cos(--B) = cos tan2 + 1 = sec2 tan(--B) = -tan0 1 + cot2 = esc2 sin (0 + 0) = sin cos + cos sin sin (0 — 0) = sin cos — cos sin cos (0 + 0) = cos cos — sin sin cos (0— 0) = cos cos + sin sin — — tan (//0i+. 0^)\ --:t—an-tan+atttaann\ tan /(/0i - 0j.)\ =-tr+a-°—tan .ttaanny, 1 1 sin 20 = 2 sin cos 2 cos2 = 1 + cos 20 cos 20= cos2 — sin2 2 sin2 = 1 - cos 20 sin45°=^V2, cos45°=^V2, tan 45° =1 = \J* sin 30° =•£, cos30°=^V3, tan 30° VI -j sin60°=^V3, cos 60°=^, tan 60° = V3 \^- 1^3 GREEK ALPHABET Letters Names Lei Names A a alpha I i iota P P rho B /? beta K K kappa 1 a sigma r y gamma lambda tau A c) delta M H mu upsilon E € epsilon N V nu <D phi z c zeta c « xi X chi H eta O O omicron 4* V psi 17 e e theta n n Pi Q omega CALCULUS WITH ANALYTIC GEOMETRY CALCULUS WITH ANALYTIC GEOMETRY George Simmons F. ProfessorofMathematics, Colorado College McGRAW-HILL BOOK COMPANY New York St. Louis San Francisco Auckland Bogota Hamburg Johannesburg London Madrid Mexico Montreal NewDelhi Panama Paris Sao Paulo Singapore Sydney Tokyo Toronto For Gertrude Clark, my the great teacher in life. must obtain it hvgreat labour sea ofbeing, h.ippv in ilic end some ifilcs in 11 Somo una i i :tud< di -i beauti du mi in U i I h th< beaut\ oftheworld Sim > I iavealwaysbeen incomprehensible ^ inderstandableonly u pro essb) onmem li is thebattleforlearning ion thai i: ab ioluli is followed al rvconceptofvictorv isdissolved in it ' -. i• i'i oldisorganization,whichtends uilibnum and samenessdescribed in the xwcll. Bolt/mann. and Gihbs meant by in theethicsofKierkegaard,whopointed In ihi our mainobligationis oestab ish •sc enclaveswill not remain there indefi- . ,r c onci establi: ic I them, i ike the uthout running as fast as we can. I in the indchnitefuture. It isthegreatest toha\cbeen. Nodefeatcandepriveusof >i.. i of time in a universe that seems