mathematics
Αα Alpha CONSTANTS Ββ Beta Γγ Gamma G (Gravitational Acceleration) = 9.81 m/s2 Δδ Delta G (Universal Gravitation) = 6.670 × 10-11 N-m2/kg2 Εε Epsilon NA (Avogadro’s Number) = 6.02 × 1023 / mol Ζζ Zeta C (Speed of Light) = 3 × 108 m/s Ηη Eta σ (Stefan Boltzmann) = 5.67 × 10-8 W/m2-K4 Θθ Theta Solar Constant = 1353 W/m2 Ιι Iota H (Planck’s C.) = 6.626 × 10-34 J-s Κκ Kappa Radius of Earth = 6.38 × 106 m Λλ Lambda Vel. To escape Earth = 11.2 km/s Μμ Mu Νν Nu ρair (@21 °C) = 1.2 kg/m3 = 0.075 lb/ft3 Ξξ Xi cp (@15.6 °C) = 1.0062 kJ/kg-K = 0.24 BTU/lb-R Οο Omicron cv (@15.6 °C) = 0.7186 kJ/kg-K = 0.1714 BTU/lb-R Ππ Phi k = 1.4 Ρρ Rho Σσ Sigma Ττ Tau Υυ Upsilon Φϕ Phi Χχ Chi Ψψ Psi Ωω Omega ρw (@15.6 °C) = 1000 kg/m3 γw (@15.6 °C) = 9.81 kN/m3 = 62.4lb/ft3 Arabic sg (water) = 1.0 1 5 Roman sg (sea water) = 1.03 cpw = 4.187 kJ/kg-K = 1 BTU/lb-R 10 I sg (mercury) = 13.6 cpice = 2.093 kJ/kg-K = 0.5 BTU/lb-R 50 V X R = 8.314 J/mol-K = 1545 ft-lbf/lbm-mol-R = 0.0821 L-atm/mol-K 100 L 500 C 1000 D M
CONVERSIONS Fundamental Standard 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 ft Quantities Fundamental Units 1 statute mile = 5280 ft Length meter (m) 1 statute mile = 1609 m Mass 1 statute mile = 80 chains Time kilogram (kg) 1 nautical mile = 1852 m second (s) 1 nautical mile = 6080 ft Temperature Kelvin (K) 1 nautical mile = 1.5156 miles Electric Current Ampere (A) 1 league = 3 nautical miles Luminous Intensity 1 Angstrom (Å) = 10-10 m Amount of Substance Candela (Cd) 1 light year = 9.46 × 1015 m Mole (mol) 1 in = 2.54 cm 1 m = 3.281 ft = 39.37 in Factor Prefix Symbol 1 rod = 5.5 yards 1024 yotta Y 1 furlong = 220 yards = 660 ft 1021 zetta Z 1 statute mile = 40 rods 1018 exa E 1 cable length = 608 ft 1015 peta P 1 fathom = 6 ft 1012 tera T 1 chain = 66 ft 109 giga G 1 span = 9 inches 106 mega M 1 cable length = 720 ft 103 kilo k 1 chain = 100 links = 66 ft 102 hecto h 1 hand = 4 inches 101 deka da 1 kilometer = 0.621 mile 10-1 deci d 1 link = 7.92 inches 10-2 centi c 1 pole = 1 perch = 1 rod 10-3 milli m 1 span = 9 inches 10-6 micro μ 1 vara = 33 1/3 inches 10-9 nano n 10-12 pico p 10-15 femto f 10-18 atto a 10-21 zepto z 10-24 yocto y
1 pound (lb) = 16 ounces CONVERSIONS 1 ounce = 28 grams 1 ounce = 20 penny wt 1 acre = 4047 m2 = 1 furlong × 1 chain 1 short ton = 2000 lb 1 acre = 660 ft × 66 ft = 43,560 ft2 1 long ton = 2240 lbs 1 mi2 = 640 acres 1 kilogram (kg) = 1000 g 1 hectare (ha) = 10,000 m2 = 2471 acres 1 kilogram (kg) = 2.205 lbs 1 are = 100 m2 1 g = 1000 mg 1 centare = 1 m2 1 tonne = 1 metric ton 1 sq rod = 25.29 m2 1 tonne = 1000 kg 1 lot = 0.046 acre 1 tonne = 2205 lbs 1 square = 100 ft2 1 kg = 2.21 lb 1 quart = 2 pints = 67.2 in3 1 kg = 35.27 oz 1 US gallon (gal) = 3.785 li 1 slug = 14.6 kg 1 US gallon (gal) = 4 quarts (qt) 1 slug = 32.2 lb 1 ft3 = 7.48 gal = 1728 in3 1 N = 100,000 dynes 1 liter (l or li) = 1000 cm3 1 kgf = 9.81N 1 gal = 3.785 li 1 lbf = 4.448 N 1 ml = 1cm3 1 picul = 133.33 lbs 1 barrel (bbl) = 42 gallons 1 quintal = 100 kg 1 ganta = 3 liters = 8 chupas 1 penny wt = 24 grains 1 cavan = 25 gantas 1 assay ton = 29,167 mg 1 tinaja = 16 gantas 1 carrat = 3.088 grains 1 truckload = 4 m3 1 scrupple = 20 grains 1 cord of wood = 128 ft3 1 dram = 30 scrupples 1 board foot = 1/12 ft3 of wood 1 stone = 14 lbs 1 barrel cement = 4 bags 1 pig = 21.5 lbs 1 m3 = 35.5 ft3 1 fother = 8 pigs 4 gills = 1 pint = 16 fld oz 2 pints = 1 quart = 52 fld oz 4 quarts = 1 gallon = 8 pints
CONVERSIONS 1 fld oz = 8 drams 1 revolution = 2π rad = 360° 1 fld oz = 2 tablespoons 1 revolution = 400 grad = 400 gons = 6400 mills 1 fld oz = 6 teaspoons = 1.805 in3 1 degree = 60 minutes = .0174 radians 1 fld oz = 29.58 cc 1 minute = 60 seconds 1 ft3 = 7.48 US gallons 1 radian = 57.32 degrees 1 US gallons = 3.785 liters π rad = 180° 1 US gallons = 1.201 British gallons 1 peck = 8 quarts °F = 1.8 °C + 32 K = °C + 273 1 bushel = 4 pecks = 1.244 ft3 R = °F + 460 ΔF° = 1.8 Δ °C 1 kerosene can = 5 gallons 1 min = 60 s 1 m/s = 3.6 kph 1 hr = 60 min = 3600 s 1 knot = 1 nautical mile/hr = 1.5156 mile/hr 1 day = 24 hr 9.81 m/s2 = 32.2 ft/s2 1 week = 7 days 15 miles/hr = 22 ft/s 1 month = 30 days (28, 29 or 31) 1 RPM = = 60 RPS = 120 1 common year = 365 days 1 leap year = 366 days 1 kg/liter = 62.4 lb/ft3 1 ppm = 1mg/L = 1mg/kg 1 year = 12 months = 52 weeks 1 decade = 10 years 1 century = 100 years 1 atm = 101.325 kPa = 14.7 psi = 29.92 inHg = 760 mmHg = 760 torr 1 MPa = 1 N/mm2 1 bar = 100 kPa
CONVERSIONS 1 BTU = 778 ft-lb = 252 cal = 1055 J 1 reyn = 1 lb-sec/in2 1 cal = 0.00397 BTU 1 poise = 1 dyne-sec/cm2 1 kcal = 4.187 kJ 1 poise =0.1 Pa-s 1 Joule = 107 erg = 1 N-m 1 erg = 1 dyne-cm 1 stoke = 1 cm2/s 1 CHU = 1.8 BTU 1eV = 1.602 × 10-19 J 1-foot water = 0.4335 psi 1 hp = 0.746 kW = 550 lbf-ft/s = 33,000 lbf-ft/min 1 gal/min = 0.063 liter/sec 1 hp = 2545 BTU/hr = 42.4 BTU/min 1 gal/min = 0.00225 ft3/sec 1 TOR = 3.517 kW = 12,000 BTU/hr 1 Pascal = 1 N/m2 1 ther (gas) 100,000 BTU 1 watt-hour = 2655.4 liters 1 dozen = 12 things 1 watt = 1 J/sec 1 gross = 12 dozens 1 kW = 1.34 hp = 56.9 BTU/min 1 great gross = 12 gross 1 hp = 0.746 kW 1 score = 20 things 1 boiler hp = 33479 BTU/hr 1 mech hp = 0.986 US hp = 75 kg-m/sec 1 metric hp = BTU/min Forces Newton Dyne Gram Kilogram Pounds 1 Newton (N) 1 100,000 102 0.102 0.2248 0.0102 2.205 × 10-6 1 Dyne 0.00001 1 1.02 × 10-6 2.205 × 10-3 1 gram (g) 9.81 × 10-3 980.67 1 0.001 2.205 1 kilogram (kg) 9.81 × 105 1000 1 1 pound (lb) 9.80665 4.45 × 105 453.6 0.4536 1 4.448
SET IDENTITIES C=A ∩ B= {x│x ∈ A and x ∈ B} • A⊂I • A⊂A • A = B if A ⊂ B and B ⊂ A C=A ∪ B= x x ∈ A or x ∈ B • Commutativity: A ∩ B = B ∩ A • Associativity: A ∩ (B ∩ C) = (A ∩ B)∩ C • Commutativity: A ∪ B = B ∪ A • Distributivity: A ∩ (B ∩ C) = (A ∩ B)(A ∩ C) • Associativity: A ∪ (B ∪ C) = (A ∪ B) ∪ C • Idempotency: A ∩ A = A • Distributivity: A ∪ (B ∪ C) = (A ∪ B)(A ∪ C) • Domination: A ∩ ∅ = ∅ • Idempotency: A∪A=A • Identity: A ∩ I = A • Domination: A ∪ I = I • Identity: A ∪ ∅ = A C = B\\A = {x│x ∈ B and x ∉ A} A′= {x ∈ I│x ∉ A} • B\\A = B\\(A ∩ B) • Complement of Union: A ∪ A’ = I • B\\A = B ∩ A’ • Complement of Intersection: A ∪ A’ = ∅ • A\\A = ∅ • A\\B = A, if A ∩ B = ∅ • (A ∪ B)’ = A’ ∩ B' • (A ∩ B)’ = A’ ∪ B' • (A\\B) ∩ C = (A ∩ C)(B ∩ C) • A’ = I\\A C=A×B= (x,y) x∈A and y∈B
SETS OF NUMBERS BASIC IDENTITIES N = {1,2,3,4…} • Identity a+0=a • Inverse a + (-a) = 0 N0 = {0,1,2,3,…} • Commutative a+b=b+a • Associative a + (b + c) = (a + b) + c • Z+ = N = {1,2,3,…} • Subtraction a – b = a + (-b) • Z- = {…,-3,-2,-1} • Z = Z- ∪ {0} ∪ Z+ Q= x x = a and a ∈ Z and b ∈ Z and b≠0 b • Identity a∙0=a Nonrepeating and non-terminating decimals • Inverse a ∙ ( 1 ) = 1, where a ≠ 0 • Commutative a R = Q ∪ Irrational numbers • Associative • Distributive a∙b=b∙a C = x + iy x ∈ R and y ∈ R , where i is imaginary • Division a ∙ (b ∙ c) = (a ∙ b) ∙ c a ∙ (b + c) = ab + ac a = a ∙ 1 b b COMPLEX NUMBERS n – natural number z – complex number bi, di – imaginary part i – imaginary unit a, c – real part r, r1, r2 – modulus of a complex number φ,φ1,φ2 – argument of a complex number i1 = i i5 = i i4n+1 = i GENERAL i2 = −1 i6 = −1 i4n+2 = −1 DEFINITION i3 = −i i7=−i i4n+3 = −i i4 = 1 i8 = 1 z = a + bi i4n = 1
COMPLEX NUMBERS • (a + bi) + (c + di) = (a + c) + (b + d)i • a + bi = a − bi • (a + bi) - (c + di) = (a - c) + (b - d)i • a = r cos φ , b = r sin φ • (a + bi) + (c+di) = (ac-bd)+(ad+bc) • a+bi = ac+bd + bc−ad i c+di c2+d2 c2+d2 • z1∙ z2 = r1( cos φ1 +i sin φ1 )∙r2( cos φ2 + i sin φ2 ) a + bi = r( cos φ +i sin φ ) • z1 ∙ z2 = r1r2 cos (φ1 + φ2 + i sin(φ1+ φ2 )] If z = a+bi , then: r = a2+b2 • Modulus b 1 1 • Argument φ =tan−1 a φ +i r r( cos sin φ ) = [ cos −φ +i sin −φ ] cos φ +i sin φ n= cos nφ +i sin nφ r (cos φ +i sin φ )=r[ cos −φ +i sin −φ ] eix= cos x +i sin x zn=[r cos φ +i sin φ ]n=rn[ cos nφ +i sin nφ ] z1 = r1( cos φ1 +i sin φ1 ) = r1 cos (φ1 +φ2 +i sin(φ1+φ2 )] n z=n r( cos φ +i sin φ )=n r cos φ+2πk +i sin φ+2πk z2 r2( cos φ2 +i sin φ2 ) r2 n n
FACTORING FORMULAS • If n is ODD, then: a2−b2=(a+b)(a−b) a3−b3=(a−b)(a2+ab+b2) an−bn=(a−b)(an−1+an−2+an−3b2+…+abn−2+bn−1) a3+b3=(a+b)(a2−ab+b2) an+bn=(a+b)(an−1−an−2+an−3b2−…−abn−2+bn−1) a4−b4= a2−b2 a2+b2 a5−b5=(a−b)(a4+a3b+a2b2+ab3+b4) • If n is EVEN, then: a5+b5=(a+b)(a4−a3b+a2b2−ab3+b4) an−bn=(a−b)(an−1+an−2+an−3b2+…+abn−2+bn−1) an+bn=(a+b)(an−1−an−2+an−3b2−…+abn−2−bn−1) PRODUCT FORMULAS (a−b)2=a2−2ab+b2 • Powers with Same Base: aman = a(m+n) a+b 2=a2+2ab+b2 • Quotient with Same Base: aamn = a(m-n) a−b 3=a3−3a2b+3ab2−b3 • Product to a Power: (a+b)3=a3+3a2b+3ab2+b3 • Power of a Power: (ab)m = ambm a−b 4=a4−4a3b+6a2b2−4ab3+b4 • Quotient of a Power: a+b 4=a4+4a3b+6a2b2+4ab3+b4 • Zero Power Rule: (am)n = amn • Negative Exponent Rule: • Fractional Exponent Rule: (ba)m = am bm a0=1,where a≠0 a(-m) = 1 am m n am a(n ) = General Formula: a+b+c+…+u+v 2=a2+b2+c2+…+u2+v2+2(ab+ac+…+au+av+bc+…+bu+bv+…+uv)
LAW OF RADICALS • n ab= n an b • n am=amn • n am b=nm ambn • m n a=mn a n a n a, • (n a)m=n am b n b • = where b≠0 n a= • 1ൗn an−1൘a , where a≠0 • n a = nm am nm am , where b≠0 a+ a2−b a− a2−b m b nm bn = bn 2± 2 • a± b= • (n am)p=n amp • (n a)n=a • 1 b= a∓ b a± a−b LOGARITHM loga 1 =0 log10 x = log x loga a =1 loga 0 = ቊ+−∞∞ if a>1 if a<1 loga (xy) = loga x + loga y loga x = loga x − loga y y loga (xn) =n loga x 1 loga n x = n loga x loge x = ln x , where e=2.718281828… loga x = logc x = logc x ∙ loga c , where c>0 , c≠1 log x = 1 ln x logc a ln 10 1 loga c = 1 ln x = log e log x logc a x=aloga x
EQUATIONS ax+b=0 , where x= − b ax2+bx+c=0 , a where x1, x2= −b± b2−4ac 2a D= b2−4ac y3+py+q=0 If x2+px+q=0, then ቊx1xx12+=xq2=−p INEQUALITIES Inequalities Interval Notation Graph a≤x≤b a, b a<x≤b (a, b] a≤x<b [a, b) a<x<b (a, b) −∞<x≤b, where x≤b (−∞, b] −∞<x<b, where x<b (−∞, b) a≤x<∞, where x≥a [a, ∞) a<x<∞, where x>a (a, ∞)
INEQUALITIES a<0 ax2+bx+c>0 a>0 D>0 D=0 D<0 • If f(x) >0, then ቊg f x ∙g x >0 g(x) x ≠0 • If f(x) <0, then ቊg f x ∙g x <0 g(x) x ≠0
Leg of a right triangle: a, b a2 + b2 = c2 Hypotenuse: c Altitude: h a2=fc, b2=gc, Medians: ma, mb, mc, Angles: α, β where f and c are projections of the legs a and b, Radius of circumscribed circle: R respectively, onto the hypotenuse c. Radius of inscribed circle: r Area: S h2 = fg, where h is the altitude from the right angle. m2a=b2− a2 , m2b=a2− b2 , 4 4 α + β = 90° where ma and mb are the medians to the legs a and b. sin α= a =cos β mc= c , c 2 cos α= b =sin β where mc is the median to the hypotenuse c. c R= c = mc a 2 tan α= b =cot β b r= a + b − c= ab a 2 +b+ cot α= =tan β a c sec α= c =cosec β ab=ch b ab ch cosec α= c =sec β S= 2 = 2 a Base: a β=90°− α 2 Legs: b Base angle: β h2=b2− a2 Vertex angle: α 4 Altitude to the base: h L=a+2b Perimeter: L ah b2 2 2 Area: S S= = sin α
Side of an equilateral triangle: a (A triangle with no two sides equal) Altitude: h Radius of circumscribed circle: R Sides of a triangle: a, b, c Radius of inscribed circle: r Semiperimeter: Perimeter: L Area: S Angles: α, β, γ Altitudes to the sides a, b, c: ha, hb, hc h= a3 Medians to the sides a, b, c: ma, mb, mc 2 Bisectors of the angles α, β, γ: ta, tb, tc Radius of circumscribed circle: R R= 2 h= a3 Radius of inscribed circle: r 3 3 Area: S r= 1 h= a 3 = R α+β+ γ=180° 3 6 2 a+b>c, L= 3a b+c>a, a+c>b. S= ah = a2 3 2 4 a−b <c, b−c <a, a2= b2+ c2−2bc cos α a−c <b. b2= a2+ c2−2ac cos β c2= a2+ b2−2ab cos γ Midline q= a , qԡa. 2
a α = b β = c γ =2R , ha=b sin γ =c sin β , sin sin sin hb=a sin γ =c sin α , hc=a sin β =b sin α . where R is the radius of the circumscribed circle. R= 2 a α = 2 b β = 2 c γ = bc = ac = ab = abc m2a= b2 + c2 a2 sin sin sin 2ha 2hb 2hc 4S 2 4 − , m2b= a2 + c2 − b2 , 2 4 r2= p−a)(p−b)(p−c , p mc2= a2 + b2 c2 2 − 4 . 1 = 1 + 1 + 1 . 2 2 2 r ha hb hc 3 3 3 AM= ma , BM= mb , CM= mc sin α = p−b)(p−c , ta2= 4bcp(p−a) , 2 bc (b+c)2 tb2= 4acp(p−b) , (a+c)2 α p(p−a) cos 2 = bc , tc2= 4abp(p−c) . (a+b)2 tan α = p−b)(p−c . S= aha = bhb = chc , 2 p(p−aሻ 2 2 2 S= ab sin γ = ac sin β = bc sin α , 2 2 2 ha= 2 p(p−a)(p−b)(p−cሻ , S= p(p−a)(p−b)(p−cሻ Heron′s Formula , a S=pr , hb= 2 p(p−a)(p−b)(p−cሻ , S= abc , b 4R hc= 2 p(p−a)(p−b)(p−cሻ . S=2R2 sin α sin β sin γ , c S=p2 α β γ tan 2 tan 2 tan 2 .
Side of a square: a Sides of a rectangle: a, b Diagonal: d Diagonal: d Radius of circumscribed circle: R Radius of circumscribed circle: R Radius of inscribed circle: Perimeter: L Perimeter: L Area: S Area: S d=a 2 d= a2 + b2 R= d = a2 R= d 2 2 2 r= a L=2(a+bሻ 2 S=ab L=4a S=a2 Sides of a rectangle: a, b Sides of a rhombus: a, b Diagonals: d1, d2 Diagonals: d1, d2 Consecutive angles: α, Consecutive angles: α, β Angle between the diagonals: φ Altitude: Altitude: h Radius of inscribed circle: r Perimeter: L Perimeter: L Area: S Area: S α+β+ γ=180° α+β+ γ=180° d12+ d22=4a2 d12+ d22=2(a2 + b2 ቁ h=a sin α = d1d2 h=b sin α =b sin β 2a r= h = d1d2 = a sin α 2 4a 2 L=2 a+b L=4a S=ah=ab sin α , S=ah=a2 sin α , 1 S= 2 d1d2 sin φ . S= 1 d1d2 . 2
Bases of a trapezoid: a, b Bases of a trapezoid: a, b Midline: q Lateral sides: c, d Altitude: h Midline: q Area: S Altitude: h q= a+b Diagonals: d1, d2 2 Angle between the diagonals: φ S= a+b h=qh Radius of inscribed circle: r 2 Radius of circumscribed circle: R Perimeter: L Bases of a trapezoid: a, b Area: S Midline: q a+b=c+d Altitude: h Diagonal: d q= a+b = c+d Radius of circumscribed circle: R 2 2 Area: S L=2 a+b =2 c+d S= a+b h= c+d h=qh 2 2 S= 1 d1d2 sin φ . 2 a+b q= 2 d= ab+c2 c2 1 b−a 2 Sides of a kite: a, b α+β+2γ=360° 4 Diagonals: d1, d2 h= − Angles: α, β, γ L=2 a+b Perimeter: L R= c ab+c2 Area: S S= d1d2 2c−a+b)(2c+a−b 2 S= a+b h=qh 2
Bases of a trapezoid: a, b Sides of a quadrilateral: a, b, c, d Leg: c Diagonals: d1, d2 Midline: q Angle between the diagonals: φ Altitude: h Internal angles: α, β, γ, δ Diagonal: d Radius of the circumscribed circle: R Radius of inscribed circle: R Perimeter: L Radius of circumscribed circle: r Semiperimeter: p Perimeter: L Area: S Area: S α+γ=β+δ=180° a+b=2c Ptolemy’s Theorem q= a+b =c ac+bd= d1d2 2 L=a+b+c+d d2=h2+c2 r= h = ab R= 1 ac+bd)(ad+bc)(ab+cd , 2 2 4 p−a)(p−b)(p−c)(p−d R= cd = cd = c 1+ c2 = c h2+c2= a+b a +6+ b where p= 1 . 2h 4r 2 ab 2h 8 b a 2 L=2 a+b =4c S= 1 d1d2 sin φ , 2 S= a+b ቀa+b) ab =qh=ch= Lr S= p−a)(p−b)(p−c)(p−d 2 h= 2 2 p= 1 . where 2 Sides of a quadrilateral: a, b, c, d a+c=b+d Diagonals: d1, d2 L=a+b+c+d=2 a+c =2(b+dሻ Angle between the diagonals: φ Radius of the inscribed circle: r r= d12d22− a−b 2 a+b−p 2 , Perimeter: L 2p Semiperimeter: p Area: S where p= L . 2 S=pr= 1 d1d2 sin φ 2
Sides of a quadrilateral: a, b, c, d Side: a Diagonals: d1, d2 Number of sides: n Angle between the diagonals: φ Internal angle: α Internal angles: α, β, γ, δ Slant height: m Perimeter: L Radius of the inscribed circle: r Area: S Radius of the circumscribed circle: R Perimeter: L α+β+γ+δ=360° Semiperimeter: p Area: S L=a+b+c+d S= 1 d1d2 sin φ 2 α= n−2 (180°ቇ 2 R= a Side: a 2 sin π Internal angle: α n Slant height: m Radius of the inscribed circle: r r= m= 2 a π = R2− a2 Radius of the circumscribed circle: R tan n 4 Perimeter: L Semiperimeter: p L=na Area: S α=120° nR2 2π 2 n r=m= a3 S= sin , 2 R=a R2− a2 4 L=6a S=pr= p S=pr= a23 3 , where p= L . 2 2 where p= L . 2
Radius: R Radius of a circle: R Diameter: d Arc length: s Chord: a Central angle (in radians): x Secant segments: e. f Central angle (in degrees): α Tangent segment: g Perimeter: L Central angle: α Area: S Inscribed angle: β Perimeter: L s=Rx Area: S s= πRα 180° L=s+2R a=2R sin α ee1= ff1 S= Rs = R2x = πR2α 2 2 2 360° a1a2=b1b2 g2= ff1 Radius of a circle: R Arc length: s Chord: a Central angle (in radians): x Central angle (in degrees): α Height of the segment: h Perimeter: L Area: S β= α a=2 2hR−h2 2 h=R− 1 4R2−a2 , h<R 2 L=2πR=πd L=s+a πd2 S= πR2= 4 = LR S= 1 sR−a(R−hሻ = R2 απ − sin α = 2 2 2 180° R2 (23xh−asin. x) , S2≈
Edge: a Lateral edge: l Diagonal: d Height: h Radius of inscribed sphere: r Lateral area: SL Radis of circumscribed sphere: R Area of base: SB Surface area: S Total surface area: S Volume: V Volume: V d=a 3 S=SL+2SB r= a Lateral Area of a Right Prism: 2 SL= a1+ a2+ a3+…+ an l R= a3 Lateral Area of an Oblique Prism: 2 SL=pl , where p is the perimeter of the cross-section. S=6a2 V=SBh V=a3 Cavalieri’s Principle: Given two solids included between parallel Edges: a, b, c planes. If every plane cross section parallel to Diagonal: d the given planes has the same area in both Surface area: S solids, then the volumes of the solids are equal. Volume: V d= a2+ b2+ c2 Triangle side length: a h= 2 a S=2(ab+ac+bcሻ Height: h 3 Area of base: SB V=abc Surface area: S SB= 3a2 Volume: V 4 S= 3a2 V= 1 SBh= a3 3 62
Side of base: a Base and top side lengths: ቊab11, ,ab22, ,ab33, ,……, ,abnn Lateral edge: b Height: h Height: h Slant height: m Slant height: m Number of sides: n Areas of bases: S1, S2 Semiperimeter of base: p Lateral surface area: SL Radius of inscribed sphere of base: r Perimeter of bases: P1, P2 Area of base: SB Scale Factor: k Lateral surface area: SL Total surface area: S Total surface area: S Volume: V Volume: V m= b2− a2 b1 = b2 = b3 =…= bn = b =k 4 a1 a2 a3 an a 4b2 sin2 π − a2 S2 =k2 n S1 h= π 2 sin n SL= 1 nam= 1 na 4b2−a2=pm SL= m(P1+ P2൯ 2 4 2 SB=pr S=SL+ S1+ S2 S=SB+SL h 3 1 1 V= S1+ S1S2+S2 3 3 V= SBh= prh hS1 b b 2 hS1 1+ k+k2 3 a a 3 V= 1+ + =
Sides of base: a, b r= a6 6 Top edge: c R= a2 2 Height: h S= 2a2 3 Lateral surface area: SL Area of base: SB R= a3 2 Total surface area: S 3 Volume: V SL= 1 a+c 4h2+ b2+b h2+ a−c 2 a 3 3+ 5 2 r= 12 SB=ab R= a 2 5+ 5 4 S=SB+SL S=5a2 3 V= bh 2a+c V= 5a3 3+ 5 6 12 The platonic solids are convex polyhedra with equivalent faces composed of congruent convex regular polygons. Edge: a Solid Number of Number Of Number Of Section Radius of inscribed Vertices Edges Faces circle: r Tetrahedron 4 6 4 3.25 Radius of circumscribed circle: R Cube 8 12 6 3.22 Surface area: S Octahedron 6 12 8 3.27 Volume: V Icosahedron 12 30 20 3.27 Dodecahedron 20 30 12 3.27
a 10 25+11 5 Radius of base: R r= 2 Diameter of base: d Height: H R= a 3 1+ 5 Lateral surface area: SL 4 Area of base: SB Total surface area: S S=3a2 5 5+2 5 Volume: V V= a3 15 + 7 5 4 SL=2πRH Radius of base: R S=SL+2SB=2πR H+R =πd H+ d Diameter of base: d 2 Height: H V=SBH=πR2H Slant height: m Lateral surface area: SL Radius of base: R Area of base: SB Total surface area: S The greatest height of a side: h1 Volume: V The shortest height of a side: h2 Lateral surface area: SL H= m2−R2 Area of base: SB Total surface area: Volume: V SL=πRm= πmd SL=πR h1+h2 2 SB=πR2 SB=πR2+πR R2+ h1− h2 2 2 S=SL + SB=πR m+R = 1 πd m+ d 2 2 S=SL + V= 1 SBH= 1 πR2H SVB==ππR2R2 h1+h2+R+ R2+ h1− h2 2 3 3 h1+h2 2
Radius of base: R, r Radius of sphere: R Height: H Radius of base: r Slant height: m Height: h Scale factor: k Area of plane face: SB Area of bases: S1, S2 Area of spherical cap: SC Lateral surface area: SL Total surface area: S Total surface area: S Volume: V Volume: V R= r2+ h2 2h H= m2− R−r 2 SB=πr2 R SC=π h2+r2 r =k S=SB+SC=π h2+2r2 =π 2Rh+r2 S2 R2 =k2 V= π h2 3R−h = π h 3r2+h2 S1 r2 6 6 = SL=πm R+r S=S1+ S2+SL=π R2+r2+m R+r Radius of sphere: R Radius of base: r1, r2 hS1 R R 2 hS1 1+k+k2 Height: h 3 r r 3 Area of spherical surface: SS V= 1+ + = Area of plane end faces: S1, S2 Total surface area: S Radius: R Volume: V Diameter: d Surface area: S S=4πR2 SS=2πRh Volume: V V= 4 πR3H= 1 πd3= 1 SR S=SS+S1+S2=π 2Rh+r21+r22 3 6 3 1 V= 6 πh 3r21+3r22+h2
Radius of sphere: R Radius: R Radius of base of spherical cap: r Dihedral angle in degrees: x Height: h Dihedral angle in radians: α Total surface area: S Area of spherical lune: SL Volume: V Total surface area: S Volume: V S=πR 2h+r V= 2 πR2h SL= πR2 α=2R2x 3 90 S=πR2+ πR2 α=πR2+2R2x 90 Note: The given formulas are correct both for V= πR3 α= 2 R3x “open” and “closed” spherical sector. 270 3 Major radius: R Semi-axes: a, b, b(a>b) Minor radius: r Surface area: S Surface area: S Volume: V Volume: V S=4π2Rr S=2πb a arcsin h be V=2π2Rr2 b+ be a a where e= b2− a2 . b a arcsin e V= 4 πb2a e 3 S=2πb b+ Semi-axes: a, b, b(a>b) where e= a2− b2 . Semi-axes: a, b, c Surface area: S a Volume: V 4 πb2a Volume: V V= 3 4 V= 3 πabc
1 rad= 180° ≈ 57°17′45\" 1′= π rad ≈ 0.000291 rad π 180⋅60 1°= π rad ≈ 0.017453 rad 1\"= π rad ≈ 0.000005 rad 180 180⋅3600 Angle 0 30 45 60 90 180 270 360 (degrees) Angle 0 π π π π π 3π 2π (radians) 6 4 3 2 2 DEFINITIONS AND GRAPHS OF TRIGONOMETRIC FUNCTIONS y= sin x , −1≤ sin x ≤1 y= cos x , −1≤ cos x ≤1 sin α = y y= tan x , x≠ 2k+1 π , −∞≤ tan x ≤∞ y= cot x , x≠kπ , −∞≤ cot x ≤∞ cos α r 2 y= cosec x , x≠kπ tan α cot α = x sec α r = y x = x y = r y= sec x , x≠ 2k+1 π x 2 cosec α = r y Angles: α , β Real numbers (point coordinates): x,y Whole number: k
Quadrant sin α cos α tan α cot α sec α cosec α + + I+ ++ + sin2 α+cos2 α =1 II + + + + III + sec2 α−tan2 α =1 IV + csc2 α−cot2 α =1 tan α ⋅ cot α =1 tan α = sin α sec α = 1 α cos α cos cot α = cos α cosec α = 1 α sin α sin α° α rad sin α cos α tan α cot α sec α cosec α 0 0 0 1 0∞ 1∞ π 1 31 32 2 30 6 2 23 3 π 22 1 1 2 2 45 4 22 π 31 31 2 2 60 3 22 3 3 π 10 ∞ 0∞ 1 90 2 120 2π 3 − 1 − 3 −1 -2 2 3 2 2 3 3 ∞ 180 π 0 -1 0 ∞ -1 270 3π -1 0∞ 0∞ -1 2 360 2π 0 1 0∞ 1∞
β sin β cos β tan β cot β sin (α±2πn) = sin α , period 2π or 360° −α − sin α + cos α − tan α − cot α cos (α±2πn) = cos α , period 2π or 360° 90°−α + cos α + sin α + cot α + tan α tan (α±πn) = tan α , period π or 180° 90°+α + cos α − sin α − cot α − tan α cot (α±πn) = cot α , period π or 180° 180°−α + sin α − cos α − tan α − cot α 180°+α − sin α − cos α + tan α + cot α sin 2α =2 sin α ⋅ cos α 270°−α − cos α − sin α + cot α + tan α cos 2α = cos2 α − sin2 α =1−2 sin2 α =2 cos2 α −1 270°+α − cos α + sin α − cot α − tan α 360°−α − sin α + cos α − tan α − cot α tan 2α = 2 tan α = cot α 2 tan α 360°+α + sin α + cos α + tan α + cot α 1− tan2 α − cot 2α = cot2 α −1 = cot α − tan α 2 cot α 2 sin a =± 1− cos2 α =± 1 (1− cos 2α )=2 cos2 α − π −1= 2 tan α 2 2 4 1+ 2 α tan2 2 cos a =± 1− sin2 α =± 1 (1+ cos 2α )=2 cos2 α −1= 1− tan2 α 2 2 1+ tan2 2 α 2 tan α = sin α =± sec2 α −1= sin 2α = 1− cos 2α =± 1− cos 2α = 2 tan α cos α 1+ cos 2α sin 2α 1+ cos 2α 1+ 2 α tan2 2 cot α = cos α =± csc2 α −1= 1+ cos 2α = sin 2α =± 1+ cos 2α = 1− tan2 α sin α sin 2α 1− cos 2α 1− cos 2α 2 2 α tan 2 sec α = 1 =± 1+ tan2 α = 1+ tan2 α csc α = 1 =± 1+ cot2 α = 1+ tan2 α cos 1− tan2 2 sin 2 2 α α α 2 α tan 2
sin (α+β) = sin α cos β + sin β cos α sin 3α =3 sin α −4 sin3 α =3 cos2 α ⋅ sin α − sin3 α sin 4α =4 sin α ⋅ cos α −8 sin3 α ⋅ cos α sin (α−y) = sin α cos β − sin β cos α sin 5α =5 sin α −20 sin3 α +16 sin5 α cos 3α =4 cos3 α −3 cos α = cos3 α −3 cos α ⋅ sin2 α cos (α+β) = cos α cos β − sin α sin β cos 4α =8 cos4 α −8 cos2 α +1 cos 5α =16 cos5 α −20 cos3 α +5 cos α cos (α−β) = cos α cos β + sin α sin β tan (α+β) = tan α + tan β 1− tan α tan β tan (α−β) = tan α − tan β 1+ tan α tan β −3 tan3 1− tan α tan β tan 3α = 3 tan α tan2 α α tan α + tan β 1−3 cot (α+β) = cot (α−β) = 1+ tan α tan β tan 4α = 4 tan α −4 tan3 α tan α − tan β 1−6 tan2 α + tan4 α tan 5α = tan5 α −10 tan3 α +5 tan α 1−10 tan2 α + 5 tan4 α α 1− cos α cot 3α = cot3 α −3 cot α 2 2 3 cot2 α −1 sin =± α 1+ cos α cot 4α = 1−6 tan2 α + tan4 α 2 2 4 tan α −4 tan3 α cos =± 1−10 tan2 α + tan4 α cot 5α = tan5 α −10 tan3 5 +5 tan α α α 1− cos α sin α 1− cos α tan 2 =± 1+ cos α = 1+ cos = sin α = csc α − cot α α cot α =± 1+ cos α = sin α = 1+ cos α = csc α + cot α 2 1− cos α 1− cos sin α α sin α = 2 tan α cos α = 1− tan2 α tan α = 2 tan α cot α = 1− tan2 α 1+ 2 1+ tan2 2 1− 2 2 2 α α α α tan2 2 2 tan2 2 tan 2
sin α + sin β =2 sin α+β cos α−β cos α + sin α = 2 cos ( π −α) = 2 sin ( π +α) 2 2 4 4 α+β α−β cos α − sin α = 2 sin ( π −α) = 2 cos ( π +α) 2 2 4 4 sin α − sin β =2 cos sin cos (α−β) α+β α−β tan α + cot β = cos α ⋅ sin β 2 2 cos α + cos β =2 cos cos cos2 α 2 α+β α−β tan α − cot β =− cos (α+β) 1+cos α =2 2 2 cos α ⋅ sin β cos α − cos β =−2 sin sin α sin2 2 1−cos α =2 tan α + tan β = sin (α+β) β cot α + cot β = sin (β+α) 1+sin α =2 cos2 ( π − α ) cos α ⋅ cos sin α ⋅ sin β 4 2 tan α − tan β = sin (α−β) β cot α − cot β = sin (β−α) 1−sin α =2 sin2 ( π − α ) cos α ⋅ cos sin α ⋅ sin β 4 2 sin α ⋅ sin β = cos (α−β) − cos (α+β) sin α ⋅ cos β = sin (α−β) + sin (α+β) cot α ⋅ cot β = cot α + cot β 2 2 tan α + tan β tan α + tan β tan α + cot β cos α ⋅ cos β = cos (α−β) + cos (α+β) tan α ⋅ tan β = cot α + cot β tan α ⋅ cot β = cot α + tan β 2 sin2 α = 1− cos 2α cos2 α = 1+ cos 2α 2 2 sin3 α = 3 sin α − sin 3α cos3 α = 3 cos α + cos 3α 4 4 sin4 α = cos 4α −4 cos 2α +3 cos4 α = cos 4α +4 cos 2α +3 8 8 sin5 α = 10 sin α −5 sin 3α + sin 5α cos5 α = 10 cos α +5 sin 3α + cos 5α 16 16 sin6 α = 10−15 cos 2α +6 cos 4α − cos 6α cos6 α = 10+15 cos 2α +6 cos 4α + cos 6α 32 32
GRAPHS OF INVERSE TRIGONOMETRIC RELATIONS BETWEEN INVERSE FUNCTIONS TRIGONOMETRIC FUNCTIONS Part 1 of 2 arcsin (−x) =− arcsin x y = arcsin x , −1≤x≤1, − π ≤ arcsin x ≤ π arcsin x = π − arccos x 2 2 2 arcsin x = arccos 1−x2 , 0≤x≤1 arcsin x =− arccos 1−x2 , −1≤x≤0 y= arccos x , −1≤x≤1, 0≤ arccos x ≤π arcsin x = arctan x ,x2<1 1−x2 arcsin x = arccot 1−x2 , 0<x≤1 x π π arcsin x = arccot 1−x2 −π , −1≤x<0 2 2 x y= arctan x −∞≤x≤∞, − < arctan x < , arccos (−x) =π− arccos x arccos x = π − arcsin x 2 arccos x = arcsin 1−x2 , 0≤x≤1 y= arccot x , −∞≤x≤∞, 0< arccot x <π arccos x =π− arcsin 1−x2 , −1≤x≤0 arccos x = arctan 1−x2 , 0<x≤1 x y= arcsec x , arccos x =π+ arctan 1−x2 , −1≤x<0 x π π x∈ −∞,−1 ∪ 1,∞ , arcsec x ∈ 0, 2 ∪ 2 ,π arccos x = arccot x , −1≤x≤1 1−x2 arctan (−x) =− arctan x arctan x = π − arccot x 2 y= arccsc x , π π arctan x = arcsin x x∈ −∞,−1 ∪ 1,∞ , arccsc x ∈ − 2 ,0 ∪ 0, 2 1+x2
RELATIONS BETWEEN INVERSE TRIGONOMETRIC FUNCTIONS Part 2 of 2 arcsin (−x) =− arcsin x arccos x =π− arcsin 1−x2 , −1≤x≤0 arctan x =− π − arctan 1 , x<0 2 x π arcsin x = 2 − arccos x arccos x = arctan 1−x2 , 0<x≤1 arctan x = arccot 1 , x>0 x x arcsin x = arccos 1−x2 , 0≤x≤1 1−x2 arctan x = arccot 1 −π , x<0 x x arccos x =π+ arctan , −1≤x<0 arcsin x =− arccos 1−x2 , −1≤x≤0 arccos x = arccot x , −1≤x≤1 arccot (−x) =π− arccot x 1−x2 arcsin x = arctan x ,x2<1 arccot x = π − arctan x 1−x2 2 arctan (−x) =− arctan x arccot x = arcsin 1 ,x>0 1+x2 arcsin x = arccot 1−x2 , 0<x≤1 arctan x = π − arccot x x 2 arccot x = π−arcsin 1 , x<0 arcsin x = arccot 1−x2 −π , −1≤x<0 arctan x = arcsin x 1+x2 x 1+x2 arccot x = arccos x arccos (−x) =π− arccos x arctan x = arccos 1 , x≥0 1+x2 1+x2 π 1 arccos x = 2 − arcsin x arctan x = −arccos 1 , x≤0 arccot x = arctan x ,x>0 arccos x = arcsin 1−x2 , 0≤x≤1 π 1+x2 1 2 1 arccot x =π+ arctan x , x<0 arctan x = − arctan x , x>0 TRIGONOMETRIC EQUATIONS OBLIQUE TRIANGLES Whole number: n a A = b B = c C a2=b2+c2−2bc cos A sin sin sin b2=a2+c2−2ac cos B sin x =a , x=(−1)narcsin a +πn c2=a2+b2−2ab cos C cos x =a , x=± arccos a +2πn tan x =a , x= arctan a +πn cot x =a , x= arccot a +πn RELATIONS TO HYPERBOLIC PYTHAGOREAN SPECIAL TRIANGLES FUNCTIONS Imaginary unit: i cot (ix) =−i coth x c2=a2+b2 sec (ix) = sech x sin (ix) =i sinh x csc (ix) =−i csch x tan (ix) =i tanh x
• det A = a1 = a1 • det A = a1 b1 = a1b2 - a2b1 a2 b2 a11 a12 a13 det A = a21 a22 a23 a31 a32 a33 det A = a11 a22 a33 + a12 a23 a31 + a13 a21 a32 - a11 a23 a32 - a12 a21 a33 - a13 a22 a31 det A= a11 a12 ··· a1j ··· a1n a··2·1 a·2··2 ······ a2j ······ a·2··n a··i1· a··i2· ······ ··· ······ a··in· an1 an2 ··· aij ··· ann ··· anj det A = σjn=1 aijCij , i = 1, 2, …, n or det A = σin=1 aijCij , j = 1, 2, …, n 1. a1 a2 = a1 b1 4. ka1 kb1 =k a1 b1 b1 b2 a2 b2 a2 b2 a2 b2 2. a1 b1 =– a2 b2 5. a1+kb1 b1 = a1 b1 a2 b2 a1 b1 a2+kb2 b2 a2 b2 3. a1 a1 =0 a2 a2
Let A = a1 a2 ;B= a1 a2 ,then A = B Let A = a1 a2 , then, Aadj = a4 a2 b1 b2 b1 b2 a3 a4 a3 a1 a11 a12 b11 b12 Let A = a1 a2 , then, Atr = a1 + a4 a21 a22 b21 b22 a3 a4 Let A = ;B= then, A ± B = a11± b11 a12± b12 a21± b21 a22± b22 A-1 = Aadj det A Let A = a1 a2 , then kA = ka1 ka2 b1 b2 kb1 kb2 ; k is scalar Let ൝aa12xx++b1by2=y=d1d2, x = DDX, Dy y= D Let A = a1 a2 ;B= b1 b2 ;C= c1 a1 b1 a3 a4 b3 b4 c2 a2 b2 Then, D= = a1b2 - a2b1 A*B= (a1∗b1)+(a1∗b3) (a2∗b2)+(a2∗b4) Dx = d1 b1 = d1b2 - d2b1 (a3∗b1)+(a3∗b3) (a4∗b2)+(a4∗b4) d2 b2 Then, A * C = a1c1 a2c2 Dy = a1 d1 = a1d2 - a2d1 a3c1 a4c2 a2 d2 Let A = a1 a2 , then AT = a1 a3 If D ≠ 0: one solution a3 a4 a2 a4 D = 0, and Dx ≠ 0 (or Dy ≠ 0): no solution D = Dx = Dy = 0: infinitely many solution
rԦ = AB = (x1 – x0) Ԧi + (y1 – y0) Ԧj + (z1 – z0) k Ԧr = AB = (x1 – x0)2 + (y1 – y0)2 + (z1 – z0)2 w=u+v w = λu w (λ + μ) u = λu + μu w= λ · u λ ( μu ) = μ (λu ) = (λμ) u λu = (λX, λY, λZ) λ ( u + v ) = λu + λv λu = uλ w = u1 + u2 + u3 + … + un Scalar Product: u · v = u · v ·cos θ Commutative Law: u + v = v + u Associative Law: ( u + v ) + w = u + ( v + w ) Angle between Two Vectors: u-v=u+(-v) If u = (X1 , Y1 , Z1) X; 1vX=2(+XY21,YY22+Z, 1ZZ2)2, then: u - u = 0 = (0, 0, 0) cos θ = X12+ Y12+ Z12 · X22+ Y22+ Z22 0 =0 Commutative Property: u · v = v · u Associative Property: ( λu ) · ( μv ) = λμu · v Distributive Property: u · ( v + w ) = u · v + u · w u · v = 0, if u , v are orthogonal ( θ = π ) 2 π u · v > 0, if 0 < θ < 2 u · v < 0, if π < θ < π 2 u·v≤ u · v u · v = u · v if u , v are parallel (θ = 0) u · u = u2 = u 2 = (X12 , Y12 , Z12) if u = (X1 , Y1 , Z1) Ԧi · Ԧi = Ԧj · Ԧj = k · k = 1 Ԧi · Ԧj = Ԧj · k = k · Ԧi = 0
Scalar Triple Product: uvw = u · (v x w) = v · (w x u) = w · (u x v) uvw = wuv = vwu =− vuw =− wvu = − uwv ku · (v x w) = k uvw u x v = w, where w = u x v ; 0 ≤ θ ≤ π 2 Ԧi Ԧj k Scalar Triple Product in Coordinate Form: w = u x v = X1 Y1 Z1 X1 Y1 Z1 X2 Y2 Z2 u · (v x w) = X2 Y2 Z2 , w=uxv= Y1 Z1 ,− X1 Z1 , X1 Y1 X3 Y3 Z3 Y2 Z2 X2 Z2 X2 Y2 Where: u = (X1, Y1, Z1), v = (X2, Y2, Z2), w = (X3, Y3, Z3) S = u x v = u · v · sin θ Angle between Two Vectors: sin θ = uxv Volume of a Parallelepiped: V = u · (v x w) u·v Non-commutative Property: u x v = - ( v x u ) Volume of a Pyramid: V = 1 u · (v x w) 6 Associative Property: ( λu ) x ( μv ) = λμu x v Vector Triple Product: u·(v x w) = (u·w) v - (u·v) Distributive Property: u x ( v + w ) = u x v + u x w u x v = 0 if u , v are parallel (θ = 0) Ԧi x Ԧi = Ԧj x Ԧj = k x k = 0 Ԧi · Ԧj = k, Ԧj · k = Ԧi, k · Ԧi = Ԧj
Ax+By+C=0 Point Slope Form: y−y1=m(x−x1) Distance Distance between Parallel Lines Slope-Intercept y=mx+b between Two d= C2−C1 Form: Points: A2+B2 y + y =1 Slope Formula: a b Areas by Coordinates: Angle Formed: m= ∆y = tan x Distance from a line to a Point ∆x Parallel Lines: tan θ= m2−m1 d= Ax1+By1+C Perpendicular 1+m1m2 A2+B2 Lines: Ax+By+C1=0; Ax+By+C2=0 m1=m2 A= 1 x1 x2 …xn x1 2 y1 y2 …yn y1 Ax+By+C1=0; Bx−Ay+C2=0 m1= 1 m2 Ax2+Bxy+Cy2+Dx+Ey+F=0 Conics Eccentricity (������ሻ Discriminant Coefficient Circle e→0 B2−4AC<0 (A=C) A=C Ellipse A≠C Parabola e<1 B2−4AC<0 (A≠C) Hyperbola A or C=0 e=1 B2−4AC=0 A&C=opposite sign e>1 B2−4AC>0 Ax2+Cy2+Dx+Ey+F=0 General x2+y2+Dx+Ey+F=0 Horizontal (x−h)2 + (y−k)2 =1 Focal c= a2−b2 Equation: (x−h)2+(y−k)2=r2 Major Axis a2 b2 Distance: Standard Vertical x−h 2 y−k 2 Eccentricity: e= c = a Equation: Major Axis: b2 + a2 =1 a d Length of Latus Rectum: LR= 2b2 a
y = a(x-h)2 + k or x = a(y-k)2 +h Ax2−Cy2+Dx+Ey+F=0 Horizontal Major Cy2+Dx+Ey+F=0 Horizontal Major (x−h)2 − (y−k)2 =1 Axis: y−k 2=±4a x−h Axis: a2 b2 Note that: Vertical Major x−k 2 y−h 2 (+) Opens Rightward Axis: a2 − b2 =1 ( - ) Opens Leftward Vertical Major Length of Latus LR= 2b2 Axis: Ax2+Dx+Ey+F=0 Rectum: a x−h 2=±4a y−k Eccentricity: Note that: e= c = a (+) Opens Upward Equation of a d ( - ) Opens Downward Asymptotes: y−k=m(x−h) LR = 4a Latus Rectum: a Focal Distance: e=1 Eccentricity :
Even: f −x =f(x) Cubic: y= x3, x ϵ R Odd: y=ax3+bx2+cx+x , x ϵ R Periodic: f −x =−f(x) Power: y= xn , n ϵ N Inverse: f x+nT =f(x) Square Composite: x=g y or y=f−1(x) Root: y= x , x ϵ [0 , ∞) Linear: Exponential: Quadratic: y =f(g x ) y= ax , a>0 , a ≠1 y=ax+b , x ϵ R Logarithmic: y= ex , a=e Hyperbolic: y= x2 , x ϵ R y=ax2+bx+c , x ϵ R y= loga x , x ϵ 0 , ∞ , a>0 , a ≠1 y= ln x , a=e , x>0 y= sinh x ex− e−x sinh x = 2 , x ϵR y= cosh x xli→ma [f x +g(x)] = xli→ma f x + xli→ma g x ex+ e−x cosh x = 2 ,xϵR xli→ma [f x −g(x)] = xli→ma f x − xli→ma g x y= tanh x ex− e−x tanh x = ex+ e−x ,xϵR xli→ma [f x ∗g(x)] = xli→ma f x ∗ xli→ma g x y= coth x xli→ma f(x) = xl→ima f(x) , if xli→ma g x ≠0 ex+ e−x g(x) xli→ma g(x) ex− e−x coth x = , x ϵ R , x≠0 y= sech x xl→ima [kf(x)] =k xl→ima f(x) sech x = 2 ,xϵR xl→ima f(g x ) =f[ xl→ima g(x)] ex+ e−x xl→ima f x =f(a) y= csch x 2 csch x = ex− e−x , x ϵ R , x≠0 xl→im0 sin x =1 xl→im0 ln (1+x) =1 x x Inverse y= arcsinh x , x ϵ R xl→im0 tan x =1 1 x Hyperbolic: y= arccosh x , x ϵ [1,∞) x x =e y= arctanh x , x ϵ (−1 , 1) xl→im∞ 1+ y= arccoth x , x ϵ −∞,−1 ∪(1,∞) y= arcsech x , x ϵ (0 , 1] xl→im0 sin−1 x =1 k x y= arccsch x , x ϵ R , x≠0 x x =ek xl→im∞ 1+ xl→im0 tan−1 x =1 xl→im0 ax =1 x
y′(x)= lim f x+∆x −f(x) x lim ∆y = dy • d (C)= 0 x→0 ∆x = ∆x→0 ∆x dx dx dy • d (x)= 1 dx dx = tan ∝ • d (ax+b)= a dx d(u+v) du dv dx = dx + dx • d ax2+bx+c = ax+b dx d(u−v) = du − dv • d xn =nxn−1 dx dx dx dx d(ku) = k du • d x−n =− n dx dx dx xn+1 Product Rule: d(u∗v) = v du +u dv • d 1 =− 1 dx dx dx dx x x2 Quotient Rule: du dv • d x= 1 dx dx dx 2x v −u d u = d n 1 dx v v2 • dx x = nn xn−1 Chain Rule: y=f g x , u=g x , • d ( ln x )= 1 dx x dy = dy du dx du ∗ dx • d loga x = 1 , a>0 , a≠1 dx ln x a Derivative of dy = 1 • d ax =ax ln a , a>0 , a≠1 Inverse Function: dx dx dx Reciprocal Rule: dy • d (ex)= ex dx dy d 1 dx y = dx • d sin x = cos x y2 dx Logarithmic y=f x , ln y= ln f(x) • d cos x = − sin x Differentiation: dx dy d dx = f(x)+ dx [ ln f(x) ] • d ( tan x )= sec2x dx • d cot x = − csc2 x dx Functions: f,g, y, u, v • d sec x = tan x∗ sec x dx Argument (Independent Variable): x • d csc x = − cot x∗ csc x Real numbers: a, b, c, d dx Angle: α
HIGHER ORDER DERIVATIVES • d ( arcsin x )= 1 dx 1−x2 d2y • d arccos x =− 1 f′′= dx2 dx 1−x2 • d ( arctan x )= 1 dny dx 1+x2 dxn • d arccot x =− 1 f(n)= =yn=(fn−1)′ dx 1+x2 (u+v)(n)=u n +v(n) • d arcsec x = 1 dx |x| x2−1 (u−v)(n)=u n −v(n) • d arccsc x =− 1 dx |x| x2−1 • d sinh x = cosh x uv ′′′=u′′′v+3u′′v′+3u′v′′+uv′′′ dx • d cosh x = sinh x (uv)(n)=u(n)v+nu(n−1)v′+ n(n−1) u(n−2)v′′+. . .+uv(n) dx 1∗2 • d tanh x = sech2 x xm (n)= m! ! xm−n dx m−n • d coth x =− csch2 x xn (n)=n! dx • d sech x =− sech x∗ tanh x loga x (n)= −1 n−1 n−1 ! dx xn ln a • d csch x =− csch x∗ coth x n−1 n−1 dx xn (n)= −1 ! d 1 ln x • dx arcsinh x = x2+1 d 1 ax (n)=axlnn a dx x2−1 • arccosh x = ex (n)=ex • d arctanh x = 1 , x <1 amx (n)=mnamx lnn a dx 1−x2 • d arccot x = 1 , x >1 sin x (n)= sin x+ nπ dx x2−1 2 • d uv =vuv−1∗ du +uv ln u∗ dv cos x (n)= cos x+ nπ dx dx dx 2
APPLICATIONS OF DERIVATIVES MULTIVARIABLE FUNCTIONS s= f t , fixed With Respect to x: ∂f =fx ; ∂z =zx v=s′=f′ t , instantaneous velocity With Respect to y: ∂x ∂x w=v′=s′′=f′′ t , instantaneous acceleration ∂f = fy ; ∂z =zy ∂x ∂x y−y0=f′(x0)(x−x0) ∂ ∂f = ∂2f =fxx ∂ ∂f = ∂2f =fxy ∂x ∂x ∂x2 ∂y ∂x ∂y∂x y−y0=− f′ 1 (x−x0) ∂ ∂f = ∂2f =fyy ∂ ∂f = ∂2f =fyx x0 ∂y ∂y ∂y2 ∂x ∂y ∂x∂y xl→imc f(x) = xl→imc f′(x) , xl→imc f x = xl→imc g x = 0 ∂2f = ∂2f g(x) g′(x) ൝∞ ∂y∂x ∂x∂y DIFFERENTIAL dy=y′dx ∂f ∂h ∂f ∂h ∂x ∂x ∂y ∂y f x+∆x =f x +f′(x)∆x =g′ h x,y ; =g′ h x,y Small Change in y: ∆y=f x+∆x −f(x) ; when f x,y =g h x,y d u+v =du+dv d u−v =du−dv h′ t = ∂f dx + ∂f dy ,h t =f x(t ,y(t)) ∂x dt ∂x dt d Cu =Cdu d uv =vdu+udv ∂z = ∂f ∂x + ∂f ∂y ; ∂z = ∂f ∂x + ∂f ∂y ∂x ∂x ∂u ∂y ∂u ∂v ∂x ∂v ∂y ∂v d u = vdu+udv v v2 ; when z=f x u,v , y(u,v))
MULTIVARIABLE FUNCTIONS DIFFERENTIAL OPERATORS ∆z≈ ∂f ∆x+ ∂f ∆y grad f=∇f= ∂f , ∂f , ∂f ∂x ∂y ∂x ∂y ∂z grad u= ∇u= ∂u , ∂u ,…, ∂u ∂x1 ∂x2 ∂xn ∂f = ∂f =0 ∂x ∂y ∂f = ∂f cos α + ∂f cos β+ ∂f cos γ ∂l ∂x ∂y ∂z ԦI cos ∝ , cos β, cos γ , cos2α+ cos2β+cos2γ=1 D= fxx(x0 , y0) fxy(x0 , y0) fyx(x0 , y0) fyy(x0 , y0) ∴D>0, fxx x0 , y0 >0, x0 , y0 − point of local minima div F=∇∗F= ∂P + ∂Q + ∂R D>0, fxx x0 , y0 <0, x0 , y0 − point of local maxima ∂x ∂y ∂z D<0, x0 , y0 − saddle point D=0, test fails Ԧi Ԧj k curl F=∇ ×F= ∂ ∂ ∂ ∂x ∂x ∂x PQR z−z0=fx xo ,y0 x−x0 +fy(x0 ,y0)(y−y0) = ∂R − ∂Q Ԧi+ ∂P − ∂R Ԧj+ ∂Q − ∂P k ∂y ∂z ∂z ∂x ∂x ∂y x−x0 = y−y0 = z−z0 ∇2f= ∂2f + d2f + ∂2y fx(x0 , y0) fy(x0 , y0) −1 ∂x2 ∂y2 ∂z2
• f x dx=f x +C if f′ x =f(x) • ( f x dx)′=f x • adx =ax+C • • f x dx=f x +C • xdx= x2 +C • 2 • kf x dx=k f x dx • x3 • f x +g x dx= f x dx + g x dx • x2dx= 3 +C • • f x −g x dx= f x dx − g x dx xPdx= xP+1 P+1 • f ax dx= 1 F ax +C +C, P≠−1 a ax+b n+1 • f ax+b dx= 1 F ax+b +C ax+b ndx= a n+1 +C, n≠−1 a • f x f′ x dx= 1 f2 x +C dx =ln x +C 2 x • f′(x) dx=ln f x +C dx = 1 ln ax+b +C f(x) ax+b a • Method of Substitution: • ax+b dx= ax + bc−ad ln cx+d +C • f x dx= f u t u′ t dt if x=u(t) • cx+d c c2 • • Integration by Parts: • dx = 1 ln| x+b | +C, a≠b • udv=uv− vdu (x+a)(x+b) a−b x+a xdx = 1 (a+bx−a ln a+bx ) +C a+bx b2 x2dx = 1 [ 1 (a+bx)3−2a a+bx +a2 ln a+bx ] +C a+bx b3 2 • dx = 1 ln a+bx +C • dx = 1 ln a+x +C x(a+bx) a x a2−x2 2a a−x • dx =− 1 + b ln a+bx +C • dx = 1 ln x−a +C x2(a+bx) ax a2 x x2−a2 2a x+a • xdx = 1 (ln a+bx + a )+C • dx =tan−1x +C (a+bx)2 b2 a+bx 1+x2 • x2dx = 1 (a+bx−2a ln a+bx − a2 )+C • dx = 1 tan−1 x +C (a+bx)2 b3 a+bx a2+x2 a a xdx 1 • dx = 1 + 1 ln a+bx +C • x2+a2 = 2 ln (x2+a2)+C x(a+bx)2 a(a+bx) a2 x • dx = 1 ln x−1 +C • dx = 1 tan−1(x b )+C, ab>0 x2−1 2 x+1 a+bx2 ab a • dx = 1 ln 1+x +C • xdx = 1 ln|x2+ a |+C 1−x2 2 1−x a+bx2 2b b
• dx = 1 ln| x2 |+C • dx = 1 ln| 2ax+b− b2−4ac |+C, b2−4ac>0 x(a+bx2) 2a a+bx2 ax2+bx+c b2−4ac a2ax+b+ b2−4ac • dx = 1 ln| a+bx |+C • dx = 2 arctan| 2ax+b |+C, b2−4ac<0 a2+b2x2 2ab a−bx ax2+bx+c 4ac−b2 4ac−b2 • dx = 2 ax+b+C • dx =2arcsin x−a +C ax+b a (x−a)(b−a) b−a • ax+bdx= 2 (ax+b)3/2+C • a+bx−cx2dx= 2cx−b a+bx−cx2+ b2−4ac arcsin 2cx−b +C • 3a 4c 8 c3 b3+4ac • • xdx = 2(ax−2b) ax+b+C • dx = 1 ln 2ax+b+2 ax2+bx+c +C,a>0 ax+b 3a2 ax2+bx+c a • • 2(3ax−2b) x ax+bdx= 15a2 (ax+b)3/2+C dx =− 1 2ax+b ax2+bx+c a 4a dx 1 ax+b− b−ac • arcsin b2−4ac +C,a<0 (x+c) ax+b b−ac ax+b+ b−ac = ln +C,b−ac>0 dx 1 ax+b • x2 +a2 dx= x x2 +a2 + a2 ln|x+ x2 +a2 |+C (x+c) ax+b ac−b ac−b • 2 2 = arctan +C,b−ac<0 x x2 +a2 dx= 1 (x2 +a2 )3/2+C 3 2(8a2−12abx+15b2x2) a4 x2 a+bxdx= 105b3 (ax+b)3/2+C • x2 x2 +a2 dx= x 2x2+a2 x2 +a2 − 8 ln|x+ x2 +a2 |+C 8 • x2 dx= 2(8a2−4abx+3b2x2) (ax+b)1/2+C • x2+a2 dx=− x2+a2 +ln|x+ x2 +a2 |+C a+bx 15b3 x2 x • x dx = 1 ln a+bx− a +C,a>0 • dx = ln x+ x2 +a2 +C a+bx a a+bx+ a x2+a2 • x dx = 2 arctan a+bx +C,a<0 • x2+a2 dx= x2 +a2 +aln| x |+C a+bx −a −a x a+ x2+a2 • a−x dx= a−x b+x +(a+b)arcsin ax++bb+C • xdx = x2 +a2 +C b+x x2+a2 • a+x dx=− a+x b−x −(a+b)arcsin ab+−bx+C b−x x2 x x2 +a2 a2 x2 +a2 |+C • 1+x dx= 1−x2+ arcsin x+C • x2+a2 dx= 2 − 2 ln|x+ 1−x
• dx =2arcsin x−a +C • ∫ dx =ln x+ x2−a2 +C (x−a)(b−a) b−a x2−a2 • a+bx−cx2dx= 2cx−b a+bx−cx2+ b2−4ac arcsin 2cx−b +C • ∫ xdx = x2−a2+C 4c 8 c3 b3+4ac x2−a2 • dx = 1 ln 2ax+b+2 ax2+bx+c +C,a>0 • ∫ x2dx = x x2−a2+ a2 ln x+ x2−a2 +C ax2+bx+c a x2−a2 2 2 • dx =− 1 arcsin 2ax+b b2−4ac +C,a<0 • ∫ dx =− 1 arcsin a +C ax2+bx+c a 4a x x2−a2 a x • x2 +a2 dx= x x2 +a2 + a2 ln|x+ x2 +a2 |+C • ∫ dx = 1 x−a +C 2 2 • (x+a) x2−a2 a x+a • x x2 +a2 dx= 1 (x2 +a2 )3/2+C 3 a4 dx 1 x+a • x2 x2 +a2 dx= x 2x2+a2 x2 +a2 − 8 ln|x+ x2 +a2 |+C ∫ (x−a) x2−a2 =− a x−a +C 8 • x2+a2 dx=− x2+a2 +ln|x+ x2 +a2 |+C dx = x2−a2 x2 x x2−a2 a2x • ∫ +C • dx = ln x+ x2 +a2 +C • x2 x2+a2 • • ∫ dx 1/2 =− a2 x +C x2−a2 x2−a2 • x2+a2 dx= x2 +a2 +aln| x |+C x a+ x2+a2 a2−x2dx= x a2−x2+ a2 x ∫ 2 2 arcsin a +C • xdx = x2 +a2 +C ∫x a2−x2dx=− 1 a2−x2 3 x2+a2 3 2 +C • x2 dx= x x2 +a2 − a2 ln|x+ x2 +a2 |+C • ∫x2 a2−x2dx= x 2x2−a2 a2−x2+ a4 arcsin x +C x2+a2 2 2 8 8 a • dx = 1 ln| x |+C • ∫ a2−x2 dx= a2−x2+ aln x +C x x2+a2 a x2+a2 x a+ a2−x2 a+ • dx = 1 ln| x |+C • ∫ a2−x2 dx=− a2−x2 −arcsin x +C x x2+a2 a x2+a2 x2 x a a+ • ∫ x2−a2 dx= x2−a2+ aarcsin a +C • ∫ dx = arcsin x +C x x 1−x2 • ∫ x2−a2 dx=− x2−a2 +ln x+ x2−a2 +C • ∫ dx = sin x +C x2 x a2−x2 a
• ∫ xdx =− a2−x2+C • ∫ dx = 1 arcsin bx+a2 +C,b>a a2−x2 x+b a2−x2 b2−a2 a x+b • ∫ x2dx =− x a2−x2+ a2 arcsin x +C • ∫ dx = 1 ln x+b +C, a2−x2 2 2 a x+b a2−x2 a2−b2 a2−b2 a2−x2+a2+bx b<a • ∫ dx =− 1 a−x +C • ∫ dx =− a2−x2 +C (x+a) a2−x2 2 a+x x2 a2−x2 a2x dx 1 a+x ∫ a2−x2 3 x 5a2−2x2 a2−x2+ 3a4 x a2−x2 2 a−x 2dx 8 8 a • ∫ =− +C • = arcsin +C x−a dx a2 a2−x2 a2−x2 x • ∫ 3/2 = +C • sinxdx=−cosx+C • ∫sinxcosxdx=− 1 cos2x+C 4 • cosxdx=sinx+C • ∫sin2xcosxdx= 1 sin3x+C • ∫sin2xdx= x − 1 sin 2x +C 3 2 4 • ∫sinxcos2xdx=− 1 cos3x+C • ∫cos2xdx= x + 1 sin2x+C 3 2 4 • ∫sin2xcos2xdx= x − 1 sin4x+C • ∫sin3xdx= 1 cos3x−cosx+C= 1 cos3x− 3 cosx+C 8 32 3 12 4 • ∫tanxdx=−ln|cosx|+C ∫cos3xdx=sinx− 1 sin3x+C= 1 3 • 3 12 sin3x+ 4 sinx+C • ∫ sin x dx= 1 +C=secx+C cos2 x cos • ∫ dx =∫cscxdx=ln tan x +C x sin x 2 • ∫ sin2 x dx= ln tan x + π − sin x +C dx x π cos x 2 4 • ∫ cos =∫secxdx=ln tan 2 + 4 +C • ∫tan2dx=tanx−x+C x • ∫ dx x =csc2 ∫ xdx =− cot x +C • ∫cotxdx=ln|sinx|+C sin2 cos x 1 • ∫ dx =∫sec2xdx=tanx+C • ∫ sin2 x dx=− sin x +C=−cscx+C cos2 x cos2 x x sin x 2 • ∫csc3x dx=− cos x + 1 ln tan x +C • ∫ dx=ln tan +cosx+C 2 sin2 x 2 2 • ∫cot2xdx=− cot x −x+C • ∫sec3x dx= sinx + 1 ln tan x + π +C 2cos2x 2 2 4 • ∫ dx =ln|tanx|+C cosxsinx
• ∫ dx =− 1 x +ln tan x + π +C • ∫csc x cot xdx =− csc x +C sin2xcosx sin 2 4 • ∫ dx = 1 x + ln tan x +C • ∫sin x cosn xdx =− cosn+1 x +C x cos2 cos 2 n+1 sin x sinn+1 • ∫ dx =tanx−cotx+C • ∫sinncos xdx = n+1 x +C sin2xcos2x ∫arcsin x dx =x arcsin x+ 1−x2 +C • sin ∫ mxsin nxdx =− sin m+n x + sin m−n x +C,m2 ≠n• 2 • 2 m+n 2 m−n ∫∫scionsmmxcxocsonsxndxxd=x−=cso2i2sn(((m(mmm++++nnn)n))x)x−+cos2si2(n(m(m(mm−−n−−n)nn)x))x+C+,Cm, 2m≠2n≠2n•••2 ∫arccos x dx =x arccos x− 1−x2 +C • • ∫arctan x dx =x arctan x− 1 ln(x2 +1)+C 2 ∫secxtanxdx=secx+C ∫arccot x dx =x arccot x+ 1 ln(x2 +1)+C 2 • sinh xdx= cosh x+C • exdx=ex+C • cosh xdx= sinh x+C • tanh xdx= ln cosh x+C • axdx= ax +C • coth xdx= ln|sinh x|+C ln a • sech2 xdx= tanh x+C eax • csch2 xdx= −coth x+C • eaxdx= a +C • sech x tanh xdx= −sech x+C • csch x coth xdx= −csch x+C • xeaxdx= eax ax−1 +C a2 • ln xdx dx=x ln x−x +C • dx =ln| ln x| +C x ln x • xn ln x dx=xn+1[ ln x − 1 2 ]+C n+1 n+1 • eax sin bx dx= a sin bx−b cos bx eax+C a2+b2 • eax cos bx dx= a cos bx+b sin bx eax+C a2+b2
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