课件分析教案成果daves math tables

1、Dave's Math TablesQuestions & CommentsIf you have a math question, there are a number of routes you may take:(1) Post your question on this site's Math Message Board. This is a web-based newsgroup (discussiongroup) for math talk, question, and answers. There are sections for algebra, tri

2、g, geometry, calculus, elementary math, and general discussions.(2) Send your question to Ask Dr. Math. This is a highly-popular, high-quality math question answering site run by Swarthmore College as part of the Math Forum.(3) I am currently very busy, so I can not please use the above Math Messaga

3、lly respond to math problems and questions; instead, where a number of qualifiedmathematicians, teachers, ands will be able to take your questions. If you do, though,have technical questions with regards to this site or need to reach me, the following methods are available:(A) Type your question/com

4、ment here. You must include your response.if you desire aSendComments(B)me at sismspec.(C) My AOL Instant Messenger screen name is C6H10CH3.(D) My ICQ is 7664967. This is another very popular instant messenger service.1 of 18/10/98 5:53 PMPlease Help:This site's Math Messagneeds people to help i

5、n answering math questions. Please see How to Help.Dave's Math Tables: Questions & Comments(Math | Questions & Comments)Number NotationHierarchy of Numbers0(zero) 1(one) 2(two) 3(three) 4(four) 5(five) 6(six) 7(seven) 8(eight) 9(nine) 101(ten) 102(hundred) 103(thousand)name million billi

6、on trillion quadrilliontillion sextillion septillion octillion nonillion decillion undecillion duodecillion tredecillion quatuordecilliondecillion sexdecillion septendecillion octodecillion novemdecillion vigintillionAmerican-French 1061091012101510181021102410271030103310361039104210451048105110541

7、05710601063English-German 10610121018102410301036104210481054106010661072107810841090109610102101081011410120googol googolplex1010010googol=10(10100)SI Prefixes1 of 38/10/98 5:01 PMNumberPrefixSymbol10 -1deci-d10 -2centi-c10 -3milli-m10 -6micro-u (greek mu)10 -9nano-n10 -12pico-p10 -15femto-f10 -18a

8、tto-a10 -21zepto-z10 -24yocto-yNumberPrefixSymbol10 1deka-da10 2hecto-h10 3kilo-k10 6mega-M10 9giga-G10 12tera-T10 15peta-P10 18exa-E10 21zeta-Z10 24yotta-YDave's Math Tables: Number Notation(Math | General | NumberNotation)Number NotationRoman NumeralsExamples:Number Base Systemsdecimal01234567

9、89101112binary01101110010111011110001001101010111100ternary012101112202122100101102110oct012345671011121314hex0123456789A B C D E F 1011121314161718192010000100011001010011101001211222002012022021222324Java Base Conversion Calculator (This converts non-integer values & negative bases too!)(For M

10、icrosoft 2+/Netscape 2+/Javascript web browsers only)Caution: due to CPU restrictions, some rounding has been known to occur for numbers spanning greater than 12 base10 digits, 13 hexadecimal digits or 52 binary digits. Just like a regular calculator, rounding can occur.2 of 38/10/98 5:01 PMfrom bas

11、eto basevalue to convert1016256calculate1001 = I2 = II3 = III4 = IV 5 = V6 = VI7 = VII8 = VIII9 = IX10 = X11 = XI12 = XII13 = XIII14 = XIV15 = XV16 = XVI17 = XVII18 = XVIII19 = XIX20 = XX21 = XXI25 = XXV 30 =40 = XL49 = XLIX50 = L51 = LI60 = LX70 = LXX80 = L90 = XC99 = XCIXI=1V=5X=10L=50C=100D=500M=

12、1 000_V=5 000_X=10 000_L=50 000_C = 100 000_D=500 000_M=1 000 000Weights and MeasuresDave's Math Tables: Weights and Measures(Math | General | Weights and Measures | Lengths)Unit Conversion Tables for Lengths & DistancesA note on the metric system:Before you use this table, convert to the ba

13、se measurement first, in that convert centi-meters to meters, convert kilo-grams to grams. In this way, I don't have to list every imaginable combination of metric units.The notation 1.23E + 4 stands for 1.23 x 10+4 = 0.000123.To use: Find the unit to convert from in the left column, and multipl

14、y it by the expression under the unit to convert to.Examples: foot = 12 inches; 2 feet = 2x12 inches.Useful Exact Length Relationships= 1760 yards = 5280 feet yard = 3 feet = 36 inchesfoot = 12 inchesinch = 2.54 centimeters1 of 18/10/98 5:02 PMfrom to=feet=inches=meters=yardsfoot120.3048(1/5280)(1/3

15、)inch(1/12)0.0254(1/63360)(1/36)meter3.280839.39.37007.6.213711.E-4 1.093613.5280633601609.3441760yard3360.9144(1/1760)Weights and MeasuresUnit Conversion Tables for AreasA note on the metric system:Before you use this table convert to the base measurement first, in that convert centi-meters to mete

16、rs, convert kilo-grams to grams. In this way, I don't have to list every imaginable combination of metric units.The notation 1.23E + 4 stands for 1.23 x 10+4 = 0.000123.To use: Find the unit to convert from in the left column, and multiply it by the expression under the unit to convert to.Exampl

17、es: foot2 = 144 inches2; 2 feet2 = 2x144 inches2.Useful Exact Area & Length Relationships2acre = (1/640)= 1760 yards = 5280 feetyard = 3 feet = 36 inches foot = 12 inchesinch = 2.54 centimetersNote that when converting area units:1 foot = 12 inches(1 foot)2 = (12 inches)2 (square both sides)1 fo

18、ot2 = 144 inches2The linear & area relationships are not the same!1 of 18/10/98 5:02 PMfrom to= acres= feet2= inches2= meters2=2= yards2acre4356062726404046.856.(1/640)4840foot2(1/43560)1440.09290304(1/27878400)(1/9)inch2(1/6272640)(1/144)6.4516E-4 3.587006E- 10(1/1296)meter22.471054.E-4 10.7639

19、1.1550.00313.861021.E-7 1.195990.2640278784002.78784E+9 2.589988.E00yard2(1/4840)912960.836127363.228305.E-7 Dave's Math Tables: Weights and Measures(Math | General | Weights and Measures | Areas)Exponential IdentitiesPowersx a x b = x (a + b) x a y a = (xy) a (x a) b = x (ab)x (a/b) = bth root

20、of (x a) = ( bth x (-a) = 1 / x ax (a - b) = x a / x b(x) ) aLogarithmsy= logb(x) if and only if x=b ylogb(1) = 0logb(b) = 1logb(x*y) = logb(x) + logb(y) logb(x/y) = logb(x) - logb(y) logb(x n) = n logb(x)logb(x) = logb(c) * logc(x) = logc(x) / logc(b)1 of 18/10/98 5:03 PMDave's Math Tables: Exp

21、onential Identities(Math | Algebra | Exponents)ee = 2.7182818284 5904523536 0287471352 66249775729574966967 627724076630353547595251664274 .e = lim (n -> 0) (1 + n)(1/n) or e = lim (n ->) (1 + 1/n)ne = 1 / k!see also Exponential Function Expansions.1 of 18/10/98 5:11 PMDave's Math Tables:

22、e(Math | OddsEnds | Constants | e)Exponential Expansions1 of 18/10/98 5:17 PMFunctionSummation ExpansionCommentsee=1 / n!= 1/1 + 1/1 + 1/2 + 1/6 + .see constant ee -1=(-1) n / n!= 1/1 - 1/1 + 1/2 - 1/6 + .e x=xn / n!= 1/1 + x/1 + x2 / 2 + x3 / 6 + .Dave's Math Tables: Exponential Expansions(Math

23、 | Calculus | Expansions | Series | Exponent)Logarithmic ExpansionsExpansions of the Logarithm FunctionExpansions Which Have Logarithm-Based Equivalents1 of 28/10/98 5:17 PMFunctionSummation ExpansionCommentsln (x)=(x-1)nn= (x-1) - (1/2)(x-1)2 + (1/3)(x-1)3 + (1/4)(x-1)4 + .Taylor Series Centered at

24、 1(0 < x <=2)ln (x)(x-1) / x)n= n= (x-1)/x + (1/2) (x-1) / x)2 + (1/3) (x-1) / x)3 + (1/4) (x-1) / x)4 + .(x > 1/2)ln (x)(x-a)n=ln(a)+ n an= ln(a) + (x-a) / a - (x-a)2 / 2a2 + (x-a)3 / 3a3 - (x-a)4 / 3a4 + .Taylor Series (0 < x <= 2a)ln (x)=2(x-1)/(x+1)(2n-1)(2n-1)= 2 (x-1)/(x+1) + (1

25、/3)( (x-1)/(x+1) )3 + (1/5) ( (x-1)/(x+1) )5+ (1/7) ( (x-1)/(x+1) )7 + . (x > 0)Dave's Math Tables: Log Expansions(Math | Calculus | Expansions | Series | Log)Logarithmic Expansions2 of 28/10/98 5:17 PMSummon ExpansionEquivalent ValueCommentsx nn= x + (1/2)x2 +(1/3)x3 + (1/4)x4 + .= - ln (x +

26、 1)(-1 < x <= 1)(-1)n xn n= - x + (1/2)x2 - (1/3)x3 + (1/4)x4 + .= - ln(x)(-1 < x <= 1)x2n-1 2n-1= x + (1/3)x3 + (1/ 5)x5 + (1/7)x7 + .= ln ( (1+x)/(1-x) ) 2(-1 < x < 1)CirclesDave's Math Tables: Circles(Math | Geometry | Circles)Definition: A circle is the locus of all points

27、equidistant from a central point.Definitions Related to Circlesa circlearc: a curved line that is part of the circumference of a circlechord: a ligment within a circle that touches 2 points on the circle.circumference: the distance around the circle.diameter: the longest distance from one end of a c

28、ircle to the other.origin: the center of the circlepi ( ): A number, 3.141592., equal to (the circumference) / (the diameter) of any circle.radius: distance from center of circle to any point on it.sector: is like a slice of pie (a circle wedge).tangent of circle: a line perpendicular to the radius

29、that touches ONLY one point on the circle.diameter = 2 x radius of circleCircumference of Circle = PI x diameter = 2 PI x radiuswhere PI = 3.141592.Area of Circle:area = PI r2Length of a Circular Arc: (with central angle)if the angle if the angleis in degrees, then length =is in radians, then length

30、 = r xx (PI/180) x rArea of Circle Sector: (with central angle)is in degrees, then area = ( /360)x PI r2 is in radians, then area = ( /2)x PI r2if the angle if the angleEquation of Circle: (cartesian coordinates)for a circle with center (j, k) and radius (r): (x-j)2 + (y-k)2 = r2Equation of Circle:

31、(polar coordinates)1 of 28/10/98 5:27 PMCirclesfor a circle with center (0, 0): r( ) = radiusfor a circle with center with polar coordinates: (c,) and radius a: r2 - 2cr cos(-) + c2 = a2Equation of a Circle: (parametric coordinates)for a circle with origin (j, k) and radius r:x(t) = r cos(t) + jy(t)

32、 = r sin(t) + k2 of 28/10/98 5:27 PMAreas, Volumes, Surface Areas(pi = 3.141592.)Areassquare = a 2rectangle = abparallelogram = bhtrapezoid = h/2 (b1 + b2)circle = pi r 2ellipse = pi r1 r2triangle = (1/2) b hequilateral triangle = (3)/2 a 2 =(3/4) a 2triangle given SAS = (1/2) a b sin Ctriangle give

33、n a,b,c =s(s-a)(s-b)(s-c) when s = (a+b+c)/2 (Heron's formula)Volumescube = a 31 of 28/10/98 5:05 PMDave's Math Tables: Areas, Volumes, Surface Areas(Math | Geometry | AreasVolumes)Areas, Volumes, Surface Areasrectangular prism = a b cirregular prism = b hcylinder = b h = pi r 2 hpyramid = (

34、1/3) b hcone = (1/3) b h = 1/3 pi r 2 hsphere = (4/3) pi r 3ellipsoid = (4/3) pi r1 r2 r3Surface Areacube = 6 a 2prism:(lateral area) = perimeter(b) L (total area) = perimeter(b) L + 2bsphere = 4 pi r 22 of 28/10/98 5:05 PMAlgebraic Graphs1 of 28/10/98 5:04 PM? ?(see also Conic Sections)Pointx2 + y2

35、 = 0 Circlex2 + y2 = r2Ellipsex2 / a2 + y2 / b2 = 1 Ellipsex2 / b2 + y2 / a2 = 1 Hyperbolax2 / a2 - y2 / b2 = 1 ParabolaParabolaHyperbolaDave's Math Tables: Algebraic Graphs(Math | OddsEnds | Graphs | Algebra)Algebraic Graphs2 of 28/10/98 5:04 PM4px = y24py = x2y2 / a2 - x2 / b2 = 1 For any of t

36、he above with a center at (j, k) instead of (0,0), replace each x term with (x-j) and each y term with (y-k) to get the desired equation.Trigonometric Indentitiessin(-x) = -sin(x)csc(-x) = -csc(x)cos(-x) = cos(x)sec(-x) = sec(x)tan(-x) = -tan(x)cot(-x) = -cot(x)tan(x y) = (tan xtan y) / (1tan x tan

37、y)sin(2x) = 2 sin x cos xcos(2x) = cos2(x) - sin2(x) = 2 cos2(x) - 1 = 1 - 2 sin2(x)tan(2x) = 2 tan(x) / (1 - tan2(x)sin2(x) = 1/2 - 1/2 cos(2x)cos2(x) = 1/2 + 1/2 cos(2x)sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )cos x - cos y = -2 sin( (x-y)/2 ) sin( (x + y)/2 )1 of 28/10/98 5:06 PMsin2(x)

38、 + cos2(x) = 1tan2(x) + 1 = sec2(x)cot2(x) + 1 = csc2(x)sin(x y) = sin x cos y cos x sin ycos(x y) = cos x cosy sin x sin ysin(theta) = a / ccsc(theta) = 1 / sin(theta) = c / acos(theta) = b / csec(theta) = 1 / cos(theta) = c / btan(theta) = sin(theta) / cos(theta) = a / bcot(theta) = 1/ tan(theta)

39、= b / aDave's Math Tables: Trigonometric Identities(Math | Trig | Identities)Trigonometric IndentitiesTrig Table of Common AnglesGiven Triangle abc, with angles A,B,C; a is opposite to A, b oppositite B, c opposite C:a/sin(A) = b/sin(B) = c/sin(C) (Law of Sines)(Law of Cosines)(a - b)/(a + b) =

40、tan 1/2(A-B) / tan 1/2(A+B) (Law of Tangents) -not neccessary with the above2 of 28/10/98 5:06 PMc2 = a2 + b2 - 2ab cos(C)b2 = a2 + c2 - 2ac cos(B) a2 = b2 + c2 - 2bc cos(A)angle030456090sin2(a)0/41/42/43/44/40/44/0cos2(a)4/4 3/4 2/4 1/40/4 1/3 2/2 3/1tan2(a)Trigonometric Graphs1 of1 8/10/98 5:35 PM

41、Dave's Math Tables: Trigonometric Graphs(Math | OddsEnds | Graphs | Trig)Series PropertiesSemi-Formal Definition of a "Series":A seriesan is the indicated sum of all values of an when n is set to each integer from a to b inclusive;namely, the indicated sum of the values aa + aa+1 + aa+

42、2 + . + ab-1 + ab.Definition of the "Sum of the Series":The "sum of the series" is the actual result when all the terms of the series are summed.Note the difference: "1 + 2 + 3" is an example of a "series," but "6" is the actual "sum of the seri

43、es."Algebraic Definition:an = aa + aa+1 + aa+2 + . + ab-1 + abSummation Arithmetic:c an = c an (constant c)an +bn =an + bnan -bn =an - bnSummation Identities on the Bounds:bccan +an =ann=an=b+1 n = a bb-can = n=aban = n=aan+c n=a-c b/cancn=a/c|(similar relations exist for subtraction and divisi

44、on as generalized below for any operation g)|bg(b)ag -1(c)n=g(a)an =n=a1 of 28/10/98 5:22 PMDave's Math Tables: Series Properties(Math | Calculus | Expansions | Series | Properties)Power Series1 of 28/10/98 5:20 PMSummationExpansionEquivalent ValueCommentsnk k=1= 1 + 2 + 3 + 4 + . + n = (n2 + n)

45、/ 2= (1/2)n2 + (1/2)nsum of 1st n integersnk 2k=1= 1 + 4 + 9 + 16 + . + n2= (1/6)n(n+1)(2n+1)= (1/3)n3 + (1/2)n2 + (1/6)nsum of 1st n squaresnk 3k=1= 1 + 8 + 27 + 64 + . + n3= (1/4)n4 + (1/2)n3 + (1/4)n2sum of 1st n cubesnk 4k=1= 1 + 16 + 81 + 256 + .+ n4= (1/5)n5 + (1/2)n4 + (1/3)n3 - (1/30)nnk 5k=

46、1= 1 + 32 + 243 + 1024+ . + n5= (1/6)n6 + (1/2)n5 + (5/12)n4 - (1/12)n2nk 6k=1= 1 + 64 + 729 + 4096+ . + n6= (1/7)n7 + (1/2)n6 + (1/2)n5 - (1/6)n3+ (1/42)nnk 7k=1= 1 + 128 + 2187 + 16384 + . + n7= (1/8)n8 + (1/2)n7 + (7/12)n6 - (7/24)n4 + (1/12)n2nk 8k=1= 1 + 256 + 6561 + 65536 + . + n8= (1/9)n9 + (

47、1/2)n8 + (2/3)n7 - (7/15)n5 + (2/9)n3 - (1/30)nDave's Math Tables: Power Summations(Math | Calculus | Expansions | Series | Power)Power Series2 of 28/10/98 5:20 PMnk 9k=1= 1 + 512 + 19683 + 262144 + . + n9= (1/10)n10 + (1/2)n9 + (3/4)n8 - (7/10)n6 + (1/2)n4 - (3/20)n2nk 10k=1= 1 + 1024 + 59049 +

48、 1048576 + . + n10= (1/11)n11 + (1/2)n10 + (5/6)n9 - n7 + n5 - (1/2)n3 + (5/66)nPower Summations #21 of 28/10/98 5:21 PMSummationExpansionEquivalent ValueComments1/n n=1= 1 + 1/2 + 1/3 + 1/4 + .diverges tosee the gamma constant1/n 2n=1= 1 + 1/4 + 1/9 + 1/16 +.= (1/6) PI 2 = 1.64493406684822.see Expa

49、nisions of PI1/n 3n=1= 1 + 1/8 + 1/27 + 1/81 +.= 1.20205690315031.see the Unproved Theorems1/n 4n=1= 1 + 1/16 + 1/81 + 1/256 + .= (1/90) PI 4 =1.08232323371113.see Expanisions of PI1/n 5n=1= 1 + 1/32 + 1/243 + 1/1024 + .= 1.03692775514333.see the Unproved Theorems1/n 6n=1= 1 + 1/64 + 1/729 + 1/4096

50、+ .= (1/945) PI 6 =1.017343061984449.see Expanisions of PI1/n 7n=1= 1 + 1/128 + 1/2187 + 1/16384 + .= 1.00834927738192.see the Unproved Theorems1/n 8n=1= 1 + 1/256 + 1/6561 + 1/65536 + .= (1/9450) PI 8 =1.00407735619794.see Expanisions of PIDave's Math Tables: Power Summations #2(Math | Calculus

51、 | Expansions | Series | Power2)Power Summations #22 of 28/10/98 5:21 PM1/n 9n=1= 1 + 1/512 + 1/19683 + 1/262144 + .= 1.00200839282608.see the Unproved Theorems1/n 10n=1= 1 + 1/1024 + 1/59049+ 1/1048576 + .= (1/93555) PI 10 =1.00099457512781.see Expanisions of PI1/(2n) nn=1= 1 + 1/(2n) 2 + 1/(2n) 3+ 1/(2n) 4 + .= (-1)n-1 (2 2n B(2n) PI 2n )/ ( 2(2n)!) see Expanisions of PIGeometric SeriesSee also the Geometric Series Convergence in the Convergence Tests.1 of 18/10/98 5:19 PMSummationExpansionConvergenceCommentsn-1r nn=0= 1 + r + r 2 + r 3 + . + r n-1(firserms)for r1,=( 1 - r n ) / (1