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math_primer - POWERS OF TEN You should be familiar with the...

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Unformatted text preview: POWERS OF TEN . You should be familiar with the usage of powers of ten. it is a compact form of Writing very large or very small numbers. For example, instead of 10 000, we write 104, where the exponent represents the number of zeros: that is, 10“ = 10 X 10 X 10 x 10 = 10 000. Likewise, a small number like 0.0001 can be expressed as 10“, where the negative exponent indicates that we are dealing with a number less than one. Some other examples of the use of powers of ten are 1000 = 10% 0.003 = 3 x 10-3 85 000 = 8.5 x 104 0.00085 = 8.5 x 10-4 3 200000 =_ 3.2 x 106 0.00002 = 2 x 10-5 If numbers written as powers of ten are multiplied, we simply add the exponents, maintaining their signs. For example, (3 X 103) x (5 x 104) = 15 x 107 = 1.5 x 105 (2 x 105) X (4 X 10‘?) = 8 X 103 (5.6 x 104) x (4.3 x 108) = 24 x 1012 When numbers written as powers of ten are divided, we can bring the power of ten from the denominator to the numerator by changing its sign. For example, géig: = 4x105 x 10-2 = 4 x103 ' ff—llg:=3x10-4x109=3x105 In general, 10m)"- : 10m 10"/10"' = ion-m (10")"' = 10m ALGEBRA When algebraic operations are performed, the laws of arithmetic apply. SymbolS such as x, y, and z are usually used to represent quantities that are not specified, and symbols such as a, b, and c are used to represent numbers. You should be familiar with the following operations: a(%)=%f’— (one (cl/b) _Ec_i- c _ adrcb (c/d) _. be _ Fractions 0 _+..._ b_d bd ax+bx=x(a+b} 12—y2=(r+y)(x—y) x2—2x——15=(x+3)(x——5) Factoring and combinations quadratic equation _b t \/b2 — 4ac th = En x 20 Multiplying powers Rootsofa '[Ifax'2+bx+,c=0, of a given quantity [ ).._, ln = logarithm to base 8 log = logarithm to base 10 in e :1 lneI = x “' ln(xy) lnx + In y Logarithmic functions ln{x/y) ___ In x _. In 9. 111(1/1') = —inx In x" = n lnx lo a = 2.302610ga log a = 0.434291n a Simultaneous Linear Equations In order to solve two simultaneous equations involving two unknowns, x and y, we solve one of the equations for x in terms of y and substitute this expression into the other equation. Example (1) 5x + y = -8 (2) 2x —-— 2y = 4 Alternate Solution: Multiply (1) by 2 and add the result to (2): Solution.- From (2), x = y + 2. Substitution of this into (1) gives 10:: + 2y = —16 5(y+2}+y=—8 ___L__23-2 =4 6y=-—18 12x=—12 y:—3 x=—1 x=y+2=m1 y:x-2=—3 GEOMETRY The Pythagorean theorem, which relates the three sides of a right triangle I The radian measure: the arc lengths is proportional to the radius r for a fixed Value of 9 (in radians) 3 = E I A * r -. Circumference of a circle . I C = 2771' Area of a circle A = nrz Area of a triangle 1 A = —-bh ‘ 2 ‘“' Surface area of a sphere Volume of a sphere Volume of a cylinder V = 1"le Equation of a straight line ! = b = y intercept .‘y mx+b m=slope=tanfl Equation of a circle of radius R centered at the origin Equation of a parabola whose vertex is at y = b .— TRIGONOMETRY The sine, cosine, and tangent functions in trigonometry are defined in terms of the ratios of the sides of a right triangle: . sinB : Side opposxte 0 __ hypotenuse c0919 = —-———-- = hypotenuse a .3 side adjacent 9 b ? a side opposite 0 t = .... mg side adjacent 6 b From‘the above definitions and the Pythagorean theorem, it follows that sinzfl + cos20 = 1 _' sinH tanfi _ cost) The cosecant, secant, and cotangent functions are defined by csc6=L secG: cotfi: 1 $1116 , cosfi tanfi The relations at the right follow directiy from the right triangle above: sinfi z (205(90“ — 0) cost? = sin(90° — 9) c015: tan(90° - 9) Some properties of trigonometric functions: sin(—8) = -sin0 cos(—6) = 0059 tan(—6) = —tan6 Relations that apply to any triangle: a+B+y=lSO° (:2 = b2 + c2 — 21):: case: Law of cosines 172 = (12 + 02 — 20c cosB 6‘2 = 02 + b2 -- 2ab cosy Law of Sines {—,— = —.-- = .— Trimuic Honda“ ' sin’fi + c0520 = 1 csczfl = 1 + cotzo secza = l + tanZB sinzg- = %(1 —. 5059) 511129 = 25mg cost? coszg— = %‘1 + 3:059) c0529 = c0530 -— 511126 1 -— c056 = Rina; _ 2tan0 i _ 1 .. cost? “”29 ‘ 1 — tanza “m2 ‘ 1 + c059 sin(A i B) = sinA cosB I cosA sinB cos(A : B] = cosA cosB : sinA sinB sinA = sagas = 25in[%(A : B}]cos[—%{A ; 3)] cosA + 0358 = 2cos[%{A + Bi]cos[%{A - 3)] cosA — 5055 = 23in[%{A + B)]sin[%(B — m] {a+b)-=5~+la~—lb+ (1 +x)"=1+nx+ Suic- Expansion 1! x2 x3 mm -—1) n{n-— 1) 2! xa+m 2! e‘=1+x+—2T+§+~- 49., w'2b2+~- 9 in radians ln{1:x}=:x—é—x¢:%x3_ sinx:x-—£.+£l— cosx=1——2x;+f—!---- tanx¢x+£+£€+n- [xl<fr/2 " 3 15 Forx< 1,the following approadmafionscanbeused: (l+xl"::1+nx sinxzx e':l+x (:05le lnfil : x) 2: ix tanxzx DIFFERENTlAL CALCULUS In various branches of science, it is sometimes necessary to use the basic tools of calculus, first invented by Newton, to describe physical phenomena. The use of calculus is fundamental in the treatment of various problems in newtom‘an mechan- ics, electricity, and magnetism. In this section, we simply state some basic P'DPC" ties and “rules of thumb” that should be a useful review to the student. First, a function must be specified which relates one variable to another (Mb ‘5 coordinate as a function of time). Suppose one of the variables is called 9 like dependent variable), the other I (the independent variable). We might have a m tion relation such as y(x)=ax3+bx2+cy+d _ If a, b, c, and d are specified constants, then y can be calculated for any ”31“” ”f z. Figure 1 We usually deal with continuous functions, that is, those for which y varies “smoothly" with x. . The derivative of y with respect to x is defined as the limit of the slopes of chords drawn between two points on the y versus at curve as An: approaches zero. Mathe- matically, we write this definition as 9-y— = lim 31 = lim My“ + "W ' yifl dx air-o Ax its-'0 Ax Where Ag and Ax are defined as Ax r: x2 — x1 and Jig = 92 — 9'1 (see Fig. 1). A useful expression to remember when y(x) :: ax", where a is a constant and n is any positive or negative number (integer or fraction), is If ytx) is a polynomial or algebraic function of x, we apply Eq. 3.2 to each term in the polynomial and take da/dx = 0, It is important to note that dy/dx does not mean dy divided bv dx, but is simply a notation of the limiting process of the derivative - - Properties of the Derivative Derivative of the Product of Two Functions. If a function y is given by the product of two functions, say. g(x) and h(x), then th ‘ - . e derivative of y iS defined as d , d dh d _ = _ _ g d-r ”i dx [glx’hml ‘ g3: 4*" ”a B. Derivative of the Sum of Two Functions. If a function y is equal to the in two functions, then the derivative of the sum is equal to the sum of the deriv ti“ of a WES: d ,. d . , dg dh 2..., C. Chain Rule of Differential Calculus. If y :2 fix) and .r is a functio other variable 3, then dy/dx can be written as the product of two d dy _ dy dz dx _ dzE n of some erivatjvg: D. The Second Derivative. The second derivative of y with respect to x is as the derivative of the function dy/dx (or, the derivative of the derivativ usually written defined e}. It is day (I (dy) dx2 =71: E Derivatives for Several Function- - ' . d « fl Eta] = 0 Emmott) _ aseczax d (we) — nae-l £(cotax) = —ocsc~‘czx E ‘ dx _d_(en.r) _ Gen: _d_(secx} = tanxsecx dx — dx d l ' t — cosax «(Hi—(cscx) = —cotxcsoc a smax — Cl dX' 1 Edicosax] = «asinax illncvt) = 3:- W Note: The letters a and n are constants. INTEGRAL CALCULUS We can think of integration as the inverse of differentiation. As an example, ponsider the expression flx) :%=Sax2 + b the result of differentiating the function y(x)=ax3+bx+c_ We can write the first expression dy = f(x)dx = (302:2 + b)dx and obtain y(x) by “summing" over all values of x. Mathematically, we write this inverse operation W) = fflxidx For the function f(x) given above, y(x) = f(3ax2 + b)dx = £113 + bx + c where c is a constant of the integration. This type of integral is called an indefinite integral since its value depends on the choice of the constant c. A general indefinite integral I (x) is defined as ' HI) = from d1(x} ‘where fix) is called the integrand and f(x) = dx - 2..., For a general continuous function f (x), the integral can be described as the area r I Under the curve bounded by f (x) and the x axis, between two specified values of' x, say. 3:} and x1“ as in Fig. 2. for) —*-l l“— AX]: Figure 2 The area of the shaded element is approximater fiAxi. If we sum all these area elernents from x1 to x2 and take the limit of this sum as .311. ——> 0, we obtain the true “8a under the curve bounded by f (x) and 1:, between the limits I] and x2: Area = lim ZfimAxi = fzzflxkix 1nttbgrals of the type defined by Eq. B8 are called definite integrab. One of the common types of integrals that arise in practical situations has the form f d xn+1 + { ¢ 1) x x = ' c n — n + l ' This result is obvious since difierentiation of the right~hand side with respect to it gives Jfir) = x" directly. If the limits of the integration are known, this integral becomes a definite integral and is written n+1 __ I n+1 32!! _.‘C2 '1 .- Idx—T (13¢ l} 4'1 2..., Partial Integration Sometimes it is useful to apply the method of partial integration to evaluate ' certain integrals. The method uses the property that fade = on -— udu where u and u are carefully chosen so as to reduce a complex integral to a simpfer one. In many cases, several reductions have to be made. Consider the example I(x) = fxze‘dx This can be evaluated by integrating by parts twice. First, if we choose a = :2. u = e‘, we get fxze’dx = x2d(e‘) = 1:28" — 2 e’xdx + 61 Now, in the second term, choose u = x, v = 6", which gives ftze‘dx 2 x283 —- 2er + Zfe‘dx + Cl or fxze‘dx = xze‘ — 2m" + 2e‘ + c2 The Perfect Differential Another useful method to remember is the use of the perfect diferenhflil P71; we should sometimes 100k for a change of variable such that the differentia o function is the differential of the independent variable appearing m the mtetr For example, consider the integral fix) = fcos2x sinxdx This becomes easy to evaluate if we rewrite the differential as d(cosx} = —sinxdx. The integral then becomes fcos2x sinxdx = — c0522: d(cosx} If we now change Variables, letting y = cosx, we get 3 3 fcos2xsinxdx= —fy2dy= w-y:’,—+e= —CO::I +5 in, Some Indefinite lntcgrals“ (an arbitrary constant should be addzd to each of theie integrals) WWW fr'dx: nx:=1 (provided n ¢ —-1) fxé’dx:%(ax—1) dx 4 _ dx — X i : T‘fx (ix—h“ fa+befl_3_acln(a+bec) dx 1 . 1 In + bx = 3111(0 + bx) fsmaxdx = “Ecosax dx 1 ‘ 1 . fm = ‘m fcosaxdx _ Esmax a? (1:28 = %tan"% ftanaxdx = —:1]—!n(cosax) = éinhecex) fa? (1;xe 7' glam: :: (02 _ x2 > D) fcotaxdx = %}n(sinax) 7 dx 1 x— a 1 _ 1 ax 1T f—fixz __ .. Zinx + a (x2 —- 02 >0) fsecaxdx .—_ Eli-:(secax + tanax) _ Eln[mn(7 + 1)] xdx _ 1 _ 1 ax 2.2—2? = —§1“{°2 I X2) . fcscaxdx _ Elnkscax «— cotax) _ Elr'1(1:an«~2n-.) f . dx = sin-1: = —cos'1£ (92 __ x2 >0} fsinzaxdx = i _ sinZax V02 — x2 a a 2 4a dx r—m 2 _ 5 sin2ax fxm=lnix+yxziazj fcosaxdx_2+ 4a dx 1 I x—Jf-d—x—T = —V‘02 — x2 fsinzax :2 —Ecotax V0? _— x“ dx 1 f ' mix = x2 i ”2 fcos2ax = Etanax Vxe : a2 2 1 ~—— tan (3de = —[tanax) - x fv‘d’ — dex 2 %(x\/a2 — x2 + (ism-1%) f a 1 fxx/a? _ xzdx = ,flaz _ xszz _ fcotzaxdx = -E(comx) _ x r'——-- — " '“1dx-x{‘-‘ ) \f1—02x2 fwa : 0de = gwxz :- a2 : a21n(x + Wu f5!“ ‘1’“ — 5‘“ a" + —a fw—‘xz I'M/F 2 02d): =1HX2 : 02%” fcos‘laxdx = x(cos‘1ax) -— LL75; _ 1 . , fend): = .159: ftan"axdx = x(tan lax) -— Emu + a-x-) flnaxdx = (find!) — x fem-2am = x(cot"ax) + 21_ain(1 + azx'cy W ' Gauss‘ probability integral and other integrals can be found on the right iron: endpaper‘ G a n! I D fits—MdzEG—‘tfi, n>—l,a>0 “ .I. It a. rfl’dz=— — o 2 6 fl! “GE, 1 r 4 3'8 dz=~ - o 4 aa °° 1 3 (2n 1) 5_ Bun—oz! I: ' ‘— '- j; 5" 6 dz 2n+1 ‘IGZMI °° l 6. zewd$=—- 0 20 ”° 1 8-“? _.._. ti: :6 (ix—2a: +1 1 14.1.1 m—"dz=a—,{e“—e"“-a(e°+e"“)} +1 15. 1 3‘9de = (—1)"+‘A.(—a) — Ada) ...
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