34 SERIES Series.—Some hand calculations, as well as computer programs of certain types of math- ematical problems, may be facilitated by the use of an appropriate series. For example, in some gear problems, the angle corresponding to a given or calculated involute function is found by using a series together with an iterative procedure such as the Newton-Raphson method described on page 33. The following are those series most commonly used for such purposes. In the series for trigonometric functions, the angles x are in radians (1 radian = 180/π degrees). The expression exp(−x 2 ) means that the base e of the natural log- arithm system is raised to the −x 2 power; e = 2.7182818. Derivatives and Integrals of Functions.—The following are formulas for obtaining the derivatives and integrals of basic mathematical functions. In these formulas, the letters a and c denotes constants; the letter x denotes a variable; and the letters u and v denote func- tions of the variable x. The expression d/dx means the derivative with respect to x, and as such applies to whatever expression in parentheses follows it. Thus, d/dx (ax) means the derivative with respect to x of the product (ax) of the constant a and the variable x. (1) sin x = x − x 3 /3! + x 5 /5! − x 7 /7! + ··· for all values of x. (2) cos x = 1 − x 2 /2! + x 4 /4! − x 6 /6! + ··· for all values of x. (3) tan x = x + x 3 /3 + 2x 5 /15 + 17x 7 /315 + 62x 9 /2835 + ··· for |x| < π/2. (4) arcsin x = x + x 3 /6 + 1 · 3 · x 5 /(2 · 4 · 5) + 1 · 3 · 5 · x 7 /(2 · 4 · 6 · 7) + ··· for |x| ≤ 1. (5) arccos x = π/2 − arcsin x (6) arctan x = x − x 3 /3 + x 5 /5 − x 7 /7 + ··· for |x| ≤ 1. (7) π/4 =1 − 1/3 + 1/5 − 1/7 + 1/9 ··· ±1/(2x − 1)ϯ ··· for all values of x. (8) e =1 + 1/1! + 2/2! + 1/3! + ··· for all values of x. (9) e x =1 + x + x 2 /2! + x 3 /3! + ··· for all values of x. (10) exp(− x 2 ) = 1 − x 2 + x 4 /2! − x 6 /3! + ··· for all values of x. (11) a x = 1 + x log e a + (x log e a) 2 /2! + (x log e a) 3 /3! + ··· for all values of x. (12) 1/(1 + x) = 1 − x + x 2 − x 3 + x 4 −··· for |x| < 1. (13) 1/(1 − x) = 1 + x + x 2 + x 3 + x 4 + ··· for |x| < 1. (14) 1/(1 + x) 2 = 1 − 2x + 3x 2 − 4x 3 + 5x 4 − ··· for |x| < 1. (15) 1/(1 − x) 2 = 1 + 2x + 3x 2 + 4x 3 + 5x 5 + ··· for |x| < 1. (16) = 1 + x/2 − x 2 /(2 · 4) + 1 · 3 · x 3 /(2 · 4 · 6) − 1 · 3 · 5 · x 4 /(2 · 4 · 6 · 8) −··· for |x| < 1. (17) = 1 − x/2 + 1 · 3 · x 2 /(2 · 4) − 1 · 3 · 5 · x 3 /(2 · 4 · 6) + ··· for |x| < 1. (18) (a + x) n = a n + na n−1 x + n(n − 1)a n−2 x 2 /2! + n(n − 1)(n − 2)a n−3 x 3 /3! + ··· for x 2 < a 2 . Formulas for Differential and Integral Calculus Derivative Value Integral Value 1 x+() 11x+()⁄ xd d c() 0 cxd ∫ cx xd d x() 1 1 xd ∫ x x d d x n () nx n 1– x n xd ∫ x n 1+ n 1+ ------------ xd d gu()() ud d gu() xd du xd ax b+ --------------- ∫ 1 a --- ax b+log xd d ux() vx()+() xd d ux() xd d vx()+ ux() vx()±()xd ∫ ux()xvx()xd ∫ ±d ∫ xd d ux() vx()×()ux() xd d vx() vx() xd d ux()+ ux()vx()xd ∫ ux()vx() vx()ux()d ∫ – Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY DERIVATIVES AND INTEGRALS 35 Formulas for Differential and Integral Calculus (Continued) Derivative Value Integral Value xd dux() vx() ---------- ⎝⎠ ⎛⎞ vx() xd d ux() ux() xd d vx()– vx() 2 --------------------------------------------------------------- xd x ------ ∫ 2 x xd d xsin() xcos xcos xd ∫ xsin xd d xcos() xsin– xsin xd ∫ xcos– xd d xtan() sec 2 x xtan xd ∫ xcoslog– xd d xcot() cosec– 2 x xcot xd ∫ xsinlog xd d xsec() xsec xtan sin 2 xxd ∫ 1 4 ---– ⎝⎠ ⎛⎞ 2x()sin 1 2 ---x+ xd d xcsc() xcsc xcot– cos 2 xxd ∫ 1 4 ---2x()sin 1 2 ---x+ xd d e x () e x e x xd ∫ e x xd d xlog() 1 x --- 1 x --- xd ∫ xlog xd d a x () a x alog a x xd ∫ a x alog ----------- xd d xasin() 1 1 x 2 – ------------------ xd b 2 x 2 – -------------------- ∫ x b ---asin xd d xacos() 1– 1 x 2 – ------------------ xd x 2 b 2 – -------------------- ∫ x b ---acosh xx 2 b 2 –+()log= xd d xatan() 1 1 x 2 + -------------- xd b 2 x 2 + ----------------- ∫ 1 b --- x b ---atan xd d xacot() 1– 1 x 2 + -------------- xd b 2 x 2 – ---------------- ∫ 1 b --- x b ---atanh 1– 2b ------ xb–() xb+() -------------------log= xd d xasec() 1 xx 2 1– --------------------- xd x 2 b 2 – ---------------- ∫ 1 b ---– x b ---acoth 1 2b ------ xb–() xb+() -------------------log= xd d xacsc() 1– xx 2 1– --------------------- xd ax 2 bx c++ ------------------------------ ∫ 2 4ac b 2 – ------------------------- 2ax b+() 4ac b 2 – -------------------------atan xd d xsinlog() xcot e ax bxsin xd ∫ bxasin bbxcos–() a 2 b 2 + ----------------------------------------------e ax xd d xcoslog() xtan– e ax bx()cos xd ∫ bx()acos bbx()sin+() a 2 b 2 + -------------------------------------------------------- e ax xd d xtanlog() 2 2xsin -------------- 1 xsin ---------- xd ∫ x 2 ---tanlog xd d xcotlog() 2– 2xsin -------------- 1 xcos ----------- xd ∫ π 4 --- x 2 ---+ ⎝⎠ ⎛⎞ tanlog xd d x() 1 2 x ---------- 1 1 xcos+ --------------------- xd ∫ x 2 ---tan xd d log 10 x() log 10 e x --------------- xlog xd ∫ xxlog x– Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY 36 ARITHMATICAL PROGRESSION GEOMETRY Arithmetical Progression An arithmetical progression is a series of numbers in which each consecutive term differs from the preceding one by a fixed amount called the common difference, d. Thus, 1, 3, 5, 7, etc., is an arithmetical progression where the difference d is 2. The difference here is added to the preceding term, and the progression is called increasing. In the series 13, 10, 7, 4, etc., the difference is ( −3), and the progression is called decreasing. In any arithmetical progression (or part of progression), let a=first term considered l=last term considered n=number of terms d=common difference S=sum of n terms Then the general formulas are In these formulas, d is positive in an increasing and negative in a decreasing progression. When any three of the preceding live quantities are given, the other two can be found by the formulas in the accompanying table of arithmetical progression. Example:In an arithmetical progression, the first term equals 5, and the last term 40. The difference is 7. Find the sum of the progression. Geometrical Progression A geometrical progression or a geometrical series is a series in which each term is derived by multiplying the preceding term by a constant multiplier called the ratio. When the ratio is greater than 1, the progression is increasing; when less than 1, it is decreasing. Thus, 2, 6, 18, 54, etc., is an increasing geometrical progression with a ratio of 3, and 24, 12, 6, etc., is a decreasing progression with a ratio of 1⁄2. In any geometrical progression (or part of progression), let a=first term l=last (or nth) term n=number of terms r=ratio of the progression S=sum of n terms Then the general formulas are When any three of the preceding five quantities are given, the other two can be found by the formulas in the accompanying table. For instance, geometrical progressions are used for finding the successive speeds in machine tool drives, and in interest calculations. Example:The lowest speed of a lathe is 20 rpm. The highest speed is 225 rpm. There are 18 speeds. Find the ratio between successive speeds. lan1–()d and+= S al+ 2 ----------- n×= S al+ 2d ----------- lda–+() 540+ 27× ---------------40 7 5–+()135== = lar n 1– and= S rl a– r 1– -------------= Ratio r l a --- n 1– 225 20 --------- 17 11.25 17 1.153== = = Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY ARITHMATICAL PROGRESSION 37 Formulas for Arithmetical Progression To Find Given Use Equation a dln dnS dlS lnS d aln anS alS lnS l adn adS anS dnS n adl adS alS dlS S adn adl aln dln aln1–()d–= a S n --- n 1– 2 ------------ d×–= a d 2 --- 1 2 ---2ld+() 2 8dS–±= a 2S n ------ l–= d la– n 1– ------------= d 2S 2an– nn 1–() ----------------------= d l 2 a 2 – 2Sl– a– -----------------------= d 2nl 2S– nn 1–() ---------------------= lan1–()d+= l d 2 ---– 1 2 --- 8dS 2ad–() 2 +±= l 2S n ------ a–= l S n --- n 1– 2 ------------ d×+= n 1 la– d ----------+= n d 2a– 2d --------------- 1 2d ------ 8dS 2ad–() 2 +±= n 2S al+ -----------= n 2ld+ 2d -------------- 1 2d ------2ld+() 2 8dS–±= S n 2 --- 2an1–()d+[]= S al+ 2 ----------- l 2 a 2 – 2d ---------------+ al+ 2d ----------- lda–+()== S n 2 --- al+()= S n 2 ---2ln1–()d–[]= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY 38 ARITHMATICAL PROGRESSION Formulas for Geometrical Progression To Find Given Use Equation a lnr nrS lrS lnS l anr arS anS nrS n alr arS alS lrS r aln anS alS lnS S anr alr aln lnr a l r n 1– -----------= a r 1–()S r n 1– -------------------= alrr1–()S–= aS a–() n 1– lS l–() n 1– = lar n 1– = l 1 r --- ar1–()S+[]= lS l–() n 1– aS a–() n 1– = l Sr 1–()r n 1– r n 1– --------------------------------= n llog alog– rlog ---------------------------1+= n ar1–()S+[]log alog– rlog -----------------------------------------------------------= n llog alog– Sa–()log Sl–()log– ------------------------------------------------------ 1+= n llog lr r 1–()S–[]log– rlog -----------------------------------------------------------1+= r l a --- n 1– = r n Sr a ----- aS– a ------------+= r Sa– Sl– ------------= r n Sr n 1– Sl– --------------- l Sl– ----------–= S ar n 1–() r 1– ----------------------= S lr a– r 1– -------------= S l n n 1– a n n 1– – l n 1– a n 1– – ---------------------------------------= S lr n 1–() r 1–()r n 1– ----------------------------= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY STRAIGHT LINES 39 Analytical Geometry Straight Line.—A straight line is a line between two points with the minimum distance. Coordinate System: It is possible to locate any point on a plane by a pair of numbers called the coordinates of the point. If P is a point on a plane, and perpendiculars are drawn from P to the coordinate axes, one perpendicular meets the X–axis at the x– coordinate of P and the other meets the Y–axis at the y–coordinate of P. The pair of numbers (x 1 , y 1 ), in that order, is called the coordinates or coordinate pair for P. Fig. 1. Coordinate Plan Distance Between Two Points: The distance d between two points P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) is given by the formula: Example 1:What is the distance AB between points A(4,5) and B(7,8)? Solution: The length of line AB is Intermediate Point: An intermediate point, P(x, y) on a line between two points, P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ), Fig. 2, can be obtained by linear interpolation as follows, where r 1 is the ratio of the distance of P 1 to P to the distance of P 1 to P 2 , and r 2 is the ratio of the distance of P 2 to P to the distance of P 1 to P 2 . If the desired point is the midpoint of line P 1 P 2 , then r 1 = r 2 = 1, and the coordinates of P are: Example 2:What is the coordinate of point P(x,y), if P divides the line defined by points A(0,0) and B(8,6) at the ratio of 5:3. Solution: 1234 1 2 3 4 −1 −2 −2 −1−3 −3 −4 −4 X Y P(x ,y ) 1 1 d P 1 P 2 ,() x 2 x 1 –() 2 y 2 y 1 –() 2 += d 74–() 2 85–() 2 +3 2 3 2 +1832==== x r 1 x 1 r 2 x 2 + r 1 r 2 + ---------------------------=andy r 1 y 1 r 2 y 2 + r 1 r 2 + ---------------------------= x x 1 x 2 + 2 ----------------=andy y 1 y 2 + 2 ----------------= x 50× 38×+ 53+ ------------------------------- 24 8 ------3===y 50× 36×+ 53+ ------------------------------- 18 8 ------2.25=== Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY 40 STRAIGHT LINES External Point: A point, Q(x, y) on the line P 1 P 2 , and beyond the two points, P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ), can be obtained by external interpolation as follows, where r 1 is the ratio of the distance of P 1 to Q to the distance of P 1 to P 2 , and r 2 is the ratio of the distance of P 2 to Q to the distance of P 1 to P 2 . Fig. 2. Finding Intermediate and External Points on a Line Equation of a line P 1 P 2 : The general equation of a line passing through points P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) is . The previous equation is frequently written in the form where is the slope of the line, m, and thus becomes where y 1 is the coordinate of the y-intercept (0, y 1 ) and x 1 is the coordinate of the x-intercept (x 1 , 0). If the line passes through point (0,0), then x 1 = y 1 = 0 and the equation becomes y = mx. The y-intercept is the y-coordinate of the point at which a line intersects the Y-axis at x = 0. The x-intercept is the x-coordinate of the point at which a line intersects the X-axis at y = 0. If a line AB intersects the X–axis at point A(a,0) and the Y–axis at point B(0,b) then the equation of line AB is Slope: The equation of a line in a Cartesian coordinate system is y = mx + b, where x and y are coordinates of a point on a line, m is the slope of the line, and b is the y-intercept. The slope is the rate at which the x coordinates are increasing or decreasing relative to the y coordinates. Another form of the equation of a line is the point-slope form (y − y 1 ) = m(x − x 1 ). The slope, m, is defined as a ratio of the change in the y coordinates, y 2 − y 1 , to the change in the x coordinates, x 2 − x 1 , x r 1 x 1 r 2 x 2 – r 1 r 2 – ---------------------------= and y r 1 y 1 r 2 y 2 – r 1 r 2 – ---------------------------= Y X O m 2 P (x ,y ) 1 1 1 P (x , y ) 2 2 2 m 1 P(x, y) Q (x, y) yy 1 – y 1 y 2 – ---------------- xx 1 – x 1 x 2 – ----------------= yy 1 – y 1 y 2 – x 1 x 2 – ---------------- xx 1 –()= y 1 y 2 – x 1 x 2 – ---------------- yy 1 – mx x 1 –()= x a --- y b ---+1= m ∆y ∆x ------ y 2 y 1 – x 2 x 1 – ----------------== Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY STRAIGHT LINES 41 Example 3:What is the equation of a line AB between points A(4,5) and B(7,8)? Solution: Example 4:Find the general equation of a line passing through the points (3, 2) and (5, 6), and its intersection point with the y-axis. First, find the slope using the equation above The line has a general form of y = 2x + b, and the value of the constant b can be determined by substituting the coordinates of a point on the line into the general form. Using point (3,2), 2 = 2 × 3 + b and rearranging, b = 2 − 6 = −4. As a check, using another point on the line, (5,6), yields equivalent results, y = 6 = 2 × 5 + b and b = 6 − 10 = −4. The equation of the line, therefore, is y = 2x − 4, indicating that line y = 2x − 4 intersects the y-axis at point (0,−4), the y-intercept. Example 5:Use the point-slope form to find the equation of the line passing through the point (3,2) and having a slope of 2. The slope of this line is positive and crosses the y-axis at the y-intercept, point (0,−4). Parallel Lines: The two lines, P 1 P 2 and Q 1 Q 2 , are parallel if both lines have the same slope, that is, if m 1 = m 2 . Perpendicular Lines: The two lines P 1 P 2 and Q 1 Q 2 are perpendicular if the product of their slopes equal −1, that is, m 1 m 2 = −1. Example 6:Find an equation of a line that passes through the point (3,4) and is (a) parallel to and (b) perpendicular to the line 2x − 3y = 16? Solution (a): Line 2x − 3y = 16 in standard form is y = 2 ⁄ 3 x − 16 ⁄ 3 , and the equation of a line passing through (3,4) is . Fig. 3. Parallel Lines Fig. 4. Perpendicular Lines yy 1 – y 1 y 2 – ---------------- xx 1 – x 1 x 2 – ----------------= y 5– 58– ------------ x 4– 47– ------------= y 5– x 4–= yx–1= m ∆y ∆x ------ 62– 53– ------------ 4 2 ---2== == y 2–()2 x 3–()= y 2x 6–2+= y 2x 4–= Y X P 1 ) x y ( 1 , 1 P 2 ) x y ( 2 , 2 O m 1 m 2 Q 4 ) x y ( 2 , 4 Q 3 ) x y ( 1 , 3 Y X P 1 ) x y ( 1 , 1 P 2 ) x y ( 2 , 2 O m 1 m 2 Q 4 ) x y ( 2 , 4 Q 3 ) x y ( 1 , 3 y 4– mx 3–()= Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY 42 COORDINATE SYSTEMS If the lines are parallel, their slopes are equal. Thus, is parallel to line 2x − 3y = −6 and passes through point (3,4). Solution (b): As illustrated in part (a), line 2x − 3y = −6 has a slope of 2 ⁄ 3 . The product of the slopes of perpendicular lines = −1, thus the slope m of a line passing through point (4,3) and perpendicular to 2x − 3y = −6 must satisfy the following: The equation of a line passing through point (4,3) and perpendicular to the line 2x − 3y = 16 is y − 4 = −3 ⁄ 2 (x − 3), which rewritten is 3x + 2y = 17. Angle Between Two Lines: For two non-perpendicular lines with slopes m 1 and m 2 , the angle between the two lines is given by Note: The straight brackets surrounding a symbol or number, as in |x|, stands for absolute value and means use the positive value of the bracketed quantity, irrespective of its sign. Example 7:Find the angle between the following two lines: 2x − y = 4 and 3x + 4y =12 Solution: The slopes are 2 and − 3 ⁄ 4 , respectively. The angle between two lines is given by Distance Between a Point and a Line: The distance between a point (x 1 ,y 1 ) and a line given by A x + B y + C = 0 is Example 8:Find the distance between the point (4,6) and the line 2x + 3y − 9 = 0. Solution: The distance between a point and the line is Coordinate Systems.—Rectangular, Cartesian Coordinates: In a Cartesian coordinate system the coordinate axes are perpendicular to one another, and the same unit of length is chosen on the two axes. This rectangular coordinate system is used in the majority of cases. Polar Coordinates: Another coordinate system is determined by a fixed point O, the ori- gin or pole, and a zero direction or axis through it, on which positive lengths can be laid off and measured, as a number line. A point P can be fixed to the zero direction line at a dis- tance r away and then rotated in a positive sense at an angle θ. The angle, θ, in polar coor- dinates can take on values from 0° to 360°. A point in polar coordinates takes the form of (r, θ). y 4– 2 3 --- x 3–()= m 1– m 1 ------ 1– 2 3 --- ------ 3 2 ---–=== θtan m 1 m 2 – 1 m 1 m 2 + ----------------------- = θtan m 1 m 2 – 1 m 1 m 2 + ----------------------- 2 3 4 ---– ⎝⎠ ⎛⎞ – 12 3 4 ---– ⎝⎠ ⎛⎞ + ------------------------ 2 3 4 ---+ 1 6 4 ---– ------------ 83+ 4 ------------ 46– 4 ------------ ------------ 11 2– ------ 11 2 ------== ==== θ 11 2 ------atan 79.70°== d Ax 1 By 1 C++ A 2 B 2 + --------------------------------------= d Ax 1 By 1 C++ A 2 B 2 + -------------------------------------- 24× 36× 9–+ 2 2 3 2 + ------------------------------------------ 8189–+ 49+ --------------------------- 17 13 ----------== == Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY COORDINATE SYSTEMS 43 Changing Coordinate Systems: For simplicity it may be assumed that the origin on a Cartesian coordinate system coincides with the pole on a polar coordinate system, and it’s axis with the x-axis. Then, if point P has polar coordinates of (r,θ) and Cartesian coordi- nates of (x, y), by trigonometry x = r × cos(θ) and y = r × sin(θ). By the Pythagorean theo- rem and trigonometry Example 1:Convert the Cartesian coordinate (3, 2) into polar coordinates. Therefore the point (3.6, 33.69) is the polar form of the Cartesian point (3, 2). Graphically, the polar and Cartesian coordinates are related in the following figure Example 2:Convert the polar form (5, 608) to Cartesian coordinates. By trigonometry, x = r × cos(θ) and y = r × sin(θ). Then x = 5 cos(608) = −1.873 and y = 5 sin(608) = −4.636. Therefore, the Cartesian point equivalent is (−1.873, −4.636). Spherical Coordinates: It is convenient in certain problems, for example, those con- cerned with spherical surfaces, to introduce non-parallel coordinates. An arbitrary point P in space can be expressed in terms of the distance r between point P and the origin O, the angle φ that OP′makes with the x–y plane, and the angle λ that the projection OP′ (of the segment OP onto the x–y plane) makes with the positive x-axis. The rectangular coordinates of a point in space can therefore be calculated by the formu- las in the following table. rx 2 y 2 += θ y x --atan= r 3 2 2 2 +94+133.6====θ 2 3 ---atan 33.69°== 1230 0 1 2 5 (3, 2) 33.78 e q u a t o r m e r i d i a n O P PЈ pole r z x y ␭ O P r z λ φ y x Machinery's Handbook 27th Edition Copyright 2004, Industrial Press, Inc., New York, NY . ux() vx()×()ux() xd d vx() vx() xd d ux()+ ux()vx()xd ∫ ux()vx() vx()ux()d ∫ – Machinery& apos;s Handbook 27th Edition Copyright 2004, Industrial Press, Inc.,. ∫ x 2 ---tan xd d log 10 x() log 10 e x --------------- xlog xd ∫ xxlog x– Machinery& apos;s Handbook 27th Edition Copyright 2004, Industrial Press, Inc.,