. . . :o) = 440 ways. . 4 ~ ~ - 5 ~ ~ - 8 X - 3 0 ~ - 2 1 = 0 (c) 3 ~ ~ - ~ ~ - l & r + l O y - 1 0 = 0 (6) 5x2-4y2- 1 0 ~ - 2 4 ~ - 5 1= O (d) 4 ~ ~ - ~ * + 8 ~ + l 6l =y O+ (U) ANSWERS TO SUPPLEMENTARY PROBLEMS 17.12 (a) circle, Fig. . . 25.87 How many different sums of money can be formed from the coins of Problem 25.86? . . . Fortunately, there’s Schaum’s. SOLUTION Each combination of 2 lines out of 4 can intersect each combination of 2 lines out of 7 to form a parallelogram. 3 8 11 \$+f=12fz=z ( 5 ) The product of two fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the fractions. . . . . - -Y* 6 - 2 - 6 --6 2.Y 3 6 - y-12 3.~42 [APPENDIX C This will always eliminate the denominators. . . . . (3) The sense of an inequality is reversed if each side is multiplied or divided by the same negative number. . . . . . Generally, banks and other businesses use computers or calculators to get the accuracy they need. . . . . The solutions are (6, -1) and (-2/3,7/3). Hence x2 - 7x + 12 > 0 is satisfied when x > 4 o r x < 3. . . . Page size 150 x 204.75 pts Since the 4 points are collinear, they form 1 line instead of 6 lines. _ - 1- --- 1 - 1 4!3! . But the formula holds for n = 1; hence it holds for n = 1+ 1 = 2. . + k2 + ( k + = k(k + 1)(2k ' ) + ( k + 6 + 368 MATHEMATICAL INDUCTION [CHAP 31 k(k + 1)(2k + 1) + 6(k + 1)* 6 - (k + 1)[(2k2 + k) + (6k + 6)L - l k + l ) ( k + 2)(2k + 3) 6 6 The right hand side of this equation = + which is the value of n(n + 1)(2n + 1)/6 when n is replaced by (k 1). arrangements of the beads on the bracelet, but half of these can be obtained from the other half simply by turning the bracelet over. The equation of the hyperbola is ( x + 1)2 ---4 (y - 5)2 -1 5 (6) The distance from the vertex (1,l) to the center (1, -3) is a, so a = 4. You will mustinclude too much info online in this document to speak what you really are trying to achieve in yourreader. . SOLUTION (a) The first student can select 4 out of 12 books in 12C4 ways. . Sign Up; Schaum's Outline of College Algebra SCHAUM’S OUTLINE OF THEORY AND PROBLEMS of COLLEGE ALGEBRA Second Edition MURRAY R. SPIEGEL, Ph.D. Former Profdsor and. Zero divided by any number # 0 (i.e., not equal to zero) is zero. . . In (1) the transverse axis V V ‘ lies on the x axis, the vertices are V ( a ,0) and V ‘ ( - a , O ) , and the foci are at F(c,O) and F’(-c,O) (see Fig. Number of ways = number of arrangements of 12 pictures taken 4 at a time = 12P4= 12.11- 10.9 = 11 880 ways. . . 24.33 Find the Richter number of an earthquake if its intensity is 3 16OOOO times as great as the reference intensity . If a person invests \$3000 at 6% per year for five years, how much will the investment be worth at the end of the five years? . . . . . . . . . 19.2 PRINCIPLES OF INEQUALITIES (1) The sense of an inequality is unchanged if each side is increased or decreased by the same real number. . The product of the factors (x - 4) and (x + 4) is negative. . Operations with Fractions . Then (1) x2 +y2 = (41)2 and (2) &xy) = 180. m, 1.3 GRAPHICAL REPRESENTATION OF REAL NUMBERS It is often useful to represent real numbers by points on a line. eBook Shop: Schaum's Outline Series: Schaum's Outline of College Algebra, Third Edition von Robert E. Moyer als Download. . Fortunately, there’s Schaum’s. 8 ~ = 2 . SOLUTION The green dyes can be chosen in (Z5- 1) ways, the blue dyes in (Z4- 1) ways, and the red dyes in Z3 ways. 10! . Missed Lectures? . . . (1) Form the partitioned matrix [AII], where A is the given n X n matrix and I is the n x n identity matrix. . Now consider a convex polygon with k + 1 sides. . . . . . . . . . . - - FUNDAMENTAL OPERATIONS WITH NUMBERS 6 z= (g) 756 36 [CHAP. . . Then, since it holds for n = 2, it holds for n = 2 1 = 3, and so on. . or. . Elementary Row Operations . . 295 PERMUTATIONS AND COMBINATIONS CHAP. . . 19.28 If a > 0, a # 1 and n is any positive integer, prove that a"+' 19.29 Show that ~ + 6a" + -. . . . Richter numbers are usually rounded to the nearest tenth or hundredth. . For more information about the software, including the sample screens, see Appendix C on page 385. a, Problem 1.2 Problem 1.15 Problem 1.16 Problem 1.17 Problem 1.19 Problem 1.21 Problem 1.23 Problem 1.25 Problem 1.26 Problem 2.1 Problem 2.2 Problem 2.11 Problem 2.12 Problem 2.13 Problem 2.14 Problem 2.16 Problem 2.17 Problem 3.6 Problem 3.7 Problem 4.3 Problem 4.5 Problem 4.7 Problem 5.13 Problem 5.15 Problem 5.16 Problem 5.19 Problem 6.6 Problem 6.7 Problem 6.8 Problem 6.10 Problem 6.11 Problem 7.1 Problem 7.3 Problem 7.6 Problem 7.19 Problem 7.20 Problem 8.11 Problem 8.12 Problem 8.13 Problem 8.14 Problem 9.6 Problem 9.8 Problem 10.10 Problem 10.12 Problem 10.13 Problem 10.14 Problem 10.15 Problem 11.19 Problem 11.21 Problem 11.29 Problem 11.33 Problem 11.34 Problem 12.29 Problem 12.31 Problem 12.33 Problem 12.34 Problem 12.35 Problem 12.36 Problem 12.39 Problem 12.40 Problem 12.42 Problem 12.43 Problem 12.44 Problem 12.45 Problem 12.47 Problem 12.50 Problem 12.51 Problem 12.56 Problem 13.34 Problem 13.35 Problem 13.39 Problem 13.41 Problem 13.42 Problem 14.10 Problem 14.11 Problem 14.14 Problem 14.15 Problem 14.16 Problem 14.17 Problem 15.26 Problem 15.28 Problem 15.29 Problem 15.33 Problem 15.34 Problem 15.35 Problem 15.36 Problem 16.30 Problem 16.31 Problem 16.32 Problem 16.33 Problem 16.34 Problem 16.35 Problem 16.36 Problem 16.37 Problem 16.40 Problem 16.41 Problem 16.43 Problem 16.45 Problem 16.48 Problem 17.1 Problem 17.12 Problem 17.14 Problem 17.15 Problem 17.18 Problem 17.19 Problem 18.17 Problem 18.18 Problem 18.19 Problem 19.21 Problem 19.23 Problem 19.30 Problem 19.32 Problem 19.34 Problem 19.35 Problem 20.43 Problem 20.44 Problem 20.47 Problem 20.51 Problem 20.55 Problem 20.60 Problem 20.72 Problem 21.5 Problem 21.6 Problem 21.8 Problem 21.9 Problem 22.1 Problem 22.2 Problem 22.58 Problem 22.60 Problem 22.62 Problem 22.72 Problem 22.78 X Problem 22.91 Problem 22.97 Problem 23.26 Problem 23.27 Problem 23.28 Problem 23.29 Problem 23.33 Problem 23.34 Problem 23.37 Problem 23.42 Problem 23.43 Problem 24.22 Problem 24.23 Problem 24.24 Problem 24.27 Problem 24.32 Problem 24.37 Problem 24.42 Problem 24.44 Problem 24.46 Problem 25.49 Problem 25.50 Problem 25.51 Problem 25.69 Problem 25.70 Problem 25.75 Problem 25.82 Problem 25.83 Problem 25.87 Problem 25.92 Problem 26.27 Problem 26.28 Problem 26.30 Problem 27.29 Problem 27.30 Problem 27.31 Problem 27.35 Problem 27.38 Problem 27.39 Problem 27.40 Problem 27.45 Problem 28.11 Problem 28.12 Problem 28.13 Problem 28.14 Problem 28.15 Problem 30.9 Problem 30.11 Problem 30.12 Problem 30.13 Problem 30.14 Problem 31.9 Problem 31.12 Problem 31.17 Problem 31.21 Problem 32.2 Problem 32.3 Problem 32.4 Problem 32.5 Problem 32.8 Problem 32.16 Problem 32.18 Problem 32.21 Chapter 1 Fundamental Operations with Numbers 1.1 FOUR OPERATIONS Four operations are fundamental in algebra, as in arithmetic. Roots . . . . . . . The first and last number n in each row are 1, while any other number in the array can be obtained by adding the two numbers to the right and left of it in the preceding row. x2 - 3x 2 Thus + By division, an improper fraction may always be written as the sum of a polynomial and a proper fraction. . . . . (e) 4(7-6) = 4(42) = 168, (4.7)6 = (28)6 = 168. 25.59 How many three-digit numbers can be formed from the digits 1,2,3,4,5if no digit is repeated in any number? . . We call the line segment between the vertices the major axis and the line segment between the covertices the minor axis. . . SOLUTION For n = 1, n3 + 1 = l3+ 1 = 1 + 1 = 2 and n2 + n = l2+ 1 = 1 + 1 = 2. . . . The book is complete in itself and can be used equally well by those who are studying college algebra for the first time as well as those who wish to review the fundamental principles and procedures of college algebra. . . 6 =I+m’ From (1): x2 + mx2 = 6, From (2): x2 + 5mx2 - 4m2x2= 10, x2 = 10 1 5m - 4m2’ + Then --6 l+m - 10 1+5m-4m2 from which m = I,\$; hence y = x/2, y = x/3. . 3 = a2 + b2, we get 36 = 4 + 62 The equation of the hyperbola is + ( x 5 ) 2 0,- 2)2 ---=I 4 32 Supplementary Problems d l & \$ 17.12 17.13 Graph each of the following equations. . . . 3! 25.79 How many different radio stations can be named with 3 different letters of the alphabet? Supplemental Problem 10.12 f and i Solve for t h e variable: f) J G = 4 i) (y + 1)*=16 SAMPLE SCREENS 392 [APPENDIX C Solution f) J G - 4 To eliminate the root, square both sides. . 3 . . . . Schaum’s Outline of Linear Algebra, Sixth Edition features: X y - 1 0 1 5 8 9 k2.11 +2 k1.89 k1.33 k0.67 0 The graph is a parabola (see Fig. . Consider that A and B put their maps together. . Terms . . . . . . . Total interest = \$20.30. 2 Dies ist ein Skript f¨ur die Vorlesungen Lineare Algebra I und II. Hence the total number of signals is 5 P 1 + 5P2 + 5P3 + 5P4 + 5P5 = 5 + 20 + 60 + 120 + 120 = 325 signals. . Matrix Equations . How many selections of three cards can be made so that ( a ) all three are red, (6) none is red? . . . . . (a) (42.8)(3.26)(8.10) (0.148)(47.6) (b) 284 (1.86)(86.7) (') (2.87)(1.88) 2453 ( d ) (67.2)(8.55) (e) 5608 (0.4536)( 11OOO) (3.92)3(72.16) (f) (8) 2.2500 (i) (h) 0.8003 0') (6) 0.005278. Solve the system of equations: 1 2 -10 5 Oll]-R2-3R1[; [3 XI+ 2x2 -x3 =0 3x1 =1 + 5x2 :I-;] 2 -1 0 -1 3/1]--R 2[; ~R1-2R21 0 5 2 [o 1 -31-11 x1 + 5x3 = 2 and x2 - 3x3 = -1 Thus, x1 = 2 - 5x3 The system has infinitely many solutions of the form (2 - 5x3, -1 and + 3x3, xg), x2 = -1 + 3x3. . . The line segment between the vertices is called the transverse axis. can not, because 70! . . 113 Chapter 14 EQUATIONS OF LINES . . . . . 26.6 + 4.3 - ( 3 ~ ~ ) ~-2b)2 ( + 4.3.2 -(3a3)( 1.2 1-2.3 = 8 1 ~-' 2~ 1 6 ~ + ~ 216a6b2 6 - 96a3b3+ 16b4 + (3a3 - 26)4 = ( 3 ~ ~4 )( 3~~ ~-26) )~( = x7 - 7x6 + 2 1 2 - 3sX4+ 35x3 - 21x2 + 7~ - 1 -26)3 + (-26)4 306 26.8 THE BINOMIAL THEOREM (1 - 2 ~ =)1 +~5(-2X) + 5.4 + -(-zX)2 1.2 = 1 - 1 0 ~40x2 - 80x3 26.9 (z+ 3 Y = (:y + 4( :r( ); [CHAP. The method of proof holds in the general case. Removing factors x and y from 1st and 2nd columns respectively. 30.13 Find the inverse, if it exists, of each matrix. . The numbers are the entries or elements of the matrix. . . CHAP. EXAMPLES 19.2. . PROPERTIES OF NUMBERS . Thus 2x-3 x2 + 5x + 4 4x2 +1 - are proper fractions. . 24.8 Find the compound interest and amount of \$2800 in 8 years at 5% compounded quarterly. . 24.9 A man expects to receive \$2000 in 10 years. . These symbols are called “inequality signs.’’ Thus since 5 is to the right of 3, 5 is greater than 3 o r 5 > 3; we may also say 3 is less than 5 and write 3 < 5 . Use Schaum's to shorten your study time-and get your best test scores! . 25.3 A student has a choice of 5 foreign languages and 4 sciences. There are 4 inversions. Download Full PDF Package. . . Using a Calculator . Download File PDF Schaums Linear Algebra Solutions Will reading dependence put on your life? (r - l ) ! . Solved Problems 25.2 Find n if ( a ) 7*,P3 = 6*n+lP3,(b) 3*,P4 = n-1P5. and y - - I - y--5 4 Simplify each APPENDIX C] SAMPLE SCREENS 393 Check Using Study Works: Remember extraneous roots may be introduced when taking t h e root of both sides. In how many ways can a committee of 3 members be selected so as to include at least 1 doctor? . . 17-21 Fig. 3 - 2= 120 numbers. . . = (n - 2)-(n - l)! ways. If these values satisfy the remaining m - n equations the system is consistent, otherwise it is inconsistent. Schaum's is the key to faster learning and higher grades in every subject. . . . When a matrix has the same number of rows as columns, it is a square matrix. ~ 2 . . 5 ~ + 0 . . SOLUTION One or more ties may be selected in (28 - 1) ways. . . Find the inverse of matrix A = r 21 1 1 -3 -2 4 1 0 0 RZ1 4 3 2 5 1 4 3 0 1 O]-R1[2 5 4 1 -3 -2 0 0 1 1 - 3 - 2 0 0 1 -RZ-2R1 R3 - R1 10 0 -3 -7 -2 1 -2 -1 -510 0 --3R2 11 [: -:-1 1 -1 :] [ 0 1 2/3 - l M U3 0 0 1 0 1/3 4/3 -513 0 1 0 1/3 4/3 -5/3 0 U3 0 1 2/3 -1/3 2/3 1 U3 -1/3 0 1 7 -11 -3 11/3 01 1 - -3R3 0 0 0 -1/3 -7/3 0 - If the matrix A is row equivalent to I, then the matrix A has an inverse and is said to be invertible. Hence & 32.2 h2+10x-3 (x+l)(x2-9) -- A +-+-B x+l x + 3 C x-3 SOLUTION 2r2 + 1 0 -~3 = A(x2 - 9) To find A, l e t x = - 1 : To find B , let x = -3: To find C, let x = 3: 32.3 2-10-3=A(1-9), 18 - 30 - 3 = B(-3 l)(-3 - 3), A = 1118. \$3.99 shipping. If t h e operations yield more solutions, t h e extra solutions are called extraneous and t h e derived equation is said t o be redundant with respect t o t h e original equation. &=-=1 5! . . . . . The associated sign is u3c2d4b1written a3b1c2d4has 2 inversions in the subscripts. SOLUTION We are asked for the present value P which will amount to A = \$2000 in 10 years. . . schaums outline of intermediate algebra Oct 31, 2020 Posted By Harold Robbins Media Publishing TEXT ID 939da521 Online PDF Ebook Epub Library overview it would have to be schaums easy outline series every book in this series is a pared down simplified and tightly focused version of its predecessor with an Schaum's reinforces the main concepts required in your course and offers hundreds of practice questions to help you suceed. 1 Check: 21 -36 = 756 rn 61 = (40 + 21)(72 - 38) --=-- (61)(34) (h) (32 - 15) 17 7 - 61.2 = 122 I Computations in arithmetic, by convention, obey the following rule: Operations of multiplication and division precede operations of addition and subtraction. . . . . . . . (a (a (a (a (a (a (a + x)O + x)1 +x)2 + x)3 + x)4 + x)5 + x)6 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 1 0 5 1 1 6 15 20 15 6 1 etc. . Central hyperbolas have their center at the origin and their vertices and foci on one axis, and are symmetric with respect to the other axis. . . . 1.2 SYSTEM OF REAL NUMBERS The system of real numbers as we know it today is a result of gradual progress, as the following indicates. . . + x2 - 4x 1 (x2+ 1)2(XZ+X+ 1) 32.6 Ax+B x2+ 1 Cx+D (x2+ 1)2 =-+- + x 2E+xx++F 1 FINDING THE PARTIAL FRACTION DECOMPOSITION Once the form of the partial fraction decomposition of a rational fraction has been determined, the next step is to find the system of equations to be solved to get the values of the constants needed in the partial fraction decomposition. . + .. ,+ n(n - 1) - ( n - r + 2) an-r+l r - 1 x +. . Author: Seymour Lipschutz .. 3000 solved problems with complete .3,000 Solved Problems in Linear Algebra - Seymour .Master linear algebra with Schaum's--the high-performance solved-problem guide.. . (a) 7n(n (6) 3n(n - l)(n - 2)(n - 3) = (n - l)(n - 2) (n - 3)(n - 4)(n - 5 ) . The formula is true for n = 1, since 1 =- 1 ( 1 + 1) 2 - 1. . Download with Google Download with Facebook. . . 19.3 ABSOLUTE VALUE INEQUALITIES The absolute value of a quantity represents the distance that the value of the expression is from zero on a number line. To do this, we choose a point on the line to represent the real number zero and call this point the origin. . . . 29.37 Transform the determinant - 2 1 2 3 3 -2 -3 2 1 2 1 2 4 3 -1 -3 into a determinant having three zeros in a row and then evaluate the determinant by use of expansion by minors. . . . . . This formula indicates that the number of selections of r out of n things is the same as the number of selections of n - r out of n things. . . 127 Parallel and Perpendicular Lines . . . . Thus x2 y2 - 3xy + 4x + 4y = 8 is symmetric in x and y . Zero has the property that any number multiplied by zero is zero. Our solutions was introduced by using a wish to function as a comprehensive on-line computerized catalogue that o3ers usage of many PDF … . . SOLVING SYMBOLICALLY 5 (X - 4)=2 has solution(s) 5 (X + I) - 7 SAMPLE SCREENS 390 2*y y b) ---2 3 6 4.y Multiply each side by the common denominator. An infinite number of solutions is obtained by assigning various values to z. . . 7 . . 24.10 What rate of interest compounded annually is the same as the rate of interest of 6% compounded semiannually? . PDF. 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 12.10 Variables . . . The solutions are (2,l) and (-2,-1). . SEYMOUR LIPSCHUTZ MARC LARS LIPSON iii. . His last position was Professor and Chairman of Mathematics at the Rensselaer Polytechnic Institute, Hartford Graduate Center. . . . 29.34 Transform the determinant - 2 4 1 3 1 - 2 2 4 3 1 - 3 2 4 3 -2 -1 into an equal determinant having three zeros in the 3rd column. . . . . . . . A partial list is given . . Assume that the theorem is true for n = k. Then a2k - 62k is divisible by a We must show that a2k+2- b2k+2 is divisible by a a2k+2 - b 2 k + 2 = a2(a2k + b. . 136 15.1 Systems of Two Linear Equations . . . SOLUTION 0.4500 - 0.4447 (0.01) 0.4500 - 0.4447 A = 5000( 1.568) A = \$7840 In N = ln(1.56 + 0.008) In N = In 1.568 N = 1.568 24.16 Find the pH of blood if the concentration of hydrogen ions is 3.98 x 10-8. . I = Prt I = \$800(0.08)(2.5) 1=\$160 EXAMPLE 24.2. . or 2(a2 + b2 + 2 )> 2(ab + bc + ca) (If a = b = c , then a2 + b2 + 2 = ab + bc + ca.) . . = (7-3)!3! . . . . . . . . 25.55 If no two books are alike, in how many ways can 2 red, 3 green, and 4 blue books be arranged on a shelf so that all the books of the same color are together? Fundamental Operations with Algebraic … . 2 Step 2 . Each wind chime requires 3 hours of work from Jean and 1 hour of work from Wesley. . . He is one of Schaum’s most prolific authors. . . . = n.(n - I)! . . SOLUTION The biology books can be arranged among themselves in 6! . . The minimum for C(x,y), if it exists, will occur at point A, B, C, or D,so we evaluate the obective function at these points. 3x -5 x2 - 3x +2 - -5 1 =-+-2 ( x - l)(x - 2) x - 1 x - 2' 3x 32.4 IDENTICALLY EQUAL POLYNOMIALS If two polynomials of degree n in the same variable x are equal for more than n values of x , the coefficients of like powers of x are equal and the two polynomials are identically equal. . RATIO. 171 17.10 For each hyperbola, write the equation in standard form and determine the center, vertices, and foci. From a2 = b2 + 2 , we get 36 = b2 + 9 and b2 = 27. . . . . Solving (1) and (2) simultaneously, the required sides are 12 and 18ft. . . . . The solution region of 2x + y > 3 and x - 2y I-1 is the shaded region of Fig. . . 25.39 An organization has 25 members, 4 of whom are doctors. Numbers formed = 9 - 9P3 = 9(9 - 8 7) = 4536 numbers. . . There is 1 inversion. a 2 b means “a is greater than or equal to b.” a Ib means “a is less than or equal to b.” O < a < 2 means “a is greater than zero but less than 2.” -2 5 x < 2 means “ x is greater than or equal to -2 but less than 2.” A n absolute inequality is true for all real values of the letters involved. . 2 3 [6 0 [ [CHAP 30 -:] by -2. Solve the system (1) x 2 + x y = 6 + ( 2 ) x2 5xy - 4y2 = 10 Method 1. . 3 2.3-4 3-4.5 31.17 a" - 6" is divisible by a - 6, for n = positive integer. + 9y2 = 36, y2 = 5 3. . (a) vertices (+4,0), foci (+2%5,0) ( 6 ) covertices (+3,0), major axis length 10 center (-3,2), vertex (2,2), c = 4 (d) vertices (3,2) and (3, -6), covertices (1, -2) and (5, -2) (c) & & 17.18 Write the equation of the ellipse in standard form and determine the center, vertices, foci, and covertices. ways, the physics books in 2! . Proportion Variation . 26.1 Combinatorial Notation 26.2 Expansion of ( a + x ) ~. It is interesting to note that these coefficients may be arranged as follows. . Hence the required number of ways = 24 - 1 = 16 - 1 = 15 ways. Hence the required number of ways = 4C1 + 4C2+ 4C3+ 4C4= 4 + 6 + 4 + 1 = 15 ways. . . . How many times the hearing threshold intensity is the intensity of a relatively quiet room? Thus y-x 2 - 5 x + 2 may be written f(x)= x2 - 5 x + 2 . . . Division may be defined in terms of multiplication. (U) (U) Ix-31>4 Solve each of these inequalities for x . . (a) vertex at origin and directrix y = 2 ( b ) vertex (-1, -3) and focus (-3, -3) SOLUTION ( a ) Since the vertex is at the origin, we have the form y 2 = 4px or x2 = 4py. * 170! . Combinations of n different things taken r at a time ,c, = nf'r = r! . 4 = 64.4 = 256 (k) Hence if one wrote 128 + 2 . = u2 b2, we get c = 5 . . . Hence, x - a is between b units below 0, -b, and b units above 0. . . . . . The solution proceeds as in Method 1. . . 5 . . How much is that money worth now considering interest at 6% compounded quarterly? This last statement is true since the square of any real number different from zero is positive. . . a, 0.3782, ~, - 5 , 315, 3 ~ 2,, -114, 6.3, 0, fi, -1817 SOLUTION If the number belongs to one or more categories this is indicated by a check mark. . . . . Negative Integral Exponent . . . . (6) A = '-1 4 3' -5 0 2. and B = [i :] SOLUTION ( a ) AB = ; not possible. SOLUTION Each selection of 4 out of 6 chemists can be associated with each selection of 3 out of 5 biologists. . 5 ' 4 . . Hence the required number of ways = 10- 10 = 100 ways. . . Write the equation of the ellipse 18x2+ 12y2- 144x + 48y + 120 = 0 in standard form. . There are 5 inversions. . . . The diagonal also forms a k-sided polygon with the other sides of the original polygon. . added addition algebraically angles answer axis base binomial Check circle coefficient Combine common Complete congruent constant CONTAINING coordinates decimal … . . . . . . . . 4 = 64.4 = 256. . Fortunately, there’s Schaum’s. . . 17-15(b)). . . 25.43 A has 3 maps and B has 9 maps. . . Since n # 0 , l we may divide by n(n - 1) to obtain 7(n - 2) = 6(n + l), n = 20. + where Al,A2,. for positive integers n. SOLUTION Step I. Download Free PDF. EXAMPLES 1.2. . . 25.62 How many odd numbers of three digits each can be formed, without the repetition of any digit in a number, from the digits ( a ) 1,2,3,4,(6) 1,2,4,6,8? . 2! . . . . . central ellipse, foci at ( t 4 , 0 ) , and vertices at ( t 5 , O ) ( b ) center at (0,3), major axis of length 12, foci at (0,6) and (0,O) (a) A central ellipse has it center at the origin, so ( h , k ) = (0,O). . SOLUTION Method 1. . . . . . 9 A=[:-:] B = [ l3 112 5 3 -1 ] C=[:-5'2 2 "1 -3 Perform the indicated operations, if possible. 261 305 Expand by the binomial formula. 379 Appendix B TABLE OF NATURAL LOGARITHMS . . 29.15 Write the minor and corresponding cofactor of the element in the second row, third column for the determinant [CHAP 29 DETERMINANTS OF ORDER n 344 SOLUTION Crossing out the row and column containing the element, the minor is given by Since the element is in the 2nd row, 3rd column and 2 + 3 = 5 is an odd number, the associated sign is minus. (These words need not have meaning.) . What is the Richter number of the Iranian earthquake? The negative number -3/2 or -1; is represented by a point R 14 units to the left of the origin. . SOLUTION a2 19.8 + 3 >2ac and b2 + d2 >2bd; hence by addition (a2 + b2)+ (c2+ d2) > 2ac + 2bd or 2 > 2ac + 2bd, i.e., 1 > ac + bd. SOLUTION Each appliance may be dealt with in 2 ways, as it can be chosen or not chosen. . . . . . . If two such numbers are added or multiplied, the result is always a natural number. . (i) The rule of (i) is applied here. . . . . . . . . . . SOLUTION Amount from principal P in 1 year at rate r = P(l + r). Finding Logarithms Using a Calculator . . . . . . . . . Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. . (a) 7! . y=x- ( a ) Hyperbola 1 (b)Two intersecting lines Fig. You can download the paper by clicking the button above. Thus we may add the numbers a , b , c, d , e by grouping ~in any order, as (a + b ) c + ( d + e ) , a + ( b c ) ( d + e), etc. . . . . Linear algebra is also used in most sciences and engineering areas, because it allows modeling many natural phenomena, and efficiently computing with such models. Solve the system (1) 2 r 2 - y 2 = 7 (2) 3x2 + 2y2 = 14 To eliminate y, multiply (1) by 2 and add to (2); then 7x2= 28, and x2 = 4 x = +2. . . . FUNCTIONS AND GRAPHS . 3 - 2 . . . . Number of ways = 12C4 - &4 + 1 = 495.70 1 = 34 650 ways. 26 n! . . - (c) The first place may be filled in 9 ways, and the last place in 5 ways (0,2,4,6,8). . . . . . . . . (3*4)! . 25.96 A box contains 7 red cards, 6 white cards and 4 blue cards. . . . 18-1. . . Subtracting (5) from (4), we have u2 - 4u - 45 = 0 , (U - 9)(u + 5) = 0 and U = 9, -5. . . . . 4+(-2)]=[: -1+2 + :] The matrix -A is called the opposite of matrix A and each entry in -A is the opposite of the corresponding entry in A. Thus 9C4denotes the number of combinations of 9 things taken 4 at a time. Symmetry . 29.38 Evaluate each determinant. . . . . 12x+11 x2+x-6 x+2 2-7x+12 32.9 32.11 +4 2+2x 32.12 32.14 x2 - 9~ - 6 X3+x2-6X 32.15 x3 x2 - 4 32.17 3x3 10x2 27x x2(x 3)2 32.18 5x2 8x 21 (x2 x 6)(x 1) 32.19 5x3+ 4x2 7x 3 (2 2x 2)(x2 - x - 1) 32.20 3x 2-1 + + + + 32*8 32.23 5x + + + + 27 2 - 3 ~ - 18 + 8-x 2x2+3x-2 32.13 1ox2 9x - 7 (x 2)(x2 - 1) =A 32,16 M- + + + + + & + + 3x2-&+9 (x - 2)3 + + + + 32.21 7x3 16x2 20x 5 (x2 2x 2)2 32.22 7x - 9 (x + l)(x - 3) - 2)(x + 2) 32.24 3x - 1 x2- 1 32.25 7x - 2 x3 - x2 - 2x + +1 + 1) -2x+9 (2x + 1)(4x2+ 9) 32.28 2x3-x+3 (x2 + 4)(x2 1) x + 10 x(x X 32.10 32.26 5x2 3x (x 2 ) ( 2 32.27 32.29 x3 - x4+3x2+x+1 32*30 (x 1)(x2 1)2 + + (x2 + 4)2 + + ANSWERS TO SUPPLEMENTARY PROBLEMS 5 32.8 6 5 -x-4 x-3 32.9 +-x + 3 x-2 32.10 3 2 -- 32.11 2 3 -+x x+2 32.12 23 -+X-6 113 ~ + 3 32.13 3 +-+-2 x + l x-1 32.14 2 2 -1 - +- 2 32.15 x+-+x-2 2 x+2 32.16 4 5 3 ++- 32.17 2 :+?+--x x2 x + 3 32.18 x x-2 x+3 5 (x+3)2 7 2x+3 x2+x+6 +-x +3 l 2x-1 x-2 x+2 (x-2)2 2x-1 32*19 x2+2x+2 5 x+2 (x-2)3 1 + x 23x+ -x-1 378 PARTIAL FRACTIONS + 32.20 -x-1 l + x-1 2+x+1 7x 2 2X+l 32*21 2 + 2 X + 2 + (x2+2x+2)2 32.23 -5/2 -+-+x 3/2 x-2 32.24 1 -+x-1 32.26 3 -+x+2 2x-1 x2+l 32.27 1 -+2x+l 32.29 X -4x -+ x2+4 (2+4)2 32.30 1 x+ 1 - 1 x+2 -+ 2 x+l -2x 4x2+1 X (x2+ 1)2 [CHAP 32 32.22 4 3 -+ x+l x-3 32.25 1 2 -+-+x x-2 32.28 3 ~ - 1 -+x2+4 -3 x+l -x+1 2 + 1 Appendix A Table of Common Logarithms ~~ N 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 m 0414 0792 1139 1461 0043 0453 0828 1173 1492 0086 0492 0864 1206 1523 0128 0531 0899 1239 1553 0170 0569 0934 1271 1584 0212 0607 0969 1303 1614 0253 0645 1004 1335 1644 0294 0682 1038 1367 1673 0334 0719 1072 1399 1703 0374 0755 1106 1430 1732 15 16 17 18 19 1761 2041 2304 2553 2788 1790 2068 2330 2577 2810 1818 2095 2355 2601 2833 1847 2122 2380 2625 2856 1875 2148 2405 2648 2878 1903 2175 2430 2672 2900 1931 2201 2455 2695 2923 1959 2227 2480 2718 2945 1987 2253 2504 2742 2967 2014 2279 2529 2765 2989 20 21 22 23 24 3010 3222 3424 3617 3802 3032 3243 3444 3636 3820 3054 3263 3464 3655 3838 3075 3284 3483 3674 3856 3096 3304 3502 3692 3874 3118 3324 3522 3711 3892 3139 3345 3541 3729 3909 3160 3365 3560 3747 3927 3181 3385 3579 3766 3945 3201 3404 3598 3784 3962 25 26 27 28 29 3979 4150 4314 4472 4624 3997 4166 4330 4487 4639 4014 4183 4346 4502 4654 4031 4200 4362 4518 4669 4048 4216 4378 4533 4683 4065 4232 4393 4548 4698 4082 4249 4409 4564 4713 4099 4265 4425 4579 4728 41 16 4281 4440 4594 4742 4133 4298 4456 4609 4757 30 31 32 33 34 4771 4914 5051 5185 5315 4786 4928 5065 5198 5328 4800 4942 5079 5211 5340 4814 4955 5092 5224 5353 4829 4969 5105 5237 5366 4843 4983 5119 5250 5378 4857 4997 5132 5263 5391 487 1 5011 5145 5276 5403 4886 5024 5159 5289 5416 4900 5038 5172 5302 5428 35 36 37 38 39 5441 5563 5682 5798 5911 5453 5575 5694 5809 5922 5465 5587 5705 5821 5933 5478 5599 5717 5832 5944 5490 5611 5729 5843 5955 5502 5623 5740 5855 5966 5514 5635 5752 5866 5977 5527 5647 5763 5877 5988 5539 5658 5775 5888 5999 555 1 5670 5786 5899 6010 40 41 42 43 44 6021 6128 6232 6335 6435 6031 6138 6243 6345 6444 6042 6149 6253 6355 6454 6053 6160 6263 6365 6464 6064 6170 6274 6375 6474 6075 6180 6284 6385 6484 6085 6191 6294 6395 6493 6096 6201 6304 6405 6503 6107 6212 6314 6415 6513 6117 6222 6325 6425 6522 N 0 1 2 3 4 5 6 7 8 9 - 379 380 TABLE OF COMMON LOGARITHMS [APPENDIX A ~ N 0 1 2 3- 4 5 6 7 8 9 45 46 47 6532 6628 672 1 6812 6902 6542 6637 6730 6821 6911 6551 6646 6739 6830 6920 6561 6656 6749 6839 6928 6571 6665 6758 6848 6937 6580 6675 6767 6857 6946 6590 !%84 6776 6866 6955 6599 6693 6785 6875 6964 6609 6702 6794 6884 6972 6618 6712 6803 6893 6981 54 6990 7076 7160 7243 7324 6998 7084 7168 725 1 7332 7007 7093 7177 7259 7340 7016 7101 7185 7267 7348 7024 7110 7183 7275 7356- 7033 7118 7202 7284 7364 7042 7126 7210 7292 7372 7050 7135 7218 7300 7380 7059 7143 7224 7308 7388 7067 7152 7235 7316 7396 55 56 57 58 59 7404 7482 7559 7634 7709 7412 7490 7566 7642 7716 7419 7497 7574 7649 7723 7427 7505 7582 7657 7731 743s 7513 7589 7664 7738 7443 7520 7597 7672 7745 7451 7528 7604 7679 7752 7459 7536 7612 7686 7760 7466 7543 76 19 7694 7767 7474 755i 7627 7701 7774 60 61 62 63 64 7782 7853 7924 7993 8062 7789 7860 7931 8OOO 8069 7796 7868 7938 8007 8075 7803 7875 7945 8014 8082 7810 7882 7952 8021 8085- 7818 7889 7959 8028 8096 7825 7896 7966 8035 8102 7832 7903 7973 8041 8109 7839 7910 7980 8048 8116 7846 7917 7987 8055 8122 65 66 67 68 69 8129 8195 8261 8325 8388 8136 8202 8267 8331 8395 8142 8209 8274 8338 8401 8149 8215 8280 8344 8407 81% 8222 8287 8351 8414 8162 8228 8293 8357 8420 8169 8235 8299 8363 8426 8176 8241 8306 8370 8432 8182 8248 8312 8376 8439 8189 8254 8319 8382 8445 70 72 73 74 845i 8513 8573 8633 8692 8457 8519 5579 8639 8698 8463 8525 8585 8645 8704 8470 8531 8591 8651 8710 8475 8537 8597 8657 8716 8482 8543 8603 8663 8722 8488 8549 8609 8669 8727 8494 8555 8615 8675 8733 8500 8561 8621 8681 8739 8506 8567 8627 8686 8745 75 76 77 78 79 875i8808 8865 8921 8976 8756 8814 8871 8927 8982 8762 8820 8876 8932 8987 8768 8825 8882 8938 8993 8774 8831 8887 8943 8998 8779 8837 8893 8949 9004 8785 8842 8899 8954 9009 8791 8848 8904 8960 9015 8797 8854 8910 8965 9 0 8802 8859 8915 8971 9025 80 82 83 84 9031 9085 9138 9191 9243 W36 9090 9143 9196 9248 9042 9096 9149 9201 9253 9047 9101 9154 92% 9258 9053 9106 9159 9212 9263 9058 9112 9165 9217 9269 9063 9117 9170 9222 9274 9069 9122 9175 9227 9279 9074 9128 9180 9232 9284 9079 9133 9186 9238 9289 N 0 1 2 3 4 5 6 7 8 9 48 49 50 51 52 c3 JJ 71 Q1 v1 ~ ~ 38 1 TABLE OF COMMON LOGARITHMS APPENDIX A] N 0 1 2 3 4 5 6 7 8 9 85 86 87 88 89 9294 9345 9395 9445 9494 9299 9350 9400 9450 9499 9304 9355 9405 9455 9504 9309 9360 9410 9460 9509 9315 9365 9415 9465 9513 9320 9370 9420 9469 9518 9325 9375 9425 9474 9523 9330 9380 9430 9479 9528 9335 9385 9435 9484 9533 9340 9390 9440 9489 9538 90 9542 9590 9638 9685 9731 9547 9595 9643 9689 9736 9552 9600 9647 9694 9741 9557 9605 9652 9699 9745 9562 9609 9657 9703 9750 9566 9614 9661 9708 9754 9571 9619 9666 9713 9759 9576 9624 9671 9717 9763 9581 9675 9722 9768 9586 9633 9680 9727 9773 98 99 9777 9823 9868 9912 9956 9782 9827 9872 9917 9961 9786 9832 9877 9921 9965 9791 9836 9881 9926 9969 9795 9841 9886 9930 9974 9800 9845 9890 9934 9978 9805 9850 9894 9939 9983 9809 9854 9899 9943 9987 9814 9859 9903 9948 9991 9818 9863 9908 9952 9996 N 0 1 2 3 4 91 92 93 94 95 96 97 %28 Appendix B N 1.o 1.1 1.2 1.3 1.4 Table of Natural Logarithms 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 ~~~~ O.oo00 0.0953 0.1823 0.2624 0.3365 0.0100 0.1044 0.1906 0.2700 0.3436 0.0198 0.1133 0.1989 0.2776 0.3507 0.0296 0.1222 0.2070 0.2852 0.3577 0.0392 0.1310 0.2151 0.2927 0.3646 0.0488 0.1398 0.2231 0.3001 0.3716 0.0583 0.1484 0.2311 0.3075 0.3784 0.0677 0.1570 0.2390 0.3148 0.3853 0.0770 0.1655 0.2469 0.3221 0.3920 0.0862 0.1740 0.2546 0.3293 0.3988 1.5 1.6 1.7 1.8 1.9 0.4055 0,4700 0.5306 0.5878 0.6419 0.4121 0.4762 0.5365 0.5933 0.6471 0.4187 0.4824 0.5423 0.5988 0.6523 0.4253 0.4886 0.5481 0.6043 0.6575 0.4318 0.4947 0.5539 0.6098 0.6627 0.4383 0.5008 0.5596 0.6152 0.6678 0.4447 0.5068 0.5653 0.6206 0.6729 0.4511 0.5128 0.5710 0.6259 0.6780 0.4574 0.5188 0.5766 0.6313 0.6831 0.4637 0.5247 0.5822 0.6366 0.6881 2.0 2.1 2.2 2.3 2.4 0.6931 0.7419 0.7885 0.8329 0.8755 0.6981 0.7467 0.7930 0.8372 0.8796 0.7031 0.7514 0.7975 0.8416 0.8838 0.7080 0.7561 0.8020 0.8459 0.8879 0.7130 0.7608 0.8065 0.8502 0.8920 0.7178 0.7655 0.8109 0.8544 0.8961 0.7227 0.7701 0.8154 0.8587 0.902 0.7275 0.7747 0.8198 0.8629 0.9042 0.7324 0.7793 0.8242 0.8671 0.9083 0.7372 0.7839 0.8286 0.8713 0.9123 2.5 2.6 2.7 2.8 2.9 0.9163 0.9555 0.9933 1.0296 1.0647 0.9203 0.9594 0.9969 1.0332 1.0682 0.9243 0.9632 1.o006 1.0367 1.0716 0.9282 0.9670 1.0043 1.MO3 1.0750 0.9322 0.9708 1.OO80 1.0438 1.0784 0.9361 0.9746 1.0116 1.0473 1.0818 0.9400 0.9783 1.0152 1.0508 1.0852 0.9439 0.9821 1.0188 1.0543 1.0886 0.9478 0.9858 1.0225 1.OS78 1.0919 0.9517 0.9895 1.0260 1.0613 1.0953 3.0 3.1 3.2 3.3 3.4 1.0986 1.1314 1.1632 1.1939 1.2238 1.1019 1.1346 1.1663 1.1970 1.2267 1.1053 1.1378 1.1694 1.2000 1.2296 1.1086 1.1410 1.1725 1.2030 1.2326 1.1119 1.1442 1.1756 1.2060 1.2355 1.1151 1.1474 1.1787 1.2090 1.2384 1.1184 1.1506 1.1817 1.2119 1.2413 1.1217 1.1537 1.1848 1.2149 1.2442 1.1249 1.1569 1.1878 1.2179 1.2470 1.1282 1.1600 1.1909 1.2208 1.2499 3.5 3.6 3.7 3.8 3.9 1.2528 1.2809 1.3083 1.3350 1.3610 1.2556 1.2837 1.3110 1.3376 1.3635 1.2585 1.2865 1.3137 1.3403 1.3661 1.2613 1.2892 1.3164 1.3429 1.3686 1.2641 1.2920 1.3191 1.3455 1.3712 1.2669 1.2947 1.3218 1.3481 1.3737 1.2698 1.2975 1.3244 1.3507 1.3762 1.2726 1.3002 1.3271 1.3533 1.3788 1.2754 1.3029 1.3297 1.3558 1.3813 1.2782 1.3056 1.3324 1.3584 1.3838 4.0 4.1 4.2 4.3 4.4 1.3863 1.4110 1.4351 1.4586 1.4816 1.3888 1.4134 1.4375 1.4609 1.4839 1.3913 1.4159 1.4398 1.4633 1.4861 1.3938 1.4183 1.4422 1.4656 1.4884 1.3962 1.4207 1.4446 1.4679 1.4907 1.3987 1.4231 1.4469 1.4702 1.4929 1.4012 1.4255 1.4493 1.4725 1.4952 1.4036 1.4279 1.4516 1.4748 1.4974 1.4061 1.4303 1.4540 1.4770 1.4996 1.4085 1.4327 1.4563 1.4793 1.5019 N - 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 382 APPENDIX B] 383 TABLE OF NATURAL LOGARITHMS N 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 4.5 4.6 4.7 4.8 4.9 1.5041 1.5261 1.5476 1.5686 1.5892 1.5063 1.5282 1.5497 1.5707 1.5913 1.5085 1.5304 1.5518 1.5728 1.5933 1.5107 1.5326 1S539 1.5748 1.5953 1.5129 1.5347 1.5560 1.5769 1.5974 1.5151 1.5369 1.5581 1.5790 1S994 1.5173 1.5390 1.5602 1.5810 1.6014 1.5195 1.5412 1.5623 1.5831 1.6034 1.5217 1S433 1S644 1.5851 1.6054 1.5239 1.5454 1S665 1.5872 1.6074 5.0 5.1 5.2 5.3 5.4 1.6094 1.6292 1.6487 1.6677 1.6864 1.6114 1.6312 1.6506 1.6696 1.6882 1.6134 1.6332 1.6525 1.6715 1.6901 1.6154 1.6351 1.6544 1.6734 1.6919 1.6174 1.6371 1.6563 1.6752 1.6938 1.6194 1.6390 1.6582 1.6771 1.6956 1.6214 1.6409 1.6601 1.6790 1.6974 1.6233 1. a 2 9 1.6620 1.6808 1.6993 1.6253 1.6448 1.6639 1.6827 1.701.1 1.6273 1.U67 1.6658 1.6845 1.7029 5.5 5.6 5.7 5.8 5.9 1.7047 1.7228 1.7405 1.7579 1.7750 1.7066 1.7246 1.7422 1.7596 1.7766 1.7084 1.7263 1.7440 1.7613 1.7783 1.7102 1.7281 1.7457 1.7630 1.7800 1.7120 1.7299 1.7475 1.7647 1.7817 1.7138 1.7317 1.7492 1.7664 1.7834 1.7156 1.7334 1.7509 1.7682 1.7851 1.7174 1.7352 1.7527 1.7699 1.7867 1.7192 1.7370 1.7544 1.7716 1.7884 1.7210 1.7387 1.7561 1.7733 1.7901 6.0 6.1 6.2 6.3 6.4 1.7918 1.8083 1.8245 1.8406 1.8563 1.7934 1.8099 1.8262 1.8421 1.8579 1.7951 1.8116 1.8278 1.8437 1.8594 1.7967 1.8132 1.8294 1.8453 1.8610 1.7984 1.8148 1.8310 1.8469 1.8625 1.8001 1.8165 1.8326 1.8485 1.8641 1.8017 1.8181 1.8342 1.8500 1.8656 1.8034 1.8197 1.8358 1.8516 1.8672 1.8050 1.8213 1.8374 1.8532 1.8687 1.8066 1.8229 1.8390 1.8547 1.8703 6.5 6.6 6.7 6.8 6.9 1.8718 1.8871 1.9021 1.9169 1.9315 1.8733 1.8886 1.9036 1.9184 1.9330 1.8749 1.8901 1.9051 1.9199 1.9344 1.8764 1.8916 1.9066 1.9213 1.9359 1.8779 1.8931 1.9081 1.9228 1.9373 1.8795 1.8946 1.9095 1.9242 1.9387 1.8810 1.8961 1.9110 1.9257 1.9402 1.8825 1.8976 1.9125 1.9272 1.9416 1.8840 1.8991 1.9140 1.9286 1.9430 1.8856 1.9006 1.9155 1.9301 1.9445 7.0 7.1 7.2 7.3 7.4 1.9459 1.9601 1.9741 1.9879 2.0015 1.9473 1.9615 1.9755 1.9892 2.0028 1.9488 1.9629 1.9769 1.9906 2.0042 1.9502 1.9643 1.9782 1.9920 2.0055 1.9516 1.9657 1.9796 1.9933 2.0069 1.9530 1.9671 1.9810 1.9947 2.0082 1.9544 1.9685 1.9824 1.9961 2.0096 1.9559 1.9699 1.9838 1.9974 2.0109 1.9573 1.9713 1.9851 1.9988 2.0122 1.9587 1.9727 1.9865 2.0001 2.0136 7.5 7.6 7.7 7.8 7.9 2.0149 2.0282 2.0412 2.0541 2.0669 2.0162 2.0295 2.0425 2.0554 2.0681 2.0176 2.0308 2.0438 2.0567 2.0694 2.0189 2.0321 2.0451 2.0580 2.0707 2.0202 2.0334 2.0464 2.0592 2.0719 2.0215 2.0347 2.0477 2.0605 2.0732 2.0229 2.0360 2.0490 2.0618 2.0744 2.0242 2.0373 2.0503 2.0631 2.0757 2.0255 2.0386 2.0516 2.0643 2.0769 2.0268 2.0399 2.0528 2.0665 2.0782 8.0 8.1 8.2 8.3 8.4 2.0794 2.0919 2.1041 2.1163 2.1282 2.0807 2.0931 2.1054 2.1175 2.1294 2.0819 2.0943 2.1066 2.1187 2.1306 2.0832 2.0956 2.1078 2.1199 2.1318 2.0844 2.0968 2.1090 2.1211 2.1330 2.0857 2.0980 2.1102 2.1223 2.1342 2.0869 2.0992 2.1114 2.1235 2.1353 2.0882 2.1005 2.1126 2.1247 2.1365 2.0894 2.1017 2.1138 2.1258 2.1377 2.0906 2.1029 2.1150 2.1270 2.1389 N 0.00 0.01 0.04 0.05 0.06 0.07 0.08 0.09 ~ - ~~ ~ 0.02 ~~ 0.03 384 TABLE OF NATURAL LOGARITHMS [APPENDIX B N 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 8.5 8.6 8.7 8.8 8.9 2.1401 2.1518 2.1633 2.1748 2.1861 2.1412 2.1529 2.1645 2.1759 2.1872 2.1424 2.1541 2.1656 2.1770 2.1883 2.1436 2.1552 2.1668 2.1782 2.1894 2.1448 2.1564 2.1679 2.1793 2.1905 2.1459 2.1576 2.1691 2.1804 2.1917 2.1471 2.1587 2.1702 2.1815 2.1928 2.1483 2.1599 2.1713 2.1827 2.1939 2.1494 2.1610 2.1725 2.1838 2.1950 2.1506 2.1622 2.1736 2.1849 2.1961 9.0 9.1 9.2 9.3 9.4 2.1972 2.2083 2.2192 2.2300 2.2407 2.1983 2.2094 2.2203 2.2311 2.2418 2.1994 2.2105 2.2214 2.2322 2.2428 2.2006 2.2116 2.2225 2.2332 2.2439 2.2017 2.2127 2.2235 2.2343 2.2450 2.2028 2.2138 2.2246 2.2354 2.2460 2.2039 2.2148 2.2257 2.2364 2.2471 2.2050 2.2159 2.2268 2.2375 2.2481 2.2061 2.2170 2.2279 2.2386 2.2492 2.2072 2.2181 2.2289 2.23% 2.2502 9.5 9.6 9.7 9.8 9.9 2.2513 2.2618 2.2721 2.2824 2.2925 2.2523 2.2628 2.2732 2.2834 2.2935 2.2534 2.2638 2.2742 2.2844 2.2946 2.2544 2.2649 2.2752 2.2854 2.2956 2.2555 2.2659 2.2762 2.2865 2.2966 2.2565 2.2670 2.2773 2.2875 2.2976 2.2576 2.2680 2.2783 2.2885 2.2986 2.2586 2.2690 2.2793 2.2895 2.2996 2.2597 2.2701 2.2803 2.2905 2.3006 2.2607 2.271 1 2.2814 2.2915 2.3016 N - 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 If N 2 10, In 10 = 2.3026 and write N in scientific notation; then use In N = ln[k - (lo"')]= Ink + m In 10 = Ink + m (2.3026), where 1 Ik < 10 and m is an integer. 127 Slope of a matrix has the solution region of Fig interpolation results in schaum's algebra pdf classroom on! 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