A numerical method for solving the equations of stability of general slip surfaces

Size: px
Start display at page:

Download "A numerical method for solving the equations of stability of general slip surfaces"

Transcription

1 A numerical method for solving the equations of stability of general slip surfaces By N. R. Morgenstern* V. E. Pricef 1. Introduction A method has been developed by Morgenstern Price (1965) for analyzing the stability of earth slopes with regard to slips along general surfaces which can be considered as a sequence of planes. The method is believed to be the first in which all the equations of statical equilibrium are satisfied. It has been used satisfactorily to determine factors of safety for several thous slip surfaces on more than forty different cross-sections of slopes. A set of equations was derived which determine the factor of safety F, an overall force ratio scaling constant A, the internal forces. In this paper we describe the numerical technique for solving these equations, which has been used in a program for a digital computer. The equations are equivalent to a pair of non-linear simultaneous equations in the unknowns A F, the principles of the method of solving them may be applied to any set of non-linear simultaneous equations, when the first derivatives with respect to the unknowns can be evaluated. Many of the preliminary experiments were performed on the Deuce computer using the Alphacode language, but most of the numerical results have been obtained on an I.C.T using a program which was written in a dialect of FORTRAN similar to FORTRAN II. The program is at present being converted to FORTRAN as specified by the American Stards Association. In 2 there is a summary of the equations which are to be solved, in 3 the principles of the numerical method of solving the equations are described, the detailed formulae being given in 4. It is necessary to evaluate some integrals which involve some slight numerical complexity, the method for these is described in 5. A practical method of solving the equations of statical equilibrium to obtain the factor of safety of earth slopes, with slip surfaces of arbitrary shape, is described. The basic mathematical problem is the solution of a pair of. simultaneous non-linear equations which arise from the boundary conditions on the solution of an ordinary differential equation. Fig. 1. L V=Vt(l) y = y(x) Potential sliding mass shearing forcej respectively in a general vertical plane with coordinate x, which intersects the slip surface at (x, y) the resultant E at (x, y,) then L=Xn Zx + L)E) - X= XfE F de A tan 4>' F~. tan. tan (1) (2) (3) (4) (5) (6) 2. Summary of the equations We consider a cross-section of the slope of unit thickness, take axes horizontal such that sliding tends to occu: ill the negative x direction Oy vertically downwa. ds, the origin being arbitrary (see Fig. 1). If E X denote the resultant total normal force (7) t The most recent version of the program works in terms of effective E X forces also incorporates earthquake loading represented by a horizontal body force expressed as some percentage of gravity. These differences only complicate the coefficients in equation (2) in no way alter the material presented here. * Lecturer in Civil Engineering, Imperial College of Science Technology, Exhibition Road, London, S.W. 1. t Reader in Numerical Analysis, The City University, St. John St., London, E.C.I. 388

2 Slip surfaces / = kx + m (8) - (9) Hence dx The unknowns in these equations are the constants A F, the functions E, X y t. All the other quantities are known functions of x depending on the chosen slip surface, the geometry of the slope, the pore pressures in the soil mass the properties of the types of soils from which it is constructed. Equation (3) is an assumption which is made concerning the internal forces which makes the problem statically determinate. A discussion of the choice of the function f(x), which must be specified, was given by Morgenstern Price (1965). By assuming that the boundaries of the different soil types are linear, that f{x) may be approximated by a sequence of linear functions, that the slip surface consists of a sequence of planes, then the potential sliding mass may be divided into n slices by vertical planes with coordinates x 0, x u..., x n such that in any slice K, L, N P are constants. We do not assume that any of these slices are necessarily thin. The boundary conditions to be satisfied are normally: E = = 0 E = = 0 M = 0 M-- For any values of A F, equation (2) may be integrated to determine E, starting with E = 0 at x 0, then X may be determined from equation (3) y, from equation (13). Thus the values of E(x n ) M(x n ) may be regarded as functions of A F, we wish to determine those values of A F which satisfy equations (11) (12). 3. Solution of the equations The basis of the technique for solving the equations is the Newton-Raphson method together with certain controls on the values of A F used. Suppose that E n {\, F) M n {\, F) denote the values of E M when x = x n. Starting with values A o F o, suppose the required solution is A o -f SA.F 0 + 8F. If 8A 8F are small, then, approximately, E n {\ 0 + 8X, F o + 8F) = E n {\, F o )+ SA^" + SA, F o + 8F) = M n (\ 0, F o ) + SA-^- when x x 0 when x = x n when x = x n the derivatives are evaluated at (A o, F o ). (10) (11) (12) (13) <^-" = 0 (14) ^F=0 05) 389 SA = SF = DA IF DF DA F " DA DA DF DF DA (16) (17) Thus starting with estimated or arbitrary values of A F, by evaluating E n, M n their derivatives with respect to A F, new estimates may be found, so on, until the changes in A F are sufficiently small. However, if this iterative process is used without additional controls to the values of A F, then even with reasonable initial approximations the process will in many cases either not converge, ox converge to give values of A F which are physically unacceptable. There are three additional methods of controlling the values of A F as follows. (a) If (L + Kx) is zero for any value of x, then it may be seen from equation (2) that E is in general infinite at that point, which is physically unacceptable. This occurs if (L + Kx) changes sign within any slice. In addition, it is reasonable to expect that (L + Kx) will not change sign at any of the points x u x 2,..., x n _i it is discontinuous, since any discontinuity in the soil characteristics, or the slope of a line, may be considered as the limit of a rapidly varying continuous quantity. Thus (Z, + Kx) should be of the same sign for the complete range of x. It has been found in practice that this function should be positive, because the contribution of 1 to L in equation (5) is often the dominant term. Thus we define a feasible region for (A, F) such that for all points in this region the value of (Z, + Kx) is positive for all x. By restricting the values of (A, F) to the feasible region, unique solutions have been found. This has been tested in several experiments by starting with different initial values of A F, always the solution has been independent of the starting values. In some of the preliminary experiments, when this control was not applied, it was found that there were often several real solutions to the equations, but that not more than one solution was physically feasible. If in any iteration (L + Kx) becomes negative for any value of x, then the current values A[ F t are replaced by K*o + A,) K^o+^i) (A 0,F 0 ) is the previous value in the feasible region. It is necessary that the initial values of A F should be in the feasible region. We have found in practice that a satisfactory starting point in general is F= 1-5 A = 0-3/f max. If during the first iteration, it is found that this point is not feasible, then, keeping the value of A fixed, an attempt is made to find a feasible value of F. If this is not possible, then A is set to zero in which case it is always possible to find a feasible F.

3 From equations (4), (5) (8) we have:.,tan <b' XfA. Hence, in order that L + Kx > 0 for all x, then F must lie in the range F max >F>F min (18) ((A - A/) tan <f>' F min = max 1 + XfA > 0 ^ (19) 1 +XfA A/) tan 1 +XfA Slip surfaces will be reduced, i.e., a value of v(< 1) can be found such that: (D(A 0 + vsa, F o + vsf) < O(A 0, F o ). (30) (20) Usually it is the lower limit to F which is most critical, since for most values of x, 1 + XfA > 0. Hence if the upper limit does not exist or if F max > F min +0-1, then we set: F=F m/n (21) KF max < F min + 0-1, then we set: F = i(f min + F max ). (22) If F mi/i > F max, then it is not possible to find a feasible F with this value of A, so we set A = 0, in which case there is no upper limit to F we set: F = max {A tan <f>') (23) (6) A restriction is applied to the magnitude of the steps SA SF obtained from equations (16) (17). If either SA > 0-5 or SF >0-5 then the subsequent values A( F] are given by A, = A o + vsa (24) F t = F o + v8f (25) v = 0-5/max ( SA, SF ). (26) In this way very large increments are prevented, which would otherwise occur usually in the early iterations, when the matrix of first derivatives is ill-conditioned. (c) The function O given by O(A, F) = E 2 n + CM 2 (27) C is any positive constant, satisfies the inequality (28) SA 8F satisfy equations (14) (15) the derivatives are evaluated at (A o, F o ). Hence if <D(A 0 + SA, F 0 + SF)> < (Ao, F o ), (29) then it is possible to reduce the size of the step so that O 390 This is done by comparing the values of O(Ai, F t ) with C>(A 0, F o ) if O has increased then X t F x are replaced by KA 0 + A,) K^o+Fi), respectively, the process repeated. The quantity C is introduced so that the contributions El CM% to <]> are of the same order of magnitude. E M are of different dimensions if, as is usual, units of lb ft are used, then on practical slopes M 2 is much greater than E 2. In these circumstances, if C is not introduced, i.e. C = 1, then the number of iterations would be significantly increased as follows. Normally within a few iterations the point (A o, F o ) would be close to the curve for which M n = 0, although the point (A o + SA, F o + 8F) would be closer to the final point both E n M n = 0, the value of M\ would be larger at (A o + SA, F o + SF) than at (A o, F o ) hence because M\ would be the dominant contribution to O the values of SA SF would be repeatedly halved until condition (30) is satisfied. At this stage (A o + vsa, F o + vsf) is very close to (A o, F o ) so several iterations would have been made giving only a slight improvement. Thus the accepted values (A, F) would follow very closely with small steps the curve M = 0 until finally E n = 0. Thus we need to use a value of C which is positive with dimensions of E 2 /M 2. When comparing <5(A,, F,) with <E>(A 0, F o ) then C is given by with the derivatives evaluated at (A o, F o ). The value of C is kept constant until a point (Ai,F,) is found for which <J> has decreased. When this occurs the value of C O are re-evaluated using the derivatives at the newly accepted point. With these three additional controls on the values of A F, in approximately 80% of practical cases the process converges in less than 10 iterations, giving values of A F correct to 3 decimal places. An upper limit is imposed in the program of 20 iterations of the type so far described. If this limit is reached, then further iterations are performed, restarting with the initial estimates for F A, in which only F is varied until the equation M n = 0 is satisfied. This is performed using the usual single variable Newton-Raphson process given by (32) If this process converges within 20 iterations, then the value of F obtained the initial estimate for A is used as the starting point for further iterations of the original type, with a limit again of 20 iterations imposed. The results of a case in which the original 20 iterations did not converge, but which subsequently converged

4 after finding a value of F for which M n = 0 is given in Table 1. From the table it may be seen that the first four iterations took the point (F, A) near to a local minimum of O at about (1-15, 1-2). The other sixteen iterations of the first batch produced only two more acceptable points, viz. (11513, ) (11537, 11883) with lower values of <1>. After the first batch had been unsuccessful, the values of F A were returned to their initial values F varied to make M n zero. This converged in four iterations to the point (1-203, 0-3) from which the Newton process applied to both variables converged in a further four iterations to the point (1-134, 0-491). As will be explained in 5, the last two iterations were needed to refine these values of A F using accurate formulae for the evaluation of M its derivatives. In a few cases the triple system of iterations described above has not converged to give solutions. Several of these have been investigated in detail it is almost certain that in these cases a solution in the feasible region does not exist. This conclusion was reached by considering a set of slip surfaces gradually changing from one for which a physically acceptable solution had been found to the required surface for which a solution could not be found. In this way the solution was traced was found to approach the boundary of the feasible region finallyattempted to cross it. 4. The calculation of E, M their derivatives We wish to obtain the formulae for determining E n, -r-p, -z-j-, M n, -J^TT, -y~^ for any values of A f. We will denote by K n L,, JV, P, the constant values of K, L, N P in the slice *,_, < x < x ; for i= 1,2,..., n. If in the /th slice x is measured from x ; _, then the solution of equation (2) is Hence E = r. a,,-, + P,x + injx 2 }. (33) L, Pfi, (34) 6, = x, *,_,. (35) Differentiating equation (34) with respect to F gives Slip surfaces Table 1 Values of F, X, E n M n for successive iterations in case which did not converge within the first batch of 20 iterations F L t I 1693 I ] A E n x 10-s M n X The derivatives of K, L, N P can easily be obtained from equations (4), (5), (6) (7). By starting with E = YF = 3A = (38) equations (34), (36) (37) may be used to determine differentiating with respect to A gives From equation (13) we have M, = M,_ x + J (X - A,E)dx, (39) Using equations (3), (8) (33) we have IL, 391 x _ \{k t x + ntj) Li + K,x ; ' -' (40)

5 hence A,} SY//> surfaces mate values, based on the trapezium rule of integration, can be obtained much more quickly. From equations (39), (3) (8) we have, approximately, (,,_, + P,x + in,x 2 )dx. (41) Differentiating equation (41) with respect to F A, ~i>f t»f J o L-, + -C (L, + K ix ) 2 (}h 4_? ', (42) Jo ~T 1 v^ {L,E, _ [ Lij +- A,-.X {AfA'/X + «7,-) /4,} J o (L, u,^-l (43) By changing the scale of x in these integrals so that the range of integration is from 0 to 1, letting H; = K,b,IL, (44) then we obtain the following recurrence relations: f 1 1 "' = "'-'+J 0 (TH 3 (45) (46) 'jx'dx, (47) the coefficients 7}, C/ y F y are independent of x. Since only values of A F are used for which L s L t + Kfi; are positive then H-, > 1 all the integrals are convergent. The evaluation of the basic -1 x^dx r x^dx integrals j Q ^ - ^ J Q fppr^a will be described in the next section. Mj = Mi_ x + ibi{(\i(kib, + m,) Ai)E t + (A,w,- A i )E i _ l }, (48) IM, 5. The evaluation of the integrals The calculation of M its derivatives takes approximately \ of the time of each iteration, but their approxi- 392 ib ' /) ~ J>E, ;)^F i -l + (A,m ; - Ai)^:- (49) (Am,- (50) These approximate formulae are used until the iterative process converges, then the exact formulae are used to refine the values of F A. Normally, about 6 of the simpler iterations are followed by 1 or 2 of the iterations in which M its derivatives are evaluated accurately. Thus there is an overall saving of approximately 50% of the computing time by using the approximate formulae first. In order to evaluate M n its derivatives accurately, it was shown in 4 that it is necessary to evaluate integrals of the following form: dx 1 + Hx 9(\+Hxf (51) (52) for values of H > 1 for; = 0, 1,..., 4. For non-zero values of H there are analytical formulae for these integrals which may easily be generated by the following recurrence relations: Bn = ~TT l g (1 + ") V 53 ) C n = 1 +H C r / f> -y+i (54) (55) (56) When H = 0, it is obvious that these formulae cannot be used. Also, for small values of H, their use introduces serious rounding-off errors since each integral is expressed as the relatively small difference between two large numbers. For small values of H, however, it can be seen that the integrs may be exped as polynomials in x, so Gaussian numerical integration formulae may be satisfactorily used.

6 By evaluating the most complicated integral C 4, for a sequence of values of H starting with 0-95, using both the recurrence relations (53),..., (56) Gaussian integration formulae of various orders, it was found that, using floating-point arithmetic with a precision of about 9 decimal places, the integrals could be obtained correct to at least 6 significant decimal places by using the analytical formulae for H < 0-4 H > 0-5, for 0-4 < #< 0-5 using the Gauss formula of order Discussion The methods described in this paper have been incorporated in a computer program wh ; ch has been used by a number of civil engineers. A description of the practical aspects of the program of some of the problems associated with it will be reported else. The method of specifying the data for an analysis is similar to that for the slip-circle program described by Little Price (1958). It is well known that the Newton-Raphson technique may be used for solving non-linear simultaneous equations, it is known that extra controls on the variables are often essential to make the process converge. The control (c) described in 3, that the sum of the squares of the residuals should be reduced by each iteration is References Slip surfaces of general application for one or more non-linear equations. It is less arbitrary than the control (b) of 3 which is advocated in N.P.L. (1961) which requires an arbitrary fixed limit to the change, or proportional change, of any of the variables. However, it should be noted that squares of the residuals should be scaled if necessary so that they are of the same order of magnitude, otherwise the restriction can increase the number of iterations considerably. The scaling may conveniently be performed by dividing each residual by the sum of the squares of its derivatives with respect to the unknowns. 7. Acknowledgement The authors gratefully acknowledge the assistance of members of the staff of the English Electric London Computing Service, in particular, Mrs. J. Skinner, with the programming of the preliminary experiments of these calculations. The authors are grateful to Messrs. Binnie Partners for permission to publish the results in Table 1, for many helpful discussions encouragement with the project. This study was supported in its early stages by a research grant awarded by the Department of Scientific Industrial Research. LITTLE, A. L., PRICE, V. E. (1958). "The Use of an Electronic Computer for Stability Analysis," Ceotechnique, Vol. 8, 3 p MORGENSTERN, N. R., PRICE, V. E. (1965). "The Analysis of the Stability of General Slip Surfaces," Geotechnique Vol 15 1, p. 79. N.P.L. (1961). Modern Computing Methods, 2nd Edition, chapter 6, 23, London, H.M.S.O. To the Editor, The Computer Journal. Sir, Many algorithms have been published in which values of 7T, 277, 7T 2 etc., are written as constants, commonly with about 9 significant figures. Sometimes comments are included that these constants are intended as approximations to 77 etc., that the number of significant figures should be adjusted to the capacity of the computer. But it is not necessary to use any explicit approximation to 77 or related values, since in ALGOL 60 the appropriate value can be generated by the statement pi: = 4 X arctan (1) Correspondence the variable pi may then be used as a component of arithmetic expressions. The accuracy with which pi approximates to n is limited only by the accuracy of the arctan function ( possible roundoff). For instance, in FOR- TRAN II-D on the IBM 1620, the corresponding statement PI = 4 0*ATANF (10) (with 28 decimal digits in the mantissae of floating-point numbers) does indeed give 77 correctly to 28 significant figures. Likewise, of course, if e is required then the statement e : exp (1) can be used. Yours faithfully, G. J. TEE University of Lancaster, 108 St. Leonardgate, Lancaster 24 November

3.1 Introduction. Solve non-linear real equation f(x) = 0 for real root or zero x. E.g. x x 1.5 =0, tan x x =0.

3.1 Introduction. Solve non-linear real equation f(x) = 0 for real root or zero x. E.g. x x 1.5 =0, tan x x =0. 3.1 Introduction Solve non-linear real equation f(x) = 0 for real root or zero x. E.g. x 3 +1.5x 1.5 =0, tan x x =0. Practical existence test for roots: by intermediate value theorem, f C[a, b] & f(a)f(b)

More information

BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES

BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES No. of Printed Pages : 5 BCS-054 BACHELOR OF COMPUTER APPLICATIONS (BCA) (Revised) Term-End Examination December, 2015 058b9 BCS-054 : COMPUTER ORIENTED NUMERICAL TECHNIQUES Time : 3 hours Maximum Marks

More information

ACM 106a: Lecture 1 Agenda

ACM 106a: Lecture 1 Agenda 1 ACM 106a: Lecture 1 Agenda Introduction to numerical linear algebra Common problems First examples Inexact computation What is this course about? 2 Typical numerical linear algebra problems Systems of

More information

- 2 ' a 2 =- + _ y'2 + _ 21/4. a 3 =! +! y'2 +! 21/4 +!. / (!:_ +! y'2) 21/ Y 2 2 c!+!v'2+!21/4). lc!+!v'2) 21/4.

- 2 ' a 2 =- + _ y'2 + _ 21/4. a 3 =! +! y'2 +! 21/4 +!. / (!:_ +! y'2) 21/ Y 2 2 c!+!v'2+!21/4). lc!+!v'2) 21/4. ....................................... ~. The Arithmetic Geometric Mean The arithmetic mean of two numbers a and b is defined as the "average" of the numbers, namely, aib while the geometric mean is given

More information

March Algebra 2 Question 1. March Algebra 2 Question 1

March Algebra 2 Question 1. March Algebra 2 Question 1 March Algebra 2 Question 1 If the statement is always true for the domain, assign that part a 3. If it is sometimes true, assign it a 2. If it is never true, assign it a 1. Your answer for this question

More information

Notes on floating point number, numerical computations and pitfalls

Notes on floating point number, numerical computations and pitfalls Notes on floating point number, numerical computations and pitfalls November 6, 212 1 Floating point numbers An n-digit floating point number in base β has the form x = ±(.d 1 d 2 d n ) β β e where.d 1

More information

NUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places.

NUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places. NUMERICAL METHODS 1. Rearranging the equation x 3 =.5 gives the iterative formula x n+1 = g(x n ), where g(x) = (2x 2 ) 1. (a) Starting with x = 1, compute the x n up to n = 6, and describe what is happening.

More information

The solution of linear differential equations in Chebyshev series

The solution of linear differential equations in Chebyshev series The solution of linear differential equations in Chebyshev series By R. E. Scraton* The numerical solution of the linear differential equation Any linear differential equation can be transformed into an

More information

x x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b)

x x2 2 + x3 3 x4 3. Use the divided-difference method to find a polynomial of least degree that fits the values shown: (b) Numerical Methods - PROBLEMS. The Taylor series, about the origin, for log( + x) is x x2 2 + x3 3 x4 4 + Find an upper bound on the magnitude of the truncation error on the interval x.5 when log( + x)

More information

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane. Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 3 Lecture 3 3.1 General remarks March 4, 2018 This

More information

MARK SCHEME for the November 2004 question paper 9709 MATHEMATICS 8719 HIGHER MATHEMATICS

MARK SCHEME for the November 2004 question paper 9709 MATHEMATICS 8719 HIGHER MATHEMATICS UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS GCE Advanced Subsidiary and Advanced Level MARK SCHEME for the November 004 question paper 9709 MATHEMATICS 879 HIGHER MATHEMATICS 9709/03, 879/03 Paper

More information

Numerical Methods. King Saud University

Numerical Methods. King Saud University Numerical Methods King Saud University Aims In this lecture, we will... find the approximate solutions of derivative (first- and second-order) and antiderivative (definite integral only). Numerical Differentiation

More information

5. Hand in the entire exam booklet and your computer score sheet.

5. Hand in the entire exam booklet and your computer score sheet. WINTER 2016 MATH*2130 Final Exam Last name: (PRINT) First name: Student #: Instructor: M. R. Garvie 19 April, 2016 INSTRUCTIONS: 1. This is a closed book examination, but a calculator is allowed. The test

More information

The residual again. The residual is our method of judging how good a potential solution x! of a system A x = b actually is. We compute. r = b - A x!

The residual again. The residual is our method of judging how good a potential solution x! of a system A x = b actually is. We compute. r = b - A x! The residual again The residual is our method of judging how good a potential solution x! of a system A x = b actually is. We compute r = b - A x! which gives us a measure of how good or bad x! is as a

More information

Version 1.0. klm. General Certificate of Education June Mathematics. Pure Core 4. Mark Scheme

Version 1.0. klm. General Certificate of Education June Mathematics. Pure Core 4. Mark Scheme Version.0 klm General Certificate of Education June 00 Mathematics MPC4 Pure Core 4 Mark Scheme Mark schemes are prepared by the Principal Examiner and considered, together with the relevant questions,

More information

1 Functions and Graphs

1 Functions and Graphs 1 Functions and Graphs 1.1 Functions Cartesian Coordinate System A Cartesian or rectangular coordinate system is formed by the intersection of a horizontal real number line, usually called the x axis,

More information

Mathematics 1 Lecture Notes Chapter 1 Algebra Review

Mathematics 1 Lecture Notes Chapter 1 Algebra Review Mathematics 1 Lecture Notes Chapter 1 Algebra Review c Trinity College 1 A note to the students from the lecturer: This course will be moving rather quickly, and it will be in your own best interests to

More information

Ordinary Differential Equations (ODEs)

Ordinary Differential Equations (ODEs) c01.tex 8/10/2010 22: 55 Page 1 PART A Ordinary Differential Equations (ODEs) Chap. 1 First-Order ODEs Sec. 1.1 Basic Concepts. Modeling To get a good start into this chapter and this section, quickly

More information

SQUARE ROOTS OF 2x2 MATRICES 1. Sam Northshield SUNY-Plattsburgh

SQUARE ROOTS OF 2x2 MATRICES 1. Sam Northshield SUNY-Plattsburgh SQUARE ROOTS OF x MATRICES Sam Northshield SUNY-Plattsburgh INTRODUCTION A B What is the square root of a matrix such as? It is not, in general, A B C D C D This is easy to see since the upper left entry

More information

EQUADIFF 1. Milan Práger; Emil Vitásek Stability of numerical processes. Terms of use:

EQUADIFF 1. Milan Práger; Emil Vitásek Stability of numerical processes. Terms of use: EQUADIFF 1 Milan Práger; Emil Vitásek Stability of numerical processes In: (ed.): Differential Equations and Their Applications, Proceedings of the Conference held in Prague in September 1962. Publishing

More information

November 20, Interpolation, Extrapolation & Polynomial Approximation

November 20, Interpolation, Extrapolation & Polynomial Approximation Interpolation, Extrapolation & Polynomial Approximation November 20, 2016 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic

More information

DRAFT - Math 101 Lecture Note - Dr. Said Algarni

DRAFT - Math 101 Lecture Note - Dr. Said Algarni 2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that

More information

Jim Lambers MAT 610 Summer Session Lecture 2 Notes

Jim Lambers MAT 610 Summer Session Lecture 2 Notes Jim Lambers MAT 610 Summer Session 2009-10 Lecture 2 Notes These notes correspond to Sections 2.2-2.4 in the text. Vector Norms Given vectors x and y of length one, which are simply scalars x and y, the

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5. Ax = b.

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5. Ax = b. CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 5 GENE H GOLUB Suppose we want to solve We actually have an approximation ξ such that 1 Perturbation Theory Ax = b x = ξ + e The question is, how

More information

Numerical Methods. King Saud University

Numerical Methods. King Saud University Numerical Methods King Saud University Aims In this lecture, we will... Introduce the topic of numerical methods Consider the Error analysis and sources of errors Introduction A numerical method which

More information

ALGEBRA I SEMESTER EXAMS PRACTICE MATERIALS SEMESTER Use the diagram below. 9.3 cm. A = (9.3 cm) (6.2 cm) = cm 2. 6.

ALGEBRA I SEMESTER EXAMS PRACTICE MATERIALS SEMESTER Use the diagram below. 9.3 cm. A = (9.3 cm) (6.2 cm) = cm 2. 6. 1. Use the diagram below. 9.3 cm A = (9.3 cm) (6.2 cm) = 57.66 cm 2 6.2 cm A rectangle s sides are measured to be 6.2 cm and 9.3 cm. What is the rectangle s area rounded to the correct number of significant

More information

(b) g(x) = 4 + 6(x 3) (x 3) 2 (= x x 2 ) M1A1 Note: Accept any alternative form that is correct. Award M1A0 for a substitution of (x + 3).

(b) g(x) = 4 + 6(x 3) (x 3) 2 (= x x 2 ) M1A1 Note: Accept any alternative form that is correct. Award M1A0 for a substitution of (x + 3). Paper. Answers. (a) METHOD f (x) q x f () q 6 q 6 f() p + 8 9 5 p METHOD f(x) (x ) + 5 x + 6x q 6, p (b) g(x) + 6(x ) (x ) ( + x x ) Note: Accept any alternative form that is correct. Award A for a substitution

More information

INTRODUCTION TO COMPUTATIONAL MATHEMATICS

INTRODUCTION TO COMPUTATIONAL MATHEMATICS INTRODUCTION TO COMPUTATIONAL MATHEMATICS Course Notes for CM 271 / AMATH 341 / CS 371 Fall 2007 Instructor: Prof. Justin Wan School of Computer Science University of Waterloo Course notes by Prof. Hans

More information

Slope stability software for soft soil engineering. D-Geo Stability. Verification Report

Slope stability software for soft soil engineering. D-Geo Stability. Verification Report Slope stability software for soft soil engineering D-Geo Stability Verification Report D-GEO STABILITY Slope stability software for soft soil engineering Verification Report Version: 16.2 Revision: 00

More information

STEP II, ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, (i) In the case a = 0, show that the equations have a solution if and only if b = 11.

STEP II, ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, (i) In the case a = 0, show that the equations have a solution if and only if b = 11. STEP II, 2003 2 Section A: Pure Mathematics 1 Consider the equations ax y z = 3, 2ax y 3z = 7, 3ax y 5z = b, where a and b are given constants. (i) In the case a = 0, show that the equations have a solution

More information

Experimental Uncertainty (Error) and Data Analysis

Experimental Uncertainty (Error) and Data Analysis Experimental Uncertainty (Error) and Data Analysis Advance Study Assignment Please contact Dr. Reuven at yreuven@mhrd.org if you have any questions Read the Theory part of the experiment (pages 2-14) and

More information

MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation

MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation MTH101 Calculus And Analytical Geometry Lecture Wise Questions and Answers For Final Term Exam Preparation Lecture No 23 to 45 Complete and Important Question and answer 1. What is the difference between

More information

Section Properties of Rational Expressions

Section Properties of Rational Expressions 88 Section. - Properties of Rational Expressions Recall that a rational number is any number that can be written as the ratio of two integers where the integer in the denominator cannot be. Rational Numbers:

More information

New test - November 03, 2015 [79 marks]

New test - November 03, 2015 [79 marks] New test - November 03, 05 [79 marks] Let f(x) = e x cosx, x. a. Show that f (x) = e x ( cosx sin x). correctly finding the derivative of e x, i.e. e x correctly finding the derivative of cosx, i.e. sin

More information

FREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS

FREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS FREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS A Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Technology in Civil Engineering By JYOTI PRAKASH SAMAL

More information

1 Delayed Renewal Processes: Exploiting Laplace Transforms

1 Delayed Renewal Processes: Exploiting Laplace Transforms IEOR 6711: Stochastic Models I Professor Whitt, Tuesday, October 22, 213 Renewal Theory: Proof of Blackwell s theorem 1 Delayed Renewal Processes: Exploiting Laplace Transforms The proof of Blackwell s

More information

Foundation Engineering Prof. Dr N.K. Samadhiya Department of Civil Engineering Indian Institute of Technology Roorkee

Foundation Engineering Prof. Dr N.K. Samadhiya Department of Civil Engineering Indian Institute of Technology Roorkee Foundation Engineering Prof. Dr N.K. Samadhiya Department of Civil Engineering Indian Institute of Technology Roorkee Module 01 Lecture - 03 Shallow Foundation So, in the last lecture, we discussed the

More information

AMSC/CMSC 466 Problem set 3

AMSC/CMSC 466 Problem set 3 AMSC/CMSC 466 Problem set 3 1. Problem 1 of KC, p180, parts (a), (b) and (c). Do part (a) by hand, with and without pivoting. Use MATLAB to check your answer. Use the command A\b to get the solution, and

More information

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS BISECTION METHOD

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS BISECTION METHOD BISECTION METHOD If a function f(x) is continuous between a and b, and f(a) and f(b) are of opposite signs, then there exists at least one root between a and b. It is shown graphically as, Let f a be negative

More information

Solution of Nonlinear Equations

Solution of Nonlinear Equations Solution of Nonlinear Equations (Com S 477/577 Notes) Yan-Bin Jia Sep 14, 017 One of the most frequently occurring problems in scientific work is to find the roots of equations of the form f(x) = 0. (1)

More information

37.3. The Poisson Distribution. Introduction. Prerequisites. Learning Outcomes

37.3. The Poisson Distribution. Introduction. Prerequisites. Learning Outcomes The Poisson Distribution 37.3 Introduction In this Section we introduce a probability model which can be used when the outcome of an experiment is a random variable taking on positive integer values and

More information

CHAPTER 1. INTRODUCTION. ERRORS.

CHAPTER 1. INTRODUCTION. ERRORS. CHAPTER 1. INTRODUCTION. ERRORS. SEC. 1. INTRODUCTION. Frequently, in fact most commonly, practical problems do not have neat analytical solutions. As examples of analytical solutions to mathematical problems,

More information

*GMF21* *32GMF2101* Further Mathematics. Unit 2 Mechanics and Statistics [GMF21] THURSDAY 11 JUNE, AFTERNOON. 2 hours.

*GMF21* *32GMF2101* Further Mathematics. Unit 2 Mechanics and Statistics [GMF21] THURSDAY 11 JUNE, AFTERNOON. 2 hours. Centre Number Candidate Number General Certificate of Secondary Education 2015 Further Mathematics Unit 2 Mechanics and Statistics *GMF21* [GMF21] *GMF21* THURSDAY 11 JUNE, AFTERNOON TIME 2 hours. INSTRUCTIONS

More information

A Brief Introduction to Numerical Methods for Differential Equations

A Brief Introduction to Numerical Methods for Differential Equations A Brief Introduction to Numerical Methods for Differential Equations January 10, 2011 This tutorial introduces some basic numerical computation techniques that are useful for the simulation and analysis

More information

Mathematical Methods for Numerical Analysis and Optimization

Mathematical Methods for Numerical Analysis and Optimization Biyani's Think Tank Concept based notes Mathematical Methods for Numerical Analysis and Optimization (MCA) Varsha Gupta Poonam Fatehpuria M.Sc. (Maths) Lecturer Deptt. of Information Technology Biyani

More information

Linear Hyperbolic Systems

Linear Hyperbolic Systems Linear Hyperbolic Systems Professor Dr E F Toro Laboratory of Applied Mathematics University of Trento, Italy eleuterio.toro@unitn.it http://www.ing.unitn.it/toro October 8, 2014 1 / 56 We study some basic

More information

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA

ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA GRADE 1 EXAMINATION NOVEMBER 017 ADVANCED PROGRAMME MATHEMATICS: PAPER I MODULE 1: CALCULUS AND ALGEBRA Time: hours 00 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists

More information

Condensed. Mathematics. General Certificate of Education Advanced Subsidiary Examination January 2012

Condensed. Mathematics. General Certificate of Education Advanced Subsidiary Examination January 2012 General Certificate of Education Advanced Subsidiary Examination January 01 Mathematics MPC1 Unit Pure Core 1 Friday 13 January 01 9.00 am to 10.30 am For this paper you must have: the blue AQA booklet

More information

EFFECTS OF WATER-LEVEL VARIATION ON THE STABILITY OF SLOPE BY LEM AND FEM

EFFECTS OF WATER-LEVEL VARIATION ON THE STABILITY OF SLOPE BY LEM AND FEM Proceedings of the 3 rd International Conference on Civil Engineering for Sustainable Development (ICCESD 2016), 12~14 February 2016, KUET, Khulna, Bangladesh (ISBN: 978-984-34-0265-3) EFFECTS OF WATER-LEVEL

More information

Integration of Ordinary Differential Equations

Integration of Ordinary Differential Equations Integration of Ordinary Differential Equations Com S 477/577 Nov 7, 00 1 Introduction The solution of differential equations is an important problem that arises in a host of areas. Many differential equations

More information

YEAR 13 - Mathematics Pure (C3) Term 1 plan

YEAR 13 - Mathematics Pure (C3) Term 1 plan Week Topic YEAR 13 - Mathematics Pure (C3) Term 1 plan 2016-2017 1-2 Algebra and functions Simplification of rational expressions including factorising and cancelling. Definition of a function. Domain

More information

Math /Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined

Math /Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined Math 400-001/Foundations of Algebra/Fall 2017 Numbers at the Foundations: Real Numbers In calculus, the derivative of a function f(x) is defined using limits. As a particular case, the derivative of f(x)

More information

defines the. The approximation f(x) L(x) is the. The point x = a is the of the approximation.

defines the. The approximation f(x) L(x) is the. The point x = a is the of the approximation. 4.5 Linearization and Newton's Method Objective SWBAT find linear approximation, use Newton's Method, estimating change with differentials, absolute relative, and percentage change, and sensitivity to

More information

CHAPTER 38 DIMENSIONAL ANALYSIS - SPURIOUS CORRELATION

CHAPTER 38 DIMENSIONAL ANALYSIS - SPURIOUS CORRELATION CHAPTER 38 DIMENSIONAL ANALYSIS - SPURIOUS CORRELATION by M S Yalm Professor, Department of Civil Engineering, and J W Kamphuis Associate Professor, Department of Civil Engineering, Queen's University

More information

Evaluating implicit equations

Evaluating implicit equations APPENDIX B Evaluating implicit equations There is a certain amount of difficulty for some students in understanding the difference between implicit and explicit equations, and in knowing how to evaluate

More information

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 31 OCTOBER 2018

SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY 31 OCTOBER 2018 SOLUTIONS FOR ADMISSIONS TEST IN MATHEMATICS, COMPUTER SCIENCE AND JOINT SCHOOLS WEDNESDAY OCTOBER 8 Mark Scheme: Each part of Question is worth marks which are awarded solely for the correct answer. Each

More information

AP Physics Summer Assignment

AP Physics Summer Assignment AP Physics Summer Assignment 1. Read College Board AP Physics C: Mechanics Course Description pg. 1-39 skip pg. 26-33 a. Link: https://secure-media.collegeboard.org/digitalservices/pdf/ap/ap-physics-c-course-description.pdf

More information

Nonlinear Equations. Chapter The Bisection Method

Nonlinear Equations. Chapter The Bisection Method Chapter 6 Nonlinear Equations Given a nonlinear function f(), a value r such that f(r) = 0, is called a root or a zero of f() For eample, for f() = e 016064, Fig?? gives the set of points satisfying y

More information

Experimental Uncertainty (Error) and Data Analysis

Experimental Uncertainty (Error) and Data Analysis E X P E R I M E N T 1 Experimental Uncertainty (Error) and Data Analysis INTRODUCTION AND OBJECTIVES Laboratory investigations involve taking measurements of physical quantities, and the process of taking

More information

Chapter 2 Derivatives

Chapter 2 Derivatives Contents Chapter 2 Derivatives Motivation to Chapter 2 2 1 Derivatives and Rates of Change 3 1.1 VIDEO - Definitions................................................... 3 1.2 VIDEO - Examples and Applications

More information

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 5. Nonlinear Equations

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 5. Nonlinear Equations Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T Heath Chapter 5 Nonlinear Equations Copyright c 2001 Reproduction permitted only for noncommercial, educational

More information

Deep Learning. Authors: I. Goodfellow, Y. Bengio, A. Courville. Chapter 4: Numerical Computation. Lecture slides edited by C. Yim. C.

Deep Learning. Authors: I. Goodfellow, Y. Bengio, A. Courville. Chapter 4: Numerical Computation. Lecture slides edited by C. Yim. C. Chapter 4: Numerical Computation Deep Learning Authors: I. Goodfellow, Y. Bengio, A. Courville Lecture slides edited by 1 Chapter 4: Numerical Computation 4.1 Overflow and Underflow 4.2 Poor Conditioning

More information

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note

Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note Math 001 - Term 171 Reading Mathematical Expressions & Arithmetic Operations Expression Reads Note x A x belongs to A,x is in A Between an element and a set. A B A is a subset of B Between two sets. φ

More information

N13/5/MATHL/HP2/ENG/TZ0/XX/M MARKSCHEME. November 2013 MATHEMATICS. Higher Level. Paper pages

N13/5/MATHL/HP2/ENG/TZ0/XX/M MARKSCHEME. November 2013 MATHEMATICS. Higher Level. Paper pages N/5/MATHL/HP/ENG/TZ0/XX/M MARKSCHEME November 0 MATHEMATICS Higher Level Paper 0 pages N/5/MATHL/HP/ENG/TZ0/XX/M This markscheme is confidential and for the exclusive use of examiners in this examination

More information

A-LEVEL Further Mathematics

A-LEVEL Further Mathematics A-LEVEL Further Mathematics F1 Mark scheme Specimen Version 1.1 Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers.

More information

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2)

King Fahd University of Petroleum and Minerals Prep-Year Math Program Math Term 161 Recitation (R1, R2) Math 001 - Term 161 Recitation (R1, R) Question 1: How many rational and irrational numbers are possible between 0 and 1? (a) 1 (b) Finite (c) 0 (d) Infinite (e) Question : A will contain how many elements

More information

PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp(

PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS. First Order Equations. p(x)dx)) = q(x) exp( PUTNAM PROBLEMS DIFFERENTIAL EQUATIONS First Order Equations 1. Linear y + p(x)y = q(x) Muliply through by the integrating factor exp( p(x)) to obtain (y exp( p(x))) = q(x) exp( p(x)). 2. Separation of

More information

CPT Solved Scanner (English) : Appendix 71

CPT Solved Scanner (English) : Appendix 71 CPT Solved Scanner (English) : Appendix 71 Paper-4: Quantitative Aptitude Chapter-1: Ratio and Proportion, Indices and Logarithm [1] (b) The integral part of a logarithms is called Characteristic and the

More information

Practice Final Exam Solutions

Practice Final Exam Solutions Important Notice: To prepare for the final exam, study past exams and practice exams, and homeworks, quizzes, and worksheets, not just this practice final. A topic not being on the practice final does

More information

Math Numerical Analysis Mid-Term Test Solutions

Math Numerical Analysis Mid-Term Test Solutions Math 400 - Numerical Analysis Mid-Term Test Solutions. Short Answers (a) A sufficient and necessary condition for the bisection method to find a root of f(x) on the interval [a,b] is f(a)f(b) < 0 or f(a)

More information

Higher order derivatives of the inverse function

Higher order derivatives of the inverse function Higher order derivatives of the inverse function Andrej Liptaj Abstract A general recursive and limit formula for higher order derivatives of the inverse function is presented. The formula is next used

More information

MA2501 Numerical Methods Spring 2015

MA2501 Numerical Methods Spring 2015 Norwegian University of Science and Technology Department of Mathematics MA5 Numerical Methods Spring 5 Solutions to exercise set 9 Find approximate values of the following integrals using the adaptive

More information

Math Day at the Beach 2016

Math Day at the Beach 2016 Multiple Choice Write your name and school and mark your answers on the answer sheet. You have 30 minutes to work on these problems. No calculator is allowed. 1. What is the median of the following five

More information

Optimal Starting Approximations. Newton's Method. By P. H. Sterbenz and C. T. Fike

Optimal Starting Approximations. Newton's Method. By P. H. Sterbenz and C. T. Fike Optimal Starting Approximations Newton's Method for By P. H. Sterbenz and C. T. Fike Abstract. Various writers have dealt with the subject of optimal starting approximations for square-root calculation

More information

Solution of Algebric & Transcendental Equations

Solution of Algebric & Transcendental Equations Page15 Solution of Algebric & Transcendental Equations Contents: o Introduction o Evaluation of Polynomials by Horner s Method o Methods of solving non linear equations o Bracketing Methods o Bisection

More information

Lecture 6.1 Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular,

Lecture 6.1 Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular, Lecture 6. Work and Energy During previous lectures we have considered many examples, which can be solved using Newtonian approach, in particular, Newton's second law. However, this is not always the most

More information

An Introduction to Differential Algebra

An Introduction to Differential Algebra An Introduction to Differential Algebra Alexander Wittig1, P. Di Lizia, R. Armellin, et al. 1 ESA Advanced Concepts Team (TEC-SF) SRL, Milan Dinamica Outline 1 Overview Five Views of Differential Algebra

More information

Roundoff Error. Monday, August 29, 11

Roundoff Error. Monday, August 29, 11 Roundoff Error A round-off error (rounding error), is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate

More information

Mathematics Higher Level

Mathematics Higher Level L.7/0 Pre-Leaving Certificate Examination, 06 Mathematics Higher Level Marking Scheme Paper Pg. Paper Pg. 36 Page of 68 exams Pre-Leaving Certificate Examination, 06 Mathematics Higher Level Paper Marking

More information

Module 4: Equations and Inequalities in One Variable

Module 4: Equations and Inequalities in One Variable Module 1: Relationships between quantities Precision- The level of detail of a measurement, determined by the unit of measure. Dimensional Analysis- A process that uses rates to convert measurements from

More information

PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435

PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435 PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435 Professor Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University DeKalb, IL. 60115 USA E mail: dattab@math.niu.edu

More information

AP Calculus Chapter 9: Infinite Series

AP Calculus Chapter 9: Infinite Series AP Calculus Chapter 9: Infinite Series 9. Sequences a, a 2, a 3, a 4, a 5,... Sequence: A function whose domain is the set of positive integers n = 2 3 4 a n = a a 2 a 3 a 4 terms of the sequence Begin

More information

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Scientific Computing: An Introductory Survey Chapter 5 Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction

More information

A-LEVEL Mathematics. Paper 3 Mark scheme. Specimen. Version 1.2

A-LEVEL Mathematics. Paper 3 Mark scheme. Specimen. Version 1.2 A-LEVEL Mathematics Paper 3 Mark scheme Specimen Version. Mark schemes are prepared by the Lead Assessment Writer and considered, together with the relevant questions, by a panel of subject teachers. This

More information

Semester Review Packet

Semester Review Packet MATH 110: College Algebra Instructor: Reyes Semester Review Packet Remarks: This semester we have made a very detailed study of four classes of functions: Polynomial functions Linear Quadratic Higher degree

More information

Systems of Equations and Inequalities. College Algebra

Systems of Equations and Inequalities. College Algebra Systems of Equations and Inequalities College Algebra System of Linear Equations There are three types of systems of linear equations in two variables, and three types of solutions. 1. An independent system

More information

Function Practice. 1. (a) attempt to form composite (M1) (c) METHOD 1 valid approach. e.g. g 1 (5), 2, f (5) f (2) = 3 A1 N2 2

Function Practice. 1. (a) attempt to form composite (M1) (c) METHOD 1 valid approach. e.g. g 1 (5), 2, f (5) f (2) = 3 A1 N2 2 1. (a) attempt to form composite e.g. ( ) 3 g 7 x, 7 x + (g f)(x) = 10 x N (b) g 1 (x) = x 3 N1 1 (c) METHOD 1 valid approach e.g. g 1 (5),, f (5) f () = 3 N METHOD attempt to form composite of f and g

More information

GENERALIZED ANNUITIES AND ASSURANCES, AND INTER-RELATIONSHIPS. BY LEIGH ROBERTS, M.Sc., ABSTRACT

GENERALIZED ANNUITIES AND ASSURANCES, AND INTER-RELATIONSHIPS. BY LEIGH ROBERTS, M.Sc., ABSTRACT GENERALIZED ANNUITIES AND ASSURANCES, AND THEIR INTER-RELATIONSHIPS BY LEIGH ROBERTS, M.Sc., A.I.A ABSTRACT By the definition of generalized assurances and annuities, the relation is shown to be the simplest

More information

PHYS-2010: General Physics I Course Lecture Notes Section V

PHYS-2010: General Physics I Course Lecture Notes Section V PHYS-2010: General Physics I Course Lecture Notes Section V Dr. Donald G. Luttermoser East Tennessee State University Edition 2.5 Abstract These class notes are designed for use of the instructor and students

More information

(Refer Slide Time: 04:21 min)

(Refer Slide Time: 04:21 min) Soil Mechanics Prof. B.V.S. Viswanathan Department of Civil Engineering Indian Institute of Technology, Bombay Lecture 44 Shear Strength of Soils Lecture No.2 Dear students today we shall go through yet

More information

Chapter 1: Precalculus Review

Chapter 1: Precalculus Review : Precalculus Review Math 115 17 January 2018 Overview 1 Important Notation 2 Exponents 3 Polynomials 4 Rational Functions 5 Cartesian Coordinates 6 Lines Notation Intervals: Interval Notation (a, b) (a,

More information

Lecture 7. Floating point arithmetic and stability

Lecture 7. Floating point arithmetic and stability Lecture 7 Floating point arithmetic and stability 2.5 Machine representation of numbers Scientific notation: 23 }{{} }{{} } 3.14159265 {{} }{{} 10 sign mantissa base exponent (significand) s m β e A floating

More information

PENRITH HIGH SCHOOL MATHEMATICS EXTENSION HSC Trial

PENRITH HIGH SCHOOL MATHEMATICS EXTENSION HSC Trial PENRITH HIGH SCHOOL MATHEMATICS EXTENSION 013 Assessor: Mr Ferguson General Instructions: HSC Trial Total marks 100 Reading time 5 minutes Working time 3 hours Write using black or blue pen. Black pen

More information

Let x be an approximate solution for Ax = b, e.g., obtained by Gaussian elimination. Let x denote the exact solution. Call. r := b A x.

Let x be an approximate solution for Ax = b, e.g., obtained by Gaussian elimination. Let x denote the exact solution. Call. r := b A x. ESTIMATION OF ERROR Let x be an approximate solution for Ax = b, e.g., obtained by Gaussian elimination. Let x denote the exact solution. Call the residual for x. Then r := b A x r = b A x = Ax A x = A

More information

2017 VCE Mathematical Methods 2 examination report

2017 VCE Mathematical Methods 2 examination report 7 VCE Mathematical Methods examination report General comments There were some excellent responses to the 7 Mathematical Methods examination and most students were able to attempt the four questions in

More information

Taylor series. Chapter Introduction From geometric series to Taylor polynomials

Taylor series. Chapter Introduction From geometric series to Taylor polynomials Chapter 2 Taylor series 2. Introduction The topic of this chapter is find approximations of functions in terms of power series, also called Taylor series. Such series can be described informally as infinite

More information

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2.

a 11 x 1 + a 12 x a 1n x n = b 1 a 21 x 1 + a 22 x a 2n x n = b 2. Chapter 1 LINEAR EQUATIONS 11 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,, a n, b are given real

More information

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES

MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can

More information

ES120 Spring 2018 Section 7 Notes

ES120 Spring 2018 Section 7 Notes ES120 Spring 2018 Section 7 Notes Matheus Fernandes March 29, 2018 Problem 1: For the beam and loading shown, (a) determine the equations of the shear and bending-moment curves, (b) draw the shear and

More information

MATHEMATICS: PAPER I

MATHEMATICS: PAPER I NATIONAL SENIOR CERTIFICATE EXAMINATION NOVEMBER 017 MATHEMATICS: PAPER I Time: 3 hours 150 marks PLEASE READ THE FOLLOWING INSTRUCTIONS CAREFULLY 1. This question paper consists of 11 pages and an Information

More information