Engineering Mathematics
|
|
- Kathleen O’Neal’
- 5 years ago
- Views:
Transcription
1 F.Y. Diploma : Sem. II [AE/CD/CE/CH/CM/CO/CR/CS/CV/CW/DE/ED/EE/EI/EJ/EN/ EP/ET/EV/EX/FE/IC/IE/IF/IS/IU/ME/MH/MI/MU/PG/PS/PT] Engineering Mathematics Time: Hrs.] Pre Question Paper Solution [Marks : 00 Q. Attempt any TEN of the following : [0] Q.(a) If ( y) + i ( + y) 7, find, y. [] (A) ( y) + i( + y) 7 ( y) + i( + y) 7 + 0i y 7 and + y 0 y 7 + y y Q.(b) Epress in the form a + ib. (A) i i i i i i i i i i i i 5 or i 5 5 i i where a, b R, i. [] Q.(c) If f() show that f( ) f(). [] (A) f(-) () 5() f() () 5() f() f() Q.(d) State whether the function f() (A) f() e e e e f() f() is even. e e is even or odd. [] Q.(e) Evaluate. [] 0 sin (A) 0 sin 0 sin 0 sin
2 Vialankar : F.Y. Diploma Engg. Mathematics Q. (f) Evaluate (A) log log [] 0 0 log log Q.(g) If y e sin cos find d. [] (A) d d d d e sin cos e cos sin sin cos e d d d e sin sin e cos cos sin cos e e sin e cos e sincos e sin cos sin cos Y OR e sin cos e sin d d d e sin sin e d d e cos sin e e cos sin Q.(h) Find d if y log ( + ). [] (A) y log( + ) d d d + + Q. (i) Find if sin, y cos. [] d (A) sin, y cos d d cos and d sin d / d d / d sin cos tan
3 Pre Question Paper Solution Q.(j) If + y find d. [] (A) + y y 0 d y d d y Q.(k) Show that root of equation 5 0 lies between and. [] (A) f() 5 f() < 0 f() 6 > 0 Therefore the root lies between and Q. (l) Find first iteration by Jacobi s method : 0 + y + z, + 0y + z, + y + 0z 5 (A) 0 + y + z, + 0y + z, + y + 0z 5 [] y z 0 z y 0 5 y z 0 Now we start with : 0 0 y 0 z y z Q. Attempt any FOUR of the following : [6] Q.(a) If f() then show that f[f()]. [] (A) f() f f[f()] f
4 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) Epress the following number in polar form i. [] (A) r / 80 tan or / / or tan / z r(cos + isin) cos0 + isin0 or cos i sin Q.(c) Find all cube root of unity. [] (A) Let z + 0i a, b 0 r 0 0 tan 0 z r(cos + i sin) cos0 + isin0 cosk + isink / [cosk + isink] / k k cos isin For k 0, / cos(0) + isin(0) + 0 For k, / cos i sin i For k, / cos i sin i
5 Pre Question Paper Solution Q.(d) Simplify using De-Moivre s theorem : (A) θ θ cos5 isin5 5 cos θisin θ 7 7 θ θ cos isin cos θisin θ θ θ cos5 isin5 5 cos θisin θ 7 7 θ θ cos isin cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos θisin θ cos isin cos isin cos isin [] Q.(e) If f() +, solve the equation f() f( ). [] (A) f() + f( ) ( ) () But f() f() or , or.5, 0.5 Q.(f) Simplify i + i 0 + i 50 + i 00. [] (A) i + i 0 + i 50 + i 00 + (i ) 5 + (i ) 5 + (i ) 5 + () 5 + () 5 + () Q. Attempt any FOUR of the following : [6] Q.(a) If f() a + b + and f(), f(), find a and b. [] (A) f() a + b + f() a() +b() + a + b + f() a() + b() + a + b + But f(), f() a + b + a + b + a + b a + b 8 a b 5
6 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) If f() log then prove that f (A) f log log log log log f() f(). [] Q.(c) Evaluate (A) 6 6. [] or 0.5 Q.(d) Evaluate (A) sin cos. [] tan sin cos tan sin cossin cos sin cos sincos sincos cos sin cos sin cos sin cos sin cos sin cos cos sin cos cos cos 6
7 Q.(e) Evaluate 5 (A) Pre Question Paper Solution. [] or.5 Q.(f) Evaluate (A) [] log log Q. Attempt any FOUR of the following : [6] Q.(a) If y sin ( ) find d. [] (A) Put sin y sin ( ) sin sin sin sin (sin) sin d. 7
8 Vialankar : F.Y. Diploma Engg. Mathematics Q.(b) Using first principle find derivative of f() a. [] (A) f() a f( + h) a + h d f h f h0 h h a a h0 h h a a h0 h h a a h0 h a log a Q.(c) If u and v are differentiable functions of and y u.v then prove that [] d dv du u v d d (A) Let y uv. Let be infinitesimal increment in and y, u, v be corresponding infinitesimal increments in y, u, v. y + y (u + u) (v + v) uv + uv + vu + uv y uv + uv + vu + uv y uv + uv + vu + uv uv uv + vu + uv As u and v are very very small. uv is negligible. y uv + vu y uv vu v u u v y v u u v 0 0 y v u u v d dv du u v d d Q.(d) Differentiate w.r.t, tan 5 6. [] 5 (A) Let y tan 6 tan Put tan A and tan B tana tanb y tan tanatanb tan tan A B A + B tan () + tan () d 9 8
9 Pre Question Paper Solution Q.(e) Find d if + y + y. [] (A) + y + y 6 y y 0 d d 6 y y d d 0 6 y y 0 d d 6 y y t Q. (f) If y tan t and t sin t t (A) y tan t Put t tan tan y tan tan tan tan y tan t t And sin t Put t tan tan sin sin (sin) tan tan t y d find d. [] Q.5 Attempt any FOUR of the following : [6] sin Q.5(a) Evaluate. [] (A) Put t as, t 0 sin sin t t0 t sin t t0 t sint t0 t sin t t0 t log log Q.5(b) Evaluate. [] log log (A) 5 9
10 Vialankar : F.Y. Diploma Engg. Mathematics Let + h or h as, h 0 log h log h0 h h log h0h h log h0 h log h0 log e log e /h /h/ Q.5c) Using Bisection method find the approimate root of + 0 ( iterations). (A) + 0 f() + f() f() the root is in (, )..5 f(.5) 0.75 the root is in (,.5)..5.5 f(.5) 0.88 the root is in (.5,.5) [] Q.5(d) Using False Position method find the root of 0 ( iterations only). [] (A) f() f() f() the root is in (, ) af b bf a.667 f(b) f a f(.667).05 the root is in (.667, )..78 f(.78) 0. the root is in (.78, )
11 Pre Question Paper Solution Q.5(e) Using Newton Raphson method find the root of 9 0 (carry out iterations). (A) 9 0 f() 9 f () f() 9 f() 5 f 9 f' 9 OR f f f 9 Start with 0, [] Q.5(f) Using Newton Raphson method find approimate value of 0 ( iterations). [] (A) Let f() 0 f() f() f() 6 f 0 (i) f 0 (ii) OR f f 0 f Start with 0, (i) (ii) Q.6 Attempt any FOUR of the following : [6] Q.6(a) If y sin 5 cos 5 show that + 5y 0. d [] (A) y sin 5 cos5 cos5 5 + sin 5 5 d
12 Vialankar : F.Y. Diploma Engg. Mathematics 5 cos5 + 5 sin 5 5 sin cos 5 d 5(sin 5 cos5) 5y 5y d 0 Q.6(b) If a ( sin ), y a ( cos ) find d and d (A) a( sin) d a(cos ) d y a( cos) d a(sin) d / d d / d at, a sin a cos sin cos sin d cos at. [] or.79 Q.6(c) Solve by Jacobi s method performing iterations : 0 + y z 7, + 0y z 8, y + 0z 5 (A) 0 + y z 7 + 0y z 8 y + 0z 5 [] 0 8 z 0 5 y 0 7 y z y z Starting with 0 0 y 0 z y 0.9 z.5.0 y z.0.00 y.00 z.00
13 Pre Question Paper Solution Q.6(d) Solve by Gauss-Seidal method ( iterations) 5 + y + z 8, + 0y z 9, 6y + 5z 5 8 y z 0 9 z 5 6y (A) y z [] Starting with 0 0 y 0 z 0. y 0.8 z y z y z Q.6(e) Solve by Gauss Eination method : + y + z, + y + z, + y + z (A) + y + z + y + z + y + z [] + 6y + 9z + y + z 5y + 7z and 6 + y + z 6 + 9y + z 7y + z 5y + 7z 9y + 7z 77 + y z 5y 08 Q.6(f) Solve by Gauss Seidal method ( iterations) 5 y 9, 5y + z, y 5z 6, Taking 0.5, y 0 0.5, z (A) 5 y 9 5 y + 0z 9 5y + z OR 5y + z y 5z y 5z 6 [] 9 y 5 z 5 6 y 5 y z
14 Vialankar : F.Y. Diploma Engg. Mathematics Starting with 0.5, y 0 0.5, z y.08 z y.006 z 0.999
S.Y. Diploma : Sem. III. Applied Mathematics. Q.1 Attempt any TEN of the following : [20] Q.1(a)
S.Y. Diploma : Sem. III Applied Mathematics Time : Hrs.] Prelim Question Paper Solution [Marks : Q. Attempt any TEN of the following : [] Q.(a) Find the gradient of the tangent of the curve y at 4. []
More informationHence a root lies between 1 and 2. Since f a is negative and f(x 0 ) is positive The root lies between a and x 0 i.e. 1 and 1.
The Bisection method or BOLZANO s method or Interval halving method: Find the positive root of x 3 x = 1 correct to four decimal places by bisection method Let f x = x 3 x 1 Here f 0 = 1 = ve, f 1 = ve,
More informationNumerical and Statistical Methods
F.Y. B.Sc.(IT) : Sem. II Numerical and Statistical Methods Time : ½ Hrs.] Prelim Question Paper Solution [Marks : 75 Q. Attempt any THREE of the following : [5] Q.(a) What is a mathematical model? With
More informationVidyalanakar F.Y. Diploma : Sem. II [AE/CE/CH/CR/CS/CV/EE/EP/FE/ME/MH/MI/PG/PT/PS] Engineering Mechanics
Vidyalanakar F.Y. Diploma : Sem. II [AE/CE/CH/CR/CS/CV/EE/EP/FE/ME/MH/MI/PG/PT/PS] Engineering Mechanics Time : 3 Hrs.] Prelim Question Paper Solution [Marks : 100 Q.1 Attempt any TEN of the following
More informationNumerical and Statistical Methods
F.Y. B.Sc.(IT) : Sem. II Numerical and Statistical Methods Time : ½ Hrs.] Prelim Question Paper Solution [Marks : 75 Q. Attempt any THREE of the following : [5] Q.(a) What is a mathematical model? With
More informationEngineering Mechanics
F.Y. Diploma : Sem. II [AE/CE/CH/CR/CS/CV/EE/EP/FE/ME/MH/MI/PG/PT/PS] Engineering Mechanics Time : 3 Hrs.] Prelim Question Paper Solution [Marks : 00 Q. Attempt any TEN of the following : [20] Q.(a) Difference
More informationChapter 9: Complex Numbers
Chapter 9: Comple Numbers 9.1 Imaginary Number 9. Comple Number - definition - argand diagram - equality of comple number 9.3 Algebraic operations on comple number - addition and subtraction - multiplication
More information= + then for all n N. n= is true, now assume the statement is. ) clearly the base case 1 ( ) ( ) ( )( ) θ θ θ θ ( θ θ θ θ)
Complex numbers mixed exercise i a We have e cos + isin hence i i ( e + e ) ( cos + isin + cos + isin ) ( cos + isin + cos sin) cos Where we have used the fact that cos cos sin sin b We have ia ia ib ib
More informationUNIT - 2 Unit-02/Lecture-01
UNIT - 2 Unit-02/Lecture-01 Solution of algebraic & transcendental equations by regula falsi method Unit-02/Lecture-01 [RGPV DEC(2013)] [7] Unit-02/Lecture-01 [RGPV JUNE(2014)] [7] Unit-02/Lecture-01 S.NO
More informationSOLUTIONS TO HOMEWORK ASSIGNMENT #2, Math 253
SOLUTIONS TO HOMEWORK ASSIGNMENT #, Math 5. Find the equation of a sphere if one of its diameters has end points (, 0, 5) and (5, 4, 7). The length of the diameter is (5 ) + ( 4 0) + (7 5) = =, so the
More informationUNIT II SOLUTION OF NON-LINEAR AND SIMULTANEOUS LINEAR EQUATION
UNIT II SOLUTION OF NON-LINEAR AND SIMULTANEOUS LINEAR EQUATION. If g x is continuous in a, b, then under what condition the iterative method x = g x has a unique solution in a, b? g x < in a, b. State
More informationC4 mark schemes - International A level (150 minute papers). First mark scheme is June 2014, second mark scheme is Specimen paper
C4 mark schemes - International A level (0 minute papers). First mark scheme is June 04, second mark scheme is Specimen paper. (a) f (.).7, f () M Sign change (and f ( ) is continuous) therefore there
More informationChemistry 456A (10:30AM Bagley 154)
Winter 0 Chemistry 456A (0:0AM Bagley 54) Problem Set B (due 9PM Friday, /0/) Q) In the previous homework we compared isothermal one-step, irreversible work with reversible isothermal work. We also compared
More informationMATHEMATICS PAPER IB COORDINATE GEOMETRY(2D &3D) AND CALCULUS. Note: This question paper consists of three sections A,B and C.
MATHEMATICS PAPER IB COORDINATE GEOMETRY(D &3D) AND CALCULUS. TIME : 3hrs Ma. Marks.75 Note: This question paper consists of three sections A,B and C. SECTION A VERY SHORT ANSWER TYPE QUESTIONS. 0X =0.
More information1) Yes 2) No 3) The energy principle does not apply in this situation. Would this be a violation of the energy principle?
Q12.1.a You put an ice cube into a styrofoam cup containing hot coffee. You would probably be surprised if the ice cube got colder and the coffee got hotter. 1) Yes 2) No 3) The energy principle does not
More informationPrelim Examination 2015/2016 (Assessing all 3 Units) MATHEMATICS. CFE Advanced Higher Grade. Time allowed - 3 hours
Prelim Eamination /6 (Assessing all Units) MATHEMATICS CFE Advanced Higher Grade Time allowed - hours Total marks Attempt ALL questions. You may use a calculator. Full credit will be given only to solutions
More informationFY B. Tech. Semester II. Complex Numbers and Calculus
FY B. Tech. Semester II Comple Numbers and Calculus Course Code FYT Course Comple numbers and Calculus (CNC) Prepared by S M Mali Date 6//7 Prerequisites Basic knowledge of results from Algebra. Knowledge
More informationM.SC. PHYSICS - II YEAR
MANONMANIAM SUNDARANAR UNIVERSITY DIRECTORATE OF DISTANCE & CONTINUING EDUCATION TIRUNELVELI 627012, TAMIL NADU M.SC. PHYSICS - II YEAR DKP26 - NUMERICAL METHODS (From the academic year 2016-17) Most Student
More informationQuestion Bank (I scheme )
Question Bank (I scheme ) Name of subject: Applied Mathematics Subject code: 22206/22224/22210/22201 Course : CH/CM/CE/EJ/IF/EE/ME Semester: II UNIT-3 (CO3) Unit Test : II (APPLICATION OF INTEGRATION)
More informationThe iteration formula for to find the root of the equation
SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY, COIMBATORE- 10 DEPARTMENT OF SCIENCE AND HUMANITIES SUBJECT: NUMERICAL METHODS & LINEAR PROGRAMMING UNIT II SOLUTIONS OF EQUATION 1. If is continuous in then under
More information1) +x 2) x 3) +y 4) y 5) +z 6) z 7) zero magnitude
Q11.1.a: What is the direction of < 0, 0, 3> x < 0, 4, 0>? 1) +x ) x 3) +y 4) y 5) +z 6) z 7) zero magnitude Q11.1.b: What is the direction of < 0, 4, 0> x < 0, 0, 3>? 1) +x ) x 3) +y 4) y 5) +z 6) z 7)
More informationMethods of Integration
Methods of Integration Essential Formulas k d = k +C sind = cos +C n d = n+ n + +C cosd = sin +C e d = e +C tand = ln sec +C d = ln +C cotd = ln sin +C + d = tan +C lnd = ln +C secd = ln sec + tan +C cscd
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers
Syllabus Objectives: 5.1 The student will eplore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More informationINVERSE TRIGONOMETRY: SA 4 MARKS
INVERSE TRIGONOMETRY: SA MARKS To prove Q. Prove that sin - tan - 7 = π 5 Ans L.H.S = Sin - tan - 7 5 = A- tan - 7 = tan - 7 tan- let A = Sin - 5 Sin A = 5 = tan - ( ( ) ) tan - 7 9 6 tan A = A = tan-
More informationCalculus Problem Sheet Prof Paul Sutcliffe. 2. State the domain and range of each of the following functions
f() 8 6 4 8 6-3 - - 3 4 5 6 f().9.8.7.6.5.4.3.. -4-3 - - 3 f() 7 6 5 4 3-3 - - Calculus Problem Sheet Prof Paul Sutcliffe. By applying the vertical line test, or otherwise, determine whether each of the
More informationMark Scheme Summer 2009
Mark Summer 009 GCE Core Mathematics C (666) Edecel is one of the leading eamining and awarding bodies in the UK and throughout the world. We provide a wide range of qualifications including academic,
More informationMAHARASHTRA STATE BOARD OF TECHNICAL EDUCATION
(ISO/IEC - 7-5 Certiied) Page No: /6 WINTER 5 EXAMINATION MODEL ANSWER Subject: ENGINEERING MATHEMATICS (EMS) Subject Code: 76 Important Instructions to eaminers: The model answer shall be the complete
More informationSOLUTION 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 informationMATH MIDTERM 4 - SOME REVIEW PROBLEMS WITH SOLUTIONS Calculus, Fall 2017 Professor: Jared Speck. Problem 1. Approximate the integral
MATH 8. - MIDTERM 4 - SOME REVIEW PROBLEMS WITH SOLUTIONS 8. Calculus, Fall 7 Professor: Jared Speck Problem. Approimate the integral 4 d using first Simpson s rule with two equal intervals and then the
More informationMathematics 132 Calculus for Physical and Life Sciences 2 Exam 3 Review Sheet April 15, 2008
Mathematics 32 Calculus for Physical and Life Sciences 2 Eam 3 Review Sheet April 5, 2008 Sample Eam Questions - Solutions This list is much longer than the actual eam will be (to give you some idea of
More informationDepartment of Mathematical x 1 x 2 1
Contents Limits. Basic Factoring Eample....................................... One-Sided Limit........................................... 3.3 Squeeze Theorem.......................................... 4.4
More informationAdvanced Higher Grade
Prelim Eamination / 5 (Assessing Units & ) MATHEMATICS Advanced Higher Grade Time allowed - hours Read Carefully. Full credit will be given only where the solution contains appropriate woring.. Calculators
More informationCalculus Problem Sheet Prof Paul Sutcliffe. 2. State the domain and range of each of the following functions
f( 8 6 4 8 6-3 - - 3 4 5 6 f(.9.8.7.6.5.4.3.. -4-3 - - 3 f( 7 6 5 4 3-3 - - Calculus Problem Sheet Prof Paul Sutcliffe. By applying the vertical line test, or otherwise, determine whether each of the following
More informationSection 7.1 Exercises
Section 7. Solving Trigonometric Equations and Identities 5 Section 7. Eercises Find all solutions on the interval sin. sin.. cos. cos Find all solutions 5. sin 9. cos 5 6. sin. 8cos 6 7. cos t 8. cos
More informationMATH 101 Midterm Examination Spring 2009
MATH Midterm Eamination Spring 9 Date: May 5, 9 Time: 7 minutes Surname: (Please, print!) Given name(s): Signature: Instructions. This is a closed book eam: No books, no notes, no calculators are allowed!.
More informationChapter 8 More About the Trigonometric Functions
Relationships Among Trigonometric Functions Section 8. 8 Chapter 8 More About the Trigonometric Functions Section 8. Relationships Among Trigonometric Functions. The amplitude of the graph of cos is while
More informationCONTINUITY AND DIFFERENTIABILITY
5. Introduction The whole of science is nothing more than a refinement of everyday thinking. ALBERT EINSTEIN This chapter is essentially a continuation of our stu of differentiation of functions in Class
More informationSection 7.1 Exercises
Section 7.1 Solving Trigonometric Equations and Identities 109 Section 7.1 Eercises Find all solutions on the interval 0 sin 1. sin 1.. cos 1. cos Find all solutions 5. sin 1 9. cos 5 6. sin 10. 8cos 6
More informationGAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 19 (LEARNER NOTES)
MATHEMATICS GRADE SESSION 9 (LEARNER NOTES) TRIGONOMETRY () Learner Note: Trigonometry is an extremely important and large part of Paper. You must ensure that you master all the basic rules and definitions
More informationSolutions to the Exercises of Chapter 8
8A Domains of Functions Solutions to the Eercises of Chapter 8 1 For 7 to make sense, we need 7 0or7 So the domain of f() is{ 7} For + 5 to make sense, +5 0 So the domain of g() is{ 5} For h() to make
More informationMAS153/MAS159. MAS153/MAS159 1 Turn Over SCHOOL OF MATHEMATICS AND STATISTICS hours. Mathematics (Materials) Mathematics For Chemists
Data provided: Formula sheet MAS53/MAS59 SCHOOL OF MATHEMATICS AND STATISTICS Mathematics (Materials Mathematics For Chemists Spring Semester 203 204 3 hours All questions are compulsory. The marks awarded
More informationTrigonometric Identities. Sum and Differences
Trigonometric Identities Sum and Differences WARNING: While viewing this pdf, the viewer may experience the following: 1.) Shock.) Confusion.) Denial 4.) Disbelief 5.) I never learned this 6.) Fear 7.)
More informationMathematics Extension 2 HSC Examination Topic: Polynomials
by Topic 995 to 006 Polynomials Page Mathematics Etension Eamination Topic: Polynomials 06 06 05 05 c Two of the zeros of P() = + 59 8 + 0 are a + ib and a + ib, where a and b are real and b > 0. Find
More informationReview exercise
Review eercise y cos sin When : 8 y and 8 gradient of normal is 8 y When : 9 y and 8 Equation of normal is y 8 8 y8 8 8 8y 8 8 8 8y 8 8 8 8y 8 8 8 y e ln( ) e ln e When : y e ln and e Equation of tangent
More informationLecture 44. Better and successive approximations x2, x3,, xn to the root are obtained from
Lecture 44 Solution of Non-Linear Equations Regula-Falsi Method Method of iteration Newton - Raphson Method Muller s Method Graeffe s Root Squaring Method Newton -Raphson Method An approximation to the
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- MARCH, 2013
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- MARCH, 2013 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationRegent College Maths Department. Further Pure 1 Numerical Solutions of Equations
Regent College Maths Department Further Pure 1 Numerical Solutions of Equations Further Pure 1 Numerical Solutions of Equations You should: Be able use interval bisection, linear interpolation and the
More information2 nd ORDER O.D.E.s SUBSTITUTIONS
nd ORDER O.D.E.s SUBSTITUTIONS Question 1 (***+) d y y 8y + 16y = d d d, y 0, Find the general solution of the above differential equation by using the transformation equation t = y. Give the answer in
More informationSection: I. u 4 du. (9x + 1) + C, 3
EXAM 3 MAT 168 Calculus II Fall 18 Name: Section: I All answers must include either supporting work or an eplanation of your reasoning. MPORTANT: These elements are considered main part of the answer and
More information) z r θ ( ) ( ) ( ) = then. Complete Solutions to Examination Questions Complete Solutions to Examination Questions 10.
Complete Solutions to Examination Questions 0 Complete Solutions to Examination Questions 0. (i We need to determine + given + j, j: + + j + j (ii The product ( ( + j6 + 6 j 8 + j is given by ( + j( j
More informationPaper Reference. Core Mathematics C3 Advanced. Thursday 11 June 2009 Morning Time: 1 hour 30 minutes. Mathematical Formulae (Orange or Green)
Centre No. Candidate No. Paper Reference(s) 6665/01 Edecel GCE Core Mathematics C3 Advanced Thursday 11 June 009 Morning Time: 1 hour 30 minutes Materials required for eamination Mathematical Formulae
More informationSRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY, COIMBATORE- 10 DEPARTMENT OF SCIENCE AND HUMANITIES B.E - EEE & CIVIL UNIT I
SRI RAMAKRISHNA INSTITUTE OF TECHNOLOGY, COIMBATORE- 10 DEPARTMENT OF SCIENCE AND HUMANITIES B.E - EEE & CIVIL SUBJECT: NUMERICAL METHODS ( SEMESTER VI ) UNIT I SOLUTIONS OF EQUATIONS AND EIGEN VALUE PROBLEMS
More informationPrevious Year Questions & Detailed Solutions
Previous Year Questions & Detailed Solutions. The rate of convergence in the Gauss-Seidal method is as fast as in Gauss Jacobi smethod ) thrice ) half-times ) twice 4) three by two times. In application
More informationTangent Line Approximations
60_009.qd //0 :8 PM Page SECTION.9 Dierentials Section.9 EXPLORATION Tangent Line Approimation Use a graphing utilit to graph. In the same viewing window, graph the tangent line to the graph o at the point,.
More information(a) Show that there is a root α of f (x) = 0 in the interval [1.2, 1.3]. (2)
. f() = 4 cosec 4 +, where is in radians. (a) Show that there is a root α of f () = 0 in the interval [.,.3]. Show that the equation f() = 0 can be written in the form = + sin 4 Use the iterative formula
More informationExample - Newton-Raphson Method
Eample - Newton-Raphson Method We now consider the following eample: minimize f( 3 3 + -- 4 4 Since f ( 3 2 + 3 3 and f ( 6 + 9 2 we form the following iteration: + n 3 ( n 3 3( n 2 ------------------------------------
More informationis the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input x at x = b.
Uses of differentials to estimate errors. Recall the derivative notation df d is the intuition: the derivative tells us the change in output y (from f(b)) in response to a change of input at = b. Eamples.
More informationPreview from Notesale.co.uk Page 2 of 42
. CONCEPTS & FORMULAS. INTRODUCTION Radian The angle subtended at centre of a circle by an arc of length equal to the radius of the circle is radian r o = o radian r r o radian = o = 6 Positive & Negative
More informationHere is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest Common Factor) first.
1 Algera and Trigonometry Notes on Topics that YOU should KNOW from your prerequisite courses! Here is a general Factoring Strategy that you should use to factor polynomials. 1. Always factor out the GCF(Greatest
More informationh(x) lim H(x) = lim Since h is nondecreasing then h(x) 0 for all x, and if h is discontinuous at a point x then H(x) > 0. Denote
Real Variables, Fall 4 Problem set 4 Solution suggestions Exercise. Let f be of bounded variation on [a, b]. Show that for each c (a, b), lim x c f(x) and lim x c f(x) exist. Prove that a monotone function
More informationB.Tech. Theory Examination (Semester IV) Engineering Mathematics III
Solved Question Paper 5-6 B.Tech. Theory Eamination (Semester IV) 5-6 Engineering Mathematics III Time : hours] [Maimum Marks : Section-A. Attempt all questions of this section. Each question carry equal
More informationP vs NP: Solutions of NP Problems Abstract The simplest solution is usually the best solution---albert Einstein
Copyright A. A. Frempong P vs NP: Solutions of NP Problems Abstract The simplest solution is usually the best solution---albert Einstein Best news. After over 30 years of debating, the debate is over.
More informationGENUINE REPLACEMENT PARTS
GENUINE REPLACEMENT PARTS SUSPENSION COMPONENTS - APPLICABLE TO ALL MARKETS REPLACEMENT SUSPE FOR ROLLS-ROYCE & B - Rolls-Royce FRONT SUSPENSION COMPONENTS REAR SUSPENSION COMPONENTS CHASSIS/VIN APPROXIMATE
More informationMath 2300 Calculus II University of Colorado
Math 3 Calculus II University of Colorado Spring Final eam review problems: ANSWER KEY. Find f (, ) for f(, y) = esin( y) ( + y ) 3/.. Consider the solid region W situated above the region apple apple,
More informationAnswer Key 1973 BC 1969 BC 24. A 14. A 24. C 25. A 26. C 27. C 28. D 29. C 30. D 31. C 13. C 12. D 12. E 3. A 32. B 27. E 34. C 14. D 25. B 26.
Answer Key 969 BC 97 BC. C. E. B. D 5. E 6. B 7. D 8. C 9. D. A. B. E. C. D 5. B 6. B 7. B 8. E 9. C. A. B. E. D. C 5. A 6. C 7. C 8. D 9. C. D. C. B. A. D 5. A 6. B 7. D 8. A 9. D. E. D. B. E. E 5. E.
More information4.3. Differentiation Rules for Sinusoidal Functions. How do the differentiation rules apply to sinusoidal functions?
.3 Differentiation Rules for Sinusoidal Functions Sinusoidal patterns occur frequentl in nature. Sinusoidal functions and compound sinusoidal functions are used to describe the patterns found in the stu
More informationMechanics of Structure
S.Y. Diploma : Sem. III [CE/CS/CR/CV] Mechanics of Structure Time: Hrs.] Prelim Question Paper Solution [Marks : 70 Q.1(a) Attempt any SIX of the following. [1] Q.1(a) Define moment of Inertia. State MI
More informationPage No.1. MTH603-Numerical Analysis_ Muhammad Ishfaq
Page No.1 File Version v1.5.3 Update: (Dated: 3-May-011) This version of file contains: Content of the Course (Done) FAQ updated version.(these must be read once because some very basic definition and
More informationNOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system.
NOTICE TO CUSTOMER: The sale of this product is intended for use of the original purchaser only and for use only on a single computer system. Duplicating, selling, or otherwise distributing this product
More informationMath 181, Exam 2, Fall 2014 Problem 1 Solution. sin 3 (x) cos(x) dx.
Math 8, Eam 2, Fall 24 Problem Solution. Integrals, Part I (Trigonometric integrals: 6 points). Evaluate the integral: sin 3 () cos() d. Solution: We begin by rewriting sin 3 () as Then, after using the
More informationSynchronous Machine Modeling
ECE 53 Session ; Page / Fall 07 Synchronous Machine Moeling Reference θ Quarature Axis B C Direct Axis Q G F D A F G Q A D C B Transient Moel for a Synchronous Machine Generator Convention ECE 53 Session
More informationx 2e e 3x 1. Find the equation of the line that passes through the two points 3,7 and 5, 2 slope-intercept form. . Write your final answer in
Algebra / Trigonometry Review (Notes for MAT0) NOTE: For more review on any of these topics just navigate to my MAT187 Precalculus page and check in the Help section for the topic(s) you wish to review!
More informationPart: Frequency and Time Domain
Numerical Methods Fourier Transform Pair Part: Frequency and Time Domain For more details on this topic Go to Clic on eyword Clic on Fourier Transform Pair You are free to Share to copy, distribute, display
More informationUSHA RAMA COLLEGE OF ENGINEERING & TECHNOLOGY
Code No: R007/R0 Set No. I B.Tech I Semester Supplementary Examinations, Feb/Mar 04 MATHEMATICAL METHODS ( Common to Civil Engineering, Electrical & Electronics Engineering, Computer Science & Engineering,
More informationCore Mathematics C3 Advanced Level
Paper Reference(s) 666/0 Edecel GCE Core Mathematics C Advanced Level Wednesda 0 Januar 00 Afternoon Time: hour 0 minutes Materials required for eamination Mathematical Formulae (Pink or Green) Items included
More informationMark Scheme (Results) Summer 2009
Mark (Results) Summer 009 GCE GCE Mathematics (6668/0) June 009 6668 Further Pure Mathematics FP (new) Mark Q (a) = rr ( + ) r ( r+ ) r ( r+ ) B aef () (b) n n r = r = = rr ( + ) r r+ = + +...... + + n
More informationS.Y. B.Sc. (IT) : Sem. III. Applied Mathematics. Q.1 Attempt the following (any THREE) [15]
S.Y. B.Sc. (IT) : Sm. III Applid Mahmaics Tim : ½ Hrs.] Prlim Qusion Papr Soluion [Marks : 75 Q. Amp h following (an THREE) 3 6 Q.(a) Rduc h mari o normal form and find is rank whr A 3 3 5 3 3 3 6 Ans.:
More informationPrecalculus Notes: Unit 6 Vectors, Parametrics, Polars, & Complex Numbers. A: Initial Point (start); B: Terminal Point (end) : ( ) ( )
Syllabus Objectives: 5.1 The student will explore methods of vector addition and subtraction. 5. The student will develop strategies for computing a vector s direction angle and magnitude given its coordinates.
More informationWrite your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
2009 Booklet No. Test Code : SIA Forenoon Questions : 30 Time : 2 hours Write your Name, Registration Number, Test Centre, Test Code and the Number of this booklet in the appropriate places on the answersheet.
More informationSection 4.3 version November 3, 2011 at 23:18 Exercises 2. The equation to be solved is
Section 4.3 version November 3, 011 at 3:18 Exercises. The equation to be solved is y (t)+y (t)+6y(t) = 0. The characteristic equation is λ +λ+6 = 0. The solutions are λ 1 = and λ = 3. Therefore, y 1 (t)
More informationPURE MATHEMATICS AM 27
AM Syllabus (014): Pure Mathematics AM SYLLABUS (014) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (014): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)
More informationMath 473: Practice Problems for Test 1, Fall 2011, SOLUTIONS
Math 473: Practice Problems for Test 1, Fall 011, SOLUTIONS Show your work: 1. (a) Compute the Taylor polynomials P n (x) for f(x) = sin x and x 0 = 0. Solution: Compute f(x) = sin x, f (x) = cos x, f
More informationMathematics Extension 1
Northern Beaches Secondary College Manly Selective Campus 04 HSC Trial Examination Mathematics Extension General Instructions Total marks 70 Reading time 5 minutes. Working time hours. Write using blue
More informationSECTION A. f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes.
SECTION A 1. State the maximal domain and range of the function f(x) = ln(x). Sketch the graph of y = f(x), indicating the coordinates of any points where the graph crosses the axes. 2. By evaluating f(0),
More informationASSIGNMENT BOOKLET. Numerical Analysis (MTE-10) (Valid from 1 st July, 2011 to 31 st March, 2012)
ASSIGNMENT BOOKLET MTE-0 Numerical Analysis (MTE-0) (Valid from st July, 0 to st March, 0) It is compulsory to submit the assignment before filling in the exam form. School of Sciences Indira Gandhi National
More informationChapter 9. Derivatives. Josef Leydold Mathematical Methods WS 2018/19 9 Derivatives 1 / 51. f x. (x 0, f (x 0 ))
Chapter 9 Derivatives Josef Leydold Mathematical Methods WS 208/9 9 Derivatives / 5 Difference Quotient Let f : R R be some function. The the ratio f = f ( 0 + ) f ( 0 ) = f ( 0) 0 is called difference
More informationSolutions to Homework Assignment #2
Solutions to Homework Assignment #. [4 marks] Evaluate each of the following limits. n i a lim n. b lim c lim d lim n i. sin πi n. a i n + b, where a and b are constants. n a There are ways to do this
More informationFP2 Mark Schemes from old P4, P5, P6 and FP1, FP2, FP3 papers (back to June 2002)
FP Mark Schemes from old P, P5, P6 and FP, FP, FP papers (back to June 00) Please note that the following pages contain mark schemes for questions from past papers. The standard of the mark schemes is
More informationPRE-LEAVING CERTIFICATE EXAMINATION, 2010
L.7 PRE-LEAVING CERTIFICATE EXAMINATION, 00 MATHEMATICS HIGHER LEVEL PAPER (300 marks) TIME : ½ HOURS Attempt SIX QUESTIONS (50 marks each). WARNING: Marks will be lost if all necessary work is not clearly
More informationPhysicsAndMathsTutor.com
PhysicsAndMathsTutor.com physicsandmathstutor.com June 2005 1. (a) Given that sin 2 θ + cos 2 θ 1, show that 1 + tan 2 θ sec 2 θ. (b) Solve, for 0 θ < 360, the equation 2 tan 2 θ + secθ = 1, giving your
More informationGOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD SCHEME OF VALUATION. Subject : MATHEMATICS Subject Code : 35
GOVERNMENT OF KARNATAKA KARNATAKA STATE PRE-UNIVERSITY EDUCATION EXAMINATION BOARD II YEAR PUC EXAMINATION MARCH APRIL 0 SCHEME OF VALUATION Subject : MATHEMATICS Subject Code : 5 PART A Write the prime
More informationC3 papers June 2007 to 2008
physicsandmathstutor.com June 007 C3 papers June 007 to 008 1. Find the exact solutions to the equations (a) ln x + ln 3 = ln 6, (b) e x + 3e x = 4. *N6109A04* physicsandmathstutor.com June 007 x + 3 9+
More informationCHAPTER 3 Applications of Differentiation
CHAPTER Applications of Differentiation Section. Etrema on an Interval.............. 0 Section. Rolle s Theorem and the Mean Value Theorem. 07 Section. Increasing and Decreasing Functions and the First
More informationCommunication about climate change on the German Baltic coast: Experience and mediated experience
Approaching national adaptation strategies to climate change in the Baltic States Tallinn 29/30 May 2012 Communication about climate change on the German Baltic coast: Experience and mediated experience
More informationJEE/BITSAT LEVEL TEST
JEE/BITSAT LEVEL TEST Booklet Code A/B/C/D Test Code : 00 Matrices & Determinants Answer Key/Hints Q. i 0 A =, then A A is equal to 0 i (a.) I (b.) -ia (c.) -I (d.) ia i 0 i 0 0 Sol. We have AA I 0 i 0
More informationGAUTENG DEPARTMENT OF EDUCATION SENIOR SECONDARY INTERVENTION PROGRAMME MATHEMATICS GRADE 12 SESSION 20 (LEARNER NOTES)
MATHEMATICS GRADE SESSION 0 (LEARNER NOTES) TRIGONOMETRY () Learner Note: Trigonometry is an extremely important and large part of Paper. You must ensure that you master all the basic rules and definitions
More informationFIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, TECHNICAL MATHEMATICS- I (Common Except DCP and CABM)
TED (10)-1002 (REVISION-2010) Reg. No.. Signature. FIRST SEMESTER DIPLOMA EXAMINATION IN ENGINEERING/ TECHNOLIGY- OCTOBER, 2010 TECHNICAL MATHEMATICS- I (Common Except DCP and CABM) (Maximum marks: 100)
More informationNUMERICAL 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 informationSBAME CALCULUS OF FINITE DIFFERENCES AND NUMERICAL ANLAYSIS-I Units : I-V
SBAME CALCULUS OF FINITE DIFFERENCES AND NUMERICAL ANLAYSIS-I Units : I-V Unit I-Syllabus Solutions of Algebraic and Transcendental equations, Bisection method, Iteration Method, Regula Falsi method, Newton
More information11.4. Differentiating ProductsandQuotients. Introduction. Prerequisites. Learning Outcomes
Differentiating ProductsandQuotients 11.4 Introduction We have seen, in the first three Sections, how standard functions like n, e a, sin a, cos a, ln a may be differentiated. In this Section we see how
More information