Engineering Mathematics Syl Lab Us

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I'ROGRAMME:

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t)REREQUISTTES:

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(lourse Syllabus adopted from theI nir crsity of 'Technology Jar4aica

COURSE DESCRIPTION

This course seeks to build in students an investigative and reasoning approach tofundamental mathematical concepts that they will need in later years. The earlier topicswill be ones they have encounte'red in high school but the emphasis at this level will beon understanding ofthe principles used.

COURSE ORIECTTVf,S

After completing this course, students should be able to:

(i) Solve and manipulate different types of algebraic equations and expressions.(ii) Define sets and use set operations to solve problems.(i i i ) Define a fi.rnction and interpret fu:nctional relationships.(iv) Use graphicaltechniques to solve problems.(v) Apply indices and logarithms to problem solving.(vj ) Solve and manipulate simple tigonometic equations, incl udi ng graphical

i nterpretati on and tigonometic i dentiti es(vii) Solve practical problems related to Ap and Gp.

COURSE CONTENT

l. Indices, Logarithmsandsurds2. Equations and Inequations3. Sets4. Relations and Functions5. Crrehs

Series and SequencesTrigonomety

Associate Degree in Engineering

Semester I, year I

Mathematics I

4 hrs/ w-cek lecture + 2 hrs tutorial/week (approx. l3 weeks)

4

CXC (General Proficiency) Grades I, Il or It | (Since Sept1998), GCE OT-evel Grades A" B or C orthe equivalent.

3.0

6.7.

t80 r7/07t2008

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2.2z . )

TJNIT I: INDICES, LOGARITHMS AND ST'RDS (8 hours)

I ' I l)el'ine the terms base, index, logarithm, power and exponent1.2 State rurd use the laws of indices and logarithms1.3 Solve cqualions using indices and logarithms1.4 Simplily and evaluate expressions involving surds1.5 lixpress a surd fiaction with a raticmal denominator

UNIT 2: EQUATIONS AND INEQUATIONS (8 hours)

2-l Change the subject of fbrmulas and equations including those involving roots andF)Wers.Solve linear in-equations in one variableSolve quadratic equation by:

(i) factorization

( ii) using the quadatic formula/completing the squarc

Sol ve simultaneots equations:

( i) two linear equations in two or three unknowns.( ii) one linear and one quadratic, or in two quadratic equations.

Perform polynomial operations: expansion, d i v i s i on and factorization.State the remaindertheorem and use it to evaluate the remainder when apolynomialexpression (x), is divided by aterm ax * b, where a, b eZ.State the factor theorem and use it to factorize a polynomial of the third c fouttldegree.

UNIT 3: SETS (4 hours)

Define a set as a collection of objercts.Define the following sets: natural numbers, whole numbers, integers, rational andirrational numbers, real numbers, complex numbers.

Apply basic concepts in set theory: empty set, subset, union, intersection,disjoint, complement, difference oftwo sets.

3.4 Solve numerical problems arising from the intersection of not more than three sets.3.5 Use Venn diagrams to represent positions from which valid conclusions can be made.3.6 Use set builder and interval notation to solve problems.

UNIT 4: RELATIONS AND FUNCTIONS [6 hours l

2.4

2.7

2.52.6

3.13.2

J . J

1 .14.24.34.4

( ourse Syllabus adopted fiom theI lnivcrsity ol' Technology Jamaica

Define relations and functionsDefine the domain and codomain of a functionDefine onto, into and swjeaive, injective, bijectiveConstruct a composite function from given functions

8 l t7/07t2008

4.54.6

Find the inverse of a functionldentify different types of functions: polynomial, rational, irrational.

exponential ; and logarithmic.

Define and illustrate sequences and seriesRepresent a given series using sigma notation

Define an arithmetic progression (AP)

Find the nth term and the sum of the first n terms of an AP

Define a geometric progression (GP)

Find the nth term and the sum of the frst n terms of a GP

Find the sumto infinrty of aGPSolve problems involving AP and GP

Derive Pascal's triangle and use it to perform binomial expansions,

Expand (1 + ax )n using the binomial theorem for rational values of n

Dstermine the values of x for which (l + ax)" converges'

Use the expansions from Pascal'siBinomial theorem to perform

approximations.

5 .1

5.2

'UNIT 5: GRAPHS (8 hours)

5 .35 .4

5 .55 .6

Graph linear firnctions of the form ay + bx + c : 0 and identiff gradients and vertical

interceptsFormulate the equation of a staight line given:

(i) the coordinates of two Pointsiiil gradientofthe line andthe coordinateofapoint lying onthe line .iiiil aparalleVperpendicular line to the given line and apoint lying on iL

Solve equations and inequations in two unknowns graphically

Draw graphs of special functions e.g. polynomial, exponential and logarithmic

formsT'ransform non-linear equations into equiva|:nt linear equations

Plot the graph of these linear transforms (in 5.5) in order to: (i) verify the

laws (ii) determine unknown parameters.

UNIT 6: SERIES & SEQUENCES (8 marks)

6 .16.26.3

6.46.5

6.66.76.86.96 .106 .116 .12

U|{IT 7: TRIGONOMETRY (10 hours)

1 .l Define the ffigonometric functions as a ratio of the sides of a right-angled triangle'

7.2 Derive anduse the following identities: (i)

$rthagorean (ii) Compound and doubleangles (iii) Factor fbrmulae

t

I

t'oursc Syllabus udoPted from the( Inilersity of Tcchnology Jamatca

82 r710712008

7.37.47.5

(i) Class Test I(ii) Class Test 2

FINAL EXAMINATION:

BRKAKDOWI\ OF HOURS

Didactic Instruction

Tutorials

Exam i nation/Assessment

52 hours (l 3 weeks @ 4 didactic lirs. /week)

26 hours

2 hours (final exam)

Prove trigonomefic identiti es .Solve tigonometic equations ingiven intmralsSkctch the family of curves y: R Sin {w t + a) by considering amplitude,period and phase shift.

{.0 INSTRUCTIONAI,/LEARNING APPROACIIES

This course is comprised of lechres and tutorials. Lectures will be enhanced wit handouts.

Studenb will be given worksheem wtrich drey re expected to atfempt on their own. Intutorial classes, worksheets will be discussed and solved.

Students are expected to play an active role in the learning process.

ASSESSMENT PROCEDTIRES

COURSE WORK:

4.O

5.O

20%200h

60%

6.0 TEXTBOOKS AIID REFERENCES

RECOMMENDED TEXT:

L

3.

Bostock, L. and Chandler, S.; Mathematics - The Core Course for

A- Level, HUBS Publishing Co.

Backhouse & Houldsworth; Pure Mathematics, Book l.

Safier, Fred; Schaum's Outline of Theory and Problems of Pre Calculus,(Schaum's Outl ine Series). Me Graw- Hil l .

('ourse Syllabus adopted from thet Jrriversity of

'lcchnology Jamaica

83 t7t07t2008

PROGRAMME:

('OURSE TITLE:

(]OURSE CODE:

DURATION:

CREDIT VALUE:

( ourse Svllabus adopted from thcllniversity of Tcchnology Jamarca

Associate Degree in Engineering

Mathematics II

Hours Lecture + 2 Hours Tutorial

COURSE DESCRIPTION

T'his course seeks to build in students an investigative and reasoning approach tofundamental mathematical concepts that they will need in later years. It builds onconcepts already leamed in high school and in semester L

COURSE OBJECTIVES

After completing this course, students should be able to:

i. Resolve vectors algebraically- Perform operations involving vector andscalar products.

ii. Represent phasors as complex numbers and resolve graphically/algebraically. Solve simple algebraic equations with complex roots.

Use matrices to solve simultaneous equations.

Evaluate limits of functions.

Use the derivative to solve practical problems on maxima and minima.

Evaluate area under curve and volume of revolution.

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2.0

84 t710712008

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l. Matrices and Dctcrminants2. Vectors3. ('omplex Numbcrs4. Calculus

tiNlT l: MATRICES AND DETERMINANTS (6 hours)

Define the dit'ferent types of matrices:(i) row, column. null, square, diagonal. identity

Perform matrix algebra of(i) addition/subtraction(ii) Multiplication

1.3 Define the determinant of a matrix.Evaluate determinants of the second, third and fourth orders.

1.4 Define the inverse of a square matrixEvaluate the inverse of matrices up to the third order.

L5 Solve systems of linear simultaneous equations up to the third orcler using:(i) matrix inversion(ii) Cramer's Rule

L l

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UNIT 2:

2 .12.22.3

2.42.52.6

3.13 .2

a aJ . J

3.4

Course Syllabus adopted from theI iniversity of

'fcchnology Jamaica

VECTORS (6 hours)

Define scalar and vector quantities.Reprcsent vectors using coordinate geometry up to $l!.Resolve vectors using.(i) Parallelogram law(ii) algebra

Define and evaluate scalar (dot) product.Define and evaluate vector (cross) product of two vectors.Solve problems involving application of vectors to engineering situations.

UNIT 3: COMPLEX NUMBERS (6 hours)

Define complex numbers algebraically and graphically.convert complex numbers from one form to another (cartesian Polar andExponential).Perform algebraic operations on complex numbers.State and apply De Moivre's theorem for rational indices.

85 l7/07t2008

"1. I

t jNI ' t { : ( 'Al ,Ct j [ ,1]S ({ hours)

l . t lvt I ' l 's4. l . I Dc lme the l imi t o t 'a lunct ion at a point .,1.1.2 l)et ine continuity ol 'a f irnction at a point.4 .1.1 Show that

f . , . ILi,n(l f r/o; - c, Lim(l * t/n)n. * e*

4.1.4, t"r*" , . l imi ts . , r"r ; ; r . rus f i rnct ions.

DITIFERENTTATION

4.2.1 Deline the derivative of a firnction llx) i.r"s:

(14 hours)

f '(x)- Liml l - l o

4.2.2 Dilferentiate functions lrom first pnnciples.4.2.3 State and use the Chain, Product rrnd Quotient rules ofdifl'erentiation,4.2.4 DilTerentiate functions including trigonomctric, cxponential andlogarithmic lunctions.4.2.5 Locate and classity stationary points of a tunction.4.2.6 Solve practical problems involving maxima and minima.4.2.7 Approximate small increments using dillbrentiation.4.2.8 Solve problems involving rates of change.

INTEGRATION (16 hours)

4.3.1 Define integration as the reverse of difTerentiation.4.3.2 f)etermine the indefinite integral of functions involving axn,

(a + bx)n, Sin(ax), Cos(ax), e* , In(ax).4.3.3 Integrate functions by first resolving into partial fractions.4.3.4 Integrate product of two functions by parts.4.3.5 [Js integration to evaluatc the area under a curve and volume ofrevolution.

INSTRUCTIONAL LEARNJNG APPROACHES

'fhis course comprises of lectures and tutorials.

Students will be given worksheets which they are expected to attempt on theirown. In tutorials. worksheets will be discussed and solved.

I

1.2

4.3

4.0

Coursc Syllahus adopted tiom the

I Jnivcrsity oi Tcchnology Jamaica

86 1710712008

II

Students are expccted to play an active role in the lcaming process.

5.0 ASSESSMENT PROCEDURES

Course Work:

Course Syllabus adopted from theUnivcrsrty of I echnology Jamaica

( i) ' Iest #1

(i i) ' l 'est #2

Irinal Examination:

IIREAKDOWN OF HOURS

I)idactic Instruction 52 hours (13 weeks @ 4 didactic hrs.

'I'utorials 26 hours (13 weeks (@2hrs. /week)

Examination/Assessment 2 hours (final exam)

TEXTBOOKS AND REFERENCES

Required Text:Bostock. L. and Chandler, S.; Mathematics - The Core Course for A- Level.ELBS Publishing Co.

RecommendedText:L Passow, Elli; Schaum's Outlines of Understanding Calculus Concepts

(Schaum's Outline Series). Mc Graw-Hill,

2. Stroud, ICA.; Engineering Mathematics (Fourth Edition), MacMillanPress Ltd.

NAME OF SYLLABUS WRITERJDEVELOPER

Lorraine Fuller, Gustwell Weir & Judith Delisser

DATE OF REVIEWPRESENTATION

November.2003

20%20%

60%

6.0

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8.0

9.0

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87 r7t07t2008

PROGRAMME:

(.OURSE TITLE:

COURSE NUMBER:

t)URATION:

(]REDIT VALUE:

I'REREQUISITES:

Associate Degree in Engineering

Engineering Mathematics 1 -Calculus

45 Hours

Three (3)

Mathematics 1' Mathematics lI' CAPE Mathematics

(Unit l&Unit 2), or GCE A'Level Mathematics

r.0

2.0

COURSE DISCRIPTIONThis course provides $udents of engineering with the basic knowledge necessary

for further dcvelopment of thc riathematical skills needed for a calccr in

engineering. They are introduced to calculus and linear algebra'

COURSE OBJICTIVIS

0verall Generrl ObiectivesUpon successful comiletion of this course' the student should:

r show profrciency in using mathematical tools for use in solving engineering

prob lems. . r , , 1_ -_ . . .

bxhibit competence in the use of mathcmatics to model and solve englneenng

problems.'fno*howtoanalyse,abstractandgeneralisemathematicalidcas'

Specific Obiectives

IIII

C'ourse S1'llabus adopted liom the

I ln i \crs l ty of Icchnology Jamaicr t

88 1710712008

A. Review of limits, continuity, differentiation, special functions

Upon successful completion of this unit, the student should be able to:

1. Determine the limit of a function as its independent variable

approaches a sPecified value.Z. dxplain the concepts of degree of continuity of a function.

3. Use special functions which arise in engineering applications

( [ inv erse] tri gonometric functions, lo gari thms, exponenti al, hyp erbolic

functions).4. Explain the concept of a derivative of a function'

5. Deiermine the derivative of various functions (simple and composite)'

Linear AlgebraUpon succissful completion of this unit, the student should be able to:

Solve a system of equations using Gaussian elimination'

Determine the rank of a matrix.Calculate determinants of 2nd and 3'd orderDetermine the eigenvalues and eigenvectors of a matrix'

Identify bilinear, quadratic, symmetric, Hermitian and skew-

Hermitian forms of matrices.

I ntegration (Continued)-Upoi ru"""riful completion of this unit, the student should be able to:

1. Understand that integration by substitution is the reverse

B.

l .2.3 .4.5 .

c.

2.3 .

application of the chain rule.tise ttte method of substitution to integrate functions'

Approximate the area under the graph of a continuous function by

sums of areas of rectangles, using the X notation'

4. Explain the TraPezium rule.5. UsL the Trapezium Rule as an approximation method for

evaluating the area under the graph of a function

6. Apply integration to the finding of areas and of volumes of

revolution.

Simple Ordinary Differential EquationsUpon successful tompletion of this unit, the student should be able to:

1. Formulate and solve differential equations of the form y'--f(x)'

2. Identify situation that can be modelled as exponential decay or

growth.3. Formulate a differential equation from a simple engineering model

involving rate of change'4. Solve first order linear differential equations with constant

coefficientS, €.8. y,:f(x), y,-lv__f(x), where & is a real constant andf

is a linear function.

89C'ourse Syllabus adoPted tiom the

I,Jnivcrsity of -fcchnology Jamatca

1710712008

E. Laplace Transformationsupon successfur completion of this unit, the student shourd be able to:1 Explain the concept of the Laplace Transformation.2' Determine the inverse Laprace transform of a function3. Explain the Laprace transiorm as a mapping frorn the time scare to

the complex plane.4' Determine the Laprace Transform of the folrowing simpre

lunctions:5' Prove that Laprace transform theorems on tierivatives.6. Prove the Laplace transform theorems on integrals.7 ' State and appry the first shifting theorem: shifting on the s-axis8' state and appry the second shifting theorern: ,hinirrg on the t-arx.9. Use the theorems on differentiatio-n and integruiro'orLaprace

transfiorms_l O. State and apply the convolutions theorem.l1'

3^"LT1* the inverse Laprace transform of a function using partial

Iractlons.12' Determine the Laprace Transform of a periodic function.

Series and Sequencesupon successful compretion of this unit, the student shourd be able to:l. write down a specific term from the formul" rorirr" nin-;;;;",

from a recunence relation, and dear with simple apprications or-Mathematical Induction.

2. Distinguish between the behaviours of bounded and unboundedsequences.

3' Distinguish between convergent and divergent sequences.4. use the sigma notation and identify the nrh i""" or u series, givcn

the sigma notation.5. Define the mth partial sum of a given series.6. Define a series as the s"qu"n"e of its partial sums.7 ' Define convergence of a series in terms of convergence of the

sequence of its partial sums.8. Define the sum of a convergent series as the rimit of the sequence

of its partial sums.

c.

3.o couRsE CONTENT AI\[D coNTExr (see specific objectives)

4.O INSTRUCTIONAL/LEARNING APPROACHES

( ourse Syllabus adopted from the[ 'nivcrsity of 1'cchnology Jamaica

-I_l

90 17t07t2008

5.0 ASSESSITENT PROCf, DURES

Coursework:(i) Assignments(ii) Monthly tests(iii) Laboratoryexercises(iv) Projects

Final Examination 600/o

6.0 BREAKDOWN OF HOURS

InstructionTestsEx amination/Assessment

36 hours3 hours2 hours final exam (2 weeks for exam)

7.0 TEXTBOOKS AND Rf,FERf,NCESRequiredKreyzig, Erwin "Advanced Engineering Mathematics" (or equivalent)

Recommended Reading

8.0 NAMf, OF SYLLABUS WRITER(SyDEVELOPER(S)

9,0 DATE OF PRESENTATION

Course Syllabus adopted from theIrniversity of fcchnology Jamaica

9 l 17t0712008

I }ROGRAMME:

(]OURSE TITLE:

C]OURSE NUMBER:

DURATION:

(:REDIT VALUE:

PREREQUISITES:

('ourse Syllabus adopted from the

t r r ivcrs i ty of Icchnology . lamatcl

(a) simply vs. multiply-connected regions.(b) interior vs. cxterior vs. boundary points.

r.0

2.0

COURSf, DESCRIPTIONThis course inhoduces the student to the calculus of more than one variable.Calculus is critical to engineering analysis and design.

COURSE OBJECTIVES

Overrll General ObjectivesUpon successful completion of this course, the student should:o Show proficiency in using mathematical tools for use in solving engineering

problems.r Exhibit competence in the use of mathematics to model and solve engineering

problems.o Know how to analyse, abstract and generalise mathematical ideas.

Specilic Objectives

A. Functions of more than one vrriebleUpon successful completion of this unit, the student should be able to:L Explain the following concepts:

r dependent variablesr independent variables. single-valuedfunctions

2. Distinguish between:

Associate Degree in Engineering

Engineering Mathematics ll-Calculus

cMP-1003

45 Hours

Three (3)

Mathematics l, Mathematics II, CAPE Mathematics

(Unit 1&Unit 2), or GCE A'Level Mathematics

. multiple valued functionsr limit pointso neighbourhood ofa point

92 1710712008

Determine the limit in a deleted, 6, neighbourhood of a point.F)etermine the iterated limit of a function.

4. Explain the concept of (uniform) continuity of a function of ntorethan one variable.

B. Partial DerivativesUpon successful completion of this unit, the student should be able to:l . Use the following notations for the partial derivative of a function.

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ri* { ri* f tr,yll* ri* { ri,n lt..,y)ir - r 16 [ / -+/O J Y-+Yo I . r - >ro )

2. Define the partial derivative of a function with respect to one of itsindependent variables.Determine the higher order partial derivatives of a multiplevariable function.Explain the concept of the differential of a function.Differentiate composite functionsApply Euler's theorem on homogeneous functions.Differentiate implicit functions of more than one variable.Use the Jacobian formulation to deterrnine the partial derivativesof implicit functions.Apply the following theorems on Jacobians:Prove the first mean value theorem for functions of two variables.Prove Taylor's theorem of the mean for functions of more than onevariable.Determine the region of convergence of a function of more thartone variable.

J .

4.5 .6.7 .8 .

Ciourse Syllabus adopted from thet )rriversity of fechnology Jamaica

9.10 .I t .

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C. Applications of Partial DerivativesUpon successful completion of this unit, the student should be able to:

Determine the tangent plane to a surface.Deterrnine the normal line to a surfaceDetermine the tangent line to a curye.Determine the normal plane to a curve.Determine the envelope of a one parameter family of curves'f)etermine the directional derivative of a function at a point.Explain and apply the concept of equipotential (level) surfaces.Apply Leibnitz's rule, in differentiating under the integral sign.Showlprove the necessary and/or sufficient conditions for afunction f(x,y) to have a relative extremum.

93 t710712008

F

Determine the relative maxima or minima of a function of morethan one variable.Show/Prove necessary conditions for extremising a functionF(x,y,z) subject to the constraint condition G(x,y,z):0.Calculate the maximum/minimum values of a function F(x,y,z)subject to constraint(s) G(x,y,z)=O.Apply partial derivatives to the determination of errors of afunction.

.

10 .

I.

II l .

t2.

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D. Complex VariablesUpon successful completion of this unit, the student should be able to:l. Delineate regions in 2 space using complex numbers, (circles, annuli,

straight lines, etc.)2. Explain the following concepts w.r.t. complex functions: continuotrs,

di fferentiable, analyic.3. Determine the derivative of a complex number.4- Apply cauchy-Riemann equations to determine the analyticity of

complex functions.5. Perform the following operations on a complex number: determine the

(a) nth root, (b) natural logarithm, (c) exponential.

Mu ltiple Variable Integrationupon successful completion of this unit, the student should be able to:l. Determine the double integral of a function F(x,y) over the region

R.2. Calculate iterated integrals.3. lnterchange the order of integration of a double integral.4. Determine the triple integral of a function F(x,y,z).5. Interchange the order of integration of a murtiple integral using the

Jacobian formulation.

Line & Surface Integrals and Integral TheoremsUpon successful completion of this unit, the student should be able to:l. Determine the line integral of a function F(x,y) along a curve.2. Express a line integral in vector form.3. Explain the properties of line integrals.4. Prove and apply Green's theorem in the plane.5' Express the necessary and sufficient conditions for an integral to

be independent of the path C joining in a region6. Determine the surface integral of a function F(x,y) over a surface

S .7. Prove and apply the divergence theorem (Green's theorem in

space).

E.

F.

('ourse syllabus adopted fiom theLJniversity of Technology Jamarca

94 17t07t2008

Prove and apply the Stokes'theorern.

Vector AnalysisUpon successful completion of this unit, the student should be able to:

J .

Explain the concept of a vector function.Define the term "continuous vector function"-

Determine the derivative of a vector, with respect to a dependentvariable.Determine the differential of a vector.Prove the following theorems of derivatives of products.Interpret vector derivatives geometrically.Define the gradient of a multiplo dcpcndcnt variable function-

Determine the divergence of a vector function.Determine the curl of a vector function.

Fourier SeriesUpon successful completion of this unit, the student should be able to:l. Understand the concept of a periodic function'2. Derive the Euler formulas.3. Determine the Fourier series of a continuous function with period 2n.4. Determine the Fourier series of a continuous f'unction with arbitrary

Distinguish between odd and even functions.Apply the simplifications in the Fourier Expansions of odd and evetrfunctions.

7. Determine the half-range expansions of given functions using eitherodd or even periodic extensions.

COURSE CONTENT AND CONTEXT (see specific objectives)

INSTRUCTI ONAI./I, EARI\ING APPROACH ES

ASSESSMf,NT PROCEDURES

( i )(ii) Monthly tests(iii) Laboratory exercises( iv) Projects

Examination

5.0 BREAKDOWN OF HOURS

3.0

3.0

4.0

('oursc S.vllabus adopted from thet Jrrrr'crsity ot l echnology Jamaica

t710712008

7.O

8.O

Course Syllabus adoPied tiom the

t)niversity of Technology .lamatoa

36 hours3 hours2 hours final exam (2 weeks for exarn)

6.0

lnstructionTestsEx amin ati ontAssessment

TEXTBOOKS AND REFERENCESRequired

Recommended Reading

NAME OF SYLLABUS WRITER(S)/DEVELOPER(S)

DATE OF PRESENTATION

96 1710712008