INTERNATIONAL ISLAMIC UNIVERSITY MALAYSIACOURSE OUTLINE

Kulliyyah Engineering

Department Science in EngineeringProgramme All ProgrammesCourse Title Engineering Calculus ICourse Code MTH 1112Status CoreLevel 1

Credit Hours 3Contact Hours Lectures: Sec.6 (TTH: 2:00-3:20, E1-3-24), Sec.3 (TTH: 3:30-4:50, E1-3-24)

Tutorial: Sec.6 (-----------------------------), Sec.3 (-----------------------------------)

Pre-requisites -Co-requisites -Instructor(s) Saad M. Saleh saadtm2000@gmail.com, Tel: 014-9360720

Consultation Hrs: (given later) or by appointmentSemester Offered Every SemesterCourse Synopsis

Complex numbers, Limits, continuity, differentiation and integration with applications, Transcendental functions, techniques of integration, infinite series and power series.

Course Objectives The objectives of this course are to:1. Present and develop a clear understanding of definitions, concepts, rules, theorems,

techniques, and applications of differential and integral calculus.2. Encourage students to think visually and numerically by generating and interpreting

graphs, using calculators and computers whenever possible.3. Strengthen the students’ number sense in order to be able to recognize an absurd answer

by encouraging them to make mental estimates, to appeal to their intuition, and to work on problems both individually and in groups.

Learning Outcomes

Upon completing this course, the students should be able to:1. Understand and apply the algebraic properties of complex numbers in Cartesian and polar

form.2. Apply the concept of limit and continuity in computing limit and determining the continuity

of a function.3. Understand the concept of derivatives and computing techniques with application such as,

linear approximation, Newton’s Method, L’Hopital’s rule, curve sketching, optimization and related rates.

4. Understand the concept and evaluating techniques of anti-derivative, fundamental theorem of calculus, and application of integration to evaluate area, volume and arc length.

5. Apply the integration techniques such as substitution, integration by part, trigonometric substitution, partial fraction and evaluate the improper integrals.

6. Identify the sequence and the infinite series and apply different tests such as nth- term test, integral test, comparison test, and the ratio test, root test for convergence and divergence.

7. Determine power series (Maclaurin, Taylor series) for various functions and apply them to generate new series.

InstructionalStrategies

Lectures and Tutorials

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Course AssessmentState weightage of each type of assessment.

LO Method Percentage

1-6 Assignments/Quizzes 10

1-3 Mid-term Examination(s) [Sunday, 13/2/2011, 10am-12noon] 45

1-7 Final Examination 45

Content Outlines

Weeks Topics Task/Reading

1 Complex Numbers: Introduction, algebraic properties, modulus and conjugates, geometric presentation, polar form and Euler’s formula, De Moivre’s Theorem.

Handout

2,3Limits and Limit and Continuity:

Review: Transcendental functions, Concept and evaluating techniques of limits and continuity for square-root and transcendental functions, intermediate value theorem, limit involving infinity, asymptotes.

Chapter 0,1

4,5 Differentiation:

Concept of derivative, derivative and differential, differentiability and continuity, differentiation techniques for algebraic and transcendental functions, implicit differentiation, Mean value theorem

Chapter 2

6,7 Applications of differentiation:

Linear approximation, Newton’s method, indeterminate form and L’Hopital’s rule, local and absolute extrema, increasing and decreasing function, concavity and inflection points, transformations on curves, curve sketching, optimization, concept of derivative as slope and rates of change.

Chapter 3

8,9 Integration:

Anti-derivatives, definite and indefinite integration, basic techniques, concept of definite integration as area and Riemann sum, fundamental theorems of calculus, integration by substitution.

Chapter 4

10 Application of integration:

Area between curves, volume of solid by slicing/disks/washers and cylindrical shells methods, arc length and surface area.

Chapter 5

11,12 Techniques of integration:

Integration by parts, Trigonometric techniques of integration, integration by partial fractions, improper integration.

Chapter 6

13, 14 Infinite series:

Infinite series and convergence, geometric series, harmonic series, tests for series of nonnegative terms: nth – term test for divergence, integral test and p-series, comparison test, alternating series: alternating series test, absolute and conditional convergence, ratio test, root test, power series: radius of convergence, Taylor and Maclaurin series, application of Taylor series.

Chapter 8

References RequiredRobert, T.S. & Roland B.M., (2006). Calculus, (3rd ed.), McGraw Hill.

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Recommended1. Anton, H., Bivens, I., & Davis, S. (2002). Calculus (7th Ed). John Wiley.2. Edwards, C.H., & Penny D.E. (2002). Calculus (6th Ed). Prentice Hall.3. Finney, R.L., Weir, M.D.L.F., & Thomas, G. (2001). Calculus

(10th ed). Addison-Wesley Publishing Company.4. Johnston, E.H., & Mathews J.C. (2002). Calculus. Addison Wesley.5. Strauss, M.J., Bradley, G.L., & Smith, K.J. (2002). Calculus (3rd ed). Prentice Hall.

Proposed Start Date (Semester) Semester II, 2009/2010

Batch of Students to be Affected Semester II, 2009/2010

Practice Assignment TutorialEx.0.1 31-74 (odd) 44 (32/0.2) 46(38/0.2)

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Ex.0.3 1-54(odd) 3,6,14,27 4,5,18,29Ex.0.4 53-64 42,54,60 39,56,64Ex.0.5 31-54(odd) 34 38Ex.0.6 23-52(odd) Using Graph of

Sketch the graph of

Using Graph of

Sketch the graph of Ex.1.2 1-6 3 6Ex 1.3 1-34(odd), 39-42, 43-48 3,7,12,13,21,27,37,39 6,14,17,28,35,42Ex 1.4 1-50(odd) 6,18,19,22,26,32,34 5,17,23,28,33Ex.1.5 1-38(odd) 5,8,22,26,38 7,15,24,30,36Ex.2.2 1-30 (odd) 8,40 9,39Ex.2.3 1-47 (odd) 12,30,34 16,32,36Ex.2.4 1-16(odd) 4,16 3,15Ex2.5 1-50(odd) 16,20,36 15,21,37Ex.2.6 1-20 (odd),25-32(odd),42-45 10,19,26,44© 14,20,31,45Ex.2.7 1-54(odd) 10,19,36,41 8,24,35,43Ex.2.8 1-16(odd), 29-38(odd) 13,20,23,34 12,19,24,36Ex.3.1 1-12(odd), 21-40(odd) 1,9(a),21,37,42 4,10(a),22,38,44Ex.3.2 1-38 (odd) 8,18,24,32,38 10,23.25,33,37Ex.3.3 5-38(odd) 11,15,20,29,32,36 12,16,26,31,34Ex.3.4 1-38(odd) 4,10,12,16,24,36 2,8,11,15,21,38Ex.3.5 1-46(odd) 2,8,10,14,24,27,42 1,6,9,13,23,30,43Ex.3.6 1-32(odd) Handout HandoutEx.3.7 1-35(odd) 4,8,10,16,24,32 3,7,9,15,25,31Ex.3.8 1-19(odd) 2,6,8,12,27 1,5,9,11,25Ex.4.1 5_30(odd),39-42 20,22,26,50 16,21,49Ex.4.2 9-22(odd),35-38 15,21 14,22Ex.4.3 5-10 6(n=4),15 5(n=4),16Ex.4.4 1-10,29-32 7,15,47 8,16,48Ex.4.5 1-20(odd), 33-36,39-44,53-58 16,30,34,41,49 18,29,35,42,50Ex.4.6 1-40(odd) 6,25,28,39 7,21,29,37Ex.4.8 17-30(odd) 11,16,24,30 12,14,26Ex.5.1 1-26(odd) 8,20,22 6,23Ex.5.2 1-12(odd),17-20,25-44(odd) 6,25,31,33 7,27,32,35Ex.5.3 9-26(odd) 18,23 17,24Ex.5.4 5-14(odd) 5,35 6,36Ex.6.1 1-48 (odd) 4,10,32,38 17,21,31,37Ex 6.2 1-48(odd) 5,7,10,17,28 8,9,25,27Ex 6.3 1-30 (odd) 2,5,9,13,22,24 1,10,14,19,21Ex 6.4 1-30 (odd) 12,18 8,28Ex.6.6 13,27,47 14,34,46

NOTE:1. Tutorial questions will be explained and solved by the tutor in regular tutorial classes.2. Quizzes will be conducted during tutorial classes. First quiz will be from Chap 0, 1, 2nd quiz will be from Chap, 2, 3

and 3rd quiz will be from Chap 4, 5. Duration of each quiz will be 45-60 minutes.

COMPLEX NUMBERS

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Moving to a greater level of abstraction, the real numbers were extended to the complex numbers C. This set of numbers arose, historically, from the question of whether a negative number can have a square root. From this problem, a new number was discovered; the square root of negative one. This number is denoted by i, a symbol assigned by Leonhard Euler. Each of the number systems is a subset of the next number system. Symbolically, . Complex numbers are frequently used in many areas such as contril theory, signal processing, fluid dynamics, quantum mechanics, etc,.

Definition: . Note that in Electrical Engineering, j is used for i because i is a notation for current.

.

Note that

Geometric Representation:

Q. Is a complex numbers? Justify your answer.

Q. Is it true that every real number is also a complex number? Justify.

Equality: Two complex numbers are equal if and only if .

If

Q. Is 2+3i = 3+ 2i . Is 1+i = i +1

Addition: If

If

Properties: z + 0 = z where 0 = (0,0)

Multiplication: If

If Properties:

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z .1 = z where 1=(1,0)

Q. If . Find

Q. If

Conjugate: then conjugate of z denoted as

Properties:

Q.

Division:

Moduli/Absolute value:

Properties:

Q. Let

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 ,  . Is it true that  = 

Q. Write at least two conditions where rather than

Q. Is it true that Q. For any two real numbers a and b , we can say . Can we compare complex numbers

. Justify your answer. What about saying

Polar Form: where

Euler Formula: Example: Write in polar form.

Properties:

Q. Interpret geometrically.Q. For what complex numbers a real number?Q. Write the following complex numbers in Polar Form, using the usual conventions for the argument:

Q. Using Euler formula, evaluate .

CHAPTER 8(Power Series, Taylor series)

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is a power series in power of (x-a), where are real

constant and “a” is the centre of the series.

is a power series in x with centre at 0 (Maclaurin series).

is called nth partial sum.

If series is convergent otherwise divergent.

Each Series is always convergent at its centre.

If converges for all points in then

i. The region is called interval of convergence. R is called radius of convergence.ii. If R = 0 then series converges only at centre. If R = then series converges everywhere or . In other

situation R can be any finite positive real number.iii. is called interval of convergence.

NOTE: If a series converges for then it must be divergent for and for , it may converge at every point or on some points or may not converge at all. HOW TO FIND RADIUS OF CONVERGENCE R ?

Find

i. If L = 0 then R = ii. If so series converges only at centre “a”.

iii. If

Q. Find radius and interval of convergence for the series

(i) (ii)

Sol: (i) , and interval of convergence is

(ii) and

interval of convergence is

Within interval of convergence, series converges absolutely and uniformly. It can be differentiated and integrated term by term and resulting series has the same radius as well as interval of convergence.

Term wise addition; =

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Term wise multiplication; =

If the series correspond to a function say then the radius of convergence in most of the cases* is

the distance of the closest singular point (point of discontinuity) from the centre of the series.

Example:

TAYLOR SERIES/POLYNOMIAL.

Taylor series, Maclaurin series and in general, the series are used to approximate a function. To approximate a function at a targeted point, we use series at another suitable (but within interval of convergence) point

as the centre of the series. Let be a given function. Assume exist and exist , then

is called Taylor series of at centre

is called Taylor polynomial of

degree n which approximate at . In case then the Taylor series/Polynomial is known as Maclaurin series/polynomial.Example:

Find the Taylor series of

Solution:

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Examples:

Applications: New Series by substitution:

Using

Change in

Similarly

New Series by differentiation and/or integration:

To evaluate the limit:

To approximate the Integral:

Largest or smallest function:

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1. 2. 3.

, ,

If we consider only up to and then

Q. Compare

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