COURSE UNIT TITLE

: CALCULUS II

DEGREE PROGRAMMES

Description of Individual Course Units

Course Unit Code Course Unit Title Type Of Course D U L ECTS
MAT 1032 CALCULUS II COMPULSORY 4 2 0 9

Offered By

Mathematics

Level of Course Unit

First Cycle Programmes (Bachelor's Degree)

Course Coordinator

ASSISTANT PROFESSOR MURAT ALTUNBULAK

Offered to

Mathematics (Evening)
Mathematics

Course Objective

The aim of this course to learn sequences and series of real numbers for Taylor series of functions and to learn multivariable calculus, that is, vector-valued functions, geometry of curves in space, partial derivatives, surfaces and tangent planes, double and triple integrals, line integrals and surface integrals. The multivariable calculus part is the main tool that you shall need in your further courses in geometry.

Learning Outcomes of the Course Unit

1   Will be able to understand theoretically the convergence of sequences and series of real numbers by their precise definitions and the tests for convergence of series.
2   Will be able to estimate a function by its Taylor polynomials when its Taylor series converges to the function.
3   Will be able to investigate the geometry of a curve traced by a particle in space by finding its tangents, velocity, acceleration, arc length and curvature.
4   Will be able to find partial derivatives of functions using the Chain Rule.
5   Will be able to find the tangent plane to a surface at a point using partial derivatives and gradients.
6   Will be able to find the local or absolute, or constrained, maxima and minima of functions of several variables using multivariable methods like Second Derivative Test and Lagrange Multipliers.
7   Will be able to evaluate double and triple integrals by iterated integrals using Fubini s theorem and by change of variables.
8   Will be able to evaluate line integrals and surface integrals.

Mode of Delivery

Face -to- Face

Prerequisites and Co-requisites

None

Recomended Optional Programme Components

None

Course Contents

Week Subject Description
1 Completeness of the real number system, the least upper bound property, sequences of real numbers, convergent sequences, Monotone Convergence Theorem
2 Limit theorems for sequences, commonly occurring limits, divergent sequences, divergence to infinity, subsequences, series of real numbers, convergent series, geometric series, divergent series
3 Basic tests for convergence of series: integral test, comparison tests, ratio and root tests; alternating series, absolute and conditional convergence
4 Power series, Taylor and Maclaurin Series, convergence of Taylor Series, Taylor s Theorem, Maclaurin series of basic transcendental functions and the binomial series
5 Vector functions and their derivatives, integrals of vector functions, arc length of curves in space, curvature of a curve, tangential and normal components of acceleration
6 Functions of several variables, limits and continuity in higher dimensions, partial derivatives, the Increment Theorem and differentiability, the Mixed Partial Derivative Theorem
7 The Chain Rule, implicit differentiation; Directional derivatives and gradient vectors; tangent planes and differentials, linearization
8 Taylor series of functions of two variables
9 Extreme Values and Saddle Points, Second Derivative Test; absolute maxima and minima on bounded closed regions, the method of Lagrange Multipliers for constrained maxima and minima
10 Midterm
11 Fubini s Theorem for calculating double integrals by iterated integrals over rectangles and over general regions, area by double integration, double integrals in polar form
12 Triple integrals, volume by triple integration, triple integrals in cylindrical coordinates and spherical coordinates; substitutions in multiple integrals, change of variables; moments and centers of mass
13 Line integrals; vector fields, work, circulation, and flux; path independence, potential functions and conservative fields, exact differential forms; Green s Theorem in the plane
14 Surfaces and area of surfaces, surface integrals and flux, Stokes Theorem, Divergence Theorem

Recomended or Required Reading

Textbook(s): Hass , J., Weir, M. D. and Thomas , G. B., Jr., University Calculus, Early Transcendentals ,International Edition, 2nd edition, Pearson, 2012.

Supplementary Book(s):
a) Spivak, M. Calculus. Corrected third ed. Cambridge University Press, 2006.
b) Hubbard, J. and Burke Hubbard, B. Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. Fourth edition. Matrix Editions, 2009.

References:
1) Callahan, James J. Advanced Calculus, A Geometric View. Springer, 2010.
2) Bressoud, D. M. Second Year Calculus, From Celestial Mechanics to Special Relativity, Springer 1991.
3) Edwards, H. M. Advanced calculus. A differential forms approach. Corrected reprint of the 1969 original. With an introduction by R. Creighton Buck. Birkhuser Boston, Inc., Boston, MA, 1994.
4) Bressoud, D. M. A Radical Approach to Real Analysis, Second edition. Mathematical Association of America, 2007.

Materials: Instructor s notes and presentations

Planned Learning Activities and Teaching Methods

Lecture Notes, Presentation, Problem Solving

Assessment Methods

SORTING NUMBER SHORT CODE LONG CODE FORMULA
1 MTE MIDTERM EXAM
2 QUZ QUIZ
3 FIN FINAL EXAM
4 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.30 + FIN * 0.40
5 RST RESIT
6 FCGR FINAL COURSE GRADE (RESIT) MTE * 0.30 + ASG * 0.30 + RST * 0.40


*** Resit Exam is Not Administered in Institutions Where Resit is not Applicable.

Further Notes About Assessment Methods

None

Assessment Criteria

To be announced.

Language of Instruction

English

Course Policies and Rules

To be announced.

Contact Details for the Lecturer(s)

e-mail: engin.mermut@deu.edu.tr
Office: (232) 301 85 82

Office Hours

To be announced.

Work Placement(s)

None

Workload Calculation

Activities Number Time (hours) Total Work Load (hours)
Tutorials 13 2 26
Lectures 13 4 52
Preparations before/after weekly lectures 12 5 60
Preparation for midterm exam 1 15 15
Preparation for final exam 1 35 35
Preparation for quiz etc. 3 7 21
Final 1 2 2
Midterm 1 2 2
Quiz etc. 3 1 3
TOTAL WORKLOAD (hours) 216

Contribution of Learning Outcomes to Programme Outcomes

PO/LOPO.1PO.2PO.3PO.4PO.5PO.6PO.7PO.8PO.9PO.10PO.11PO.12PO.13
LO.15555
LO.25555
LO.355555
LO.45555
LO.555554
LO.655554
LO.75555
LO.855555

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