# Advanced Calculus I (MATH251) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Advanced Calculus I MATH251 3. Semester 3 2 0 4 8
Pre-requisite Course(s)
MATH 122 Analytic Geometry II, MATH 136 Mathematical Analysis II
Course Language English Compulsory Departmental Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Discussion, Problem Solving. This course is desined to introduce higher-level aspects of the calculus through a rigorous development of the fundamental ideas in the topic and to achieve a further development of the math student’s ability to deal with abstract mathematics and proofs. The students who succeeded in this course; Comprehend and use the functions of several variables, Comprehend and use the functions of several variables, Find extreme values for functions of two and three variables, Model and solve optimization problems with side conditions. Vector and matrix algebra, functions of several variables: limit, continuity, partial derivatives, chain rule; implicit functions. inverse functions, directional derivatives, maxima and minima of functions of several variables, extrema for functions with side conditions.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Vectors and Matrix Algebra (a very brief review). pp. 1-31, 50-56, 60-66
2 Functions of several variables, pp. 77-78
3 Domain and Regions, Functional Notation, pp. 78-81
4 Limits and continuity, pp. 82-87
5 Partial Derivatives, Total differential (fundamental lemma), pp. 88-93
6 Differential of functions of n variables (The Jacobian matrix), pp. 94-100
7 Midterm
8 Derivatives and differentials of composite functions, pp. 101-105
9 The general chain rule, Implicit functions, Proof of a case of the implicit function theorem, pp. 106-121
10 Inverse functions (curvilinear coordinates), Geometrical applications (tangent plane, tangent line, etc.) pp. 122-134
11 The directional derivatives, Partial derivatives of higher order, pp. 135-142
12 Higher derivatives of composite functions, The Laplacian in polar, cylindrical, and spherical coordinates, pp. 143-145
13 Higher derivatives of implicit functions, Maxima and minima of functions of several variables, pp. 146-158
14 Extrema for functions with side conditions (Lagrange Multipliers). pp. 159-160
15 Review
16 Final

### Sources

Course Book 1. W. Kaplan, Advanced Calculus. Addison-Wesley, 1993 2. H. Helson. Honors Calculus 3. B. Demidovich. Problem book in mathematical analysis

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 5 10
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 50
Final Exam/Final Jury 1 40
Toplam 8 100
 Percentage of Semester Work 60 40 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area X
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research X
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary X
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study X
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework X
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility X
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation X
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2) X
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge X
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach. X
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications. X

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours) 16 3 48
Laboratory
Application 16 2 32
Special Course Internship
Field Work
Study Hours Out of Class 16 4 64
Presentation/Seminar Prepration
Project
Report
Homework Assignments 5 3 15
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 10 20
Prepration of Final Exams/Final Jury 1 21 21