Course Objectives
At the completion of this course, students will be able to
1) Calculate polynomial approximations of one-variable functions, limits of sequences, geometric series, Maclaurin expansion, Taylor expansion, and Euler's formula.
2) Calculate limits of two-variable functions, partial derivatives, total derivatives, equations of tangent planes, derivatives of composite functions, second-order partial derivatives, maxima and minima, derivatives of implicit functions, constrained extremum problems, and envelopes.
Rubric
| Ideal Level of Achievement (Very Good) | Standard Level of Achievement (Good) | Unacceptable Level of Achievement (Fail) |
| Evaluation 1 | Can calculate polynomial approximations of one-variable functions, limits of sequences, geometric series, Maclaurin expansion, Taylor expansion, and Euler's formula with more than 80% correct answers. | Can calculate polynomial approximations of one-variable functions, limits of sequences, geometric series, Maclaurin expansion, Taylor expansion, and Euler's formula with more than 60% correct answers. | Can't calculate polynomial approximations of one-variable functions, limits of sequences, geometric series, Maclaurin expansion, Taylor expansion, and Euler's formula. |
| Evaluation 2 | Can calculate limits of two-variable functions, partial derivatives, total derivatives, equations of tangent planes, derivatives of composite functions, second-order partial derivatives, maxima and minima, derivatives of implicit functions, constrained extremum problems, and envelopes with more than 80% correct answers. | Can calculate limits of two-variable functions, partial derivatives, total derivatives, equations of tangent planes, derivatives of composite functions, second-order partial derivatives, maxima and minima, derivatives of implicit functions, constrained extremum problems, and envelopes with more than 60% correct answers. | Can't Calculate limits of two-variable functions, partial derivatives, total derivatives, equations of tangent planes, derivatives of composite functions, second-order partial derivatives, maxima and minima, derivatives of implicit functions, constrained extremum problems, and envelopes. |
Assigned Department Objectives
Diploma policy 3
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Teaching Method
Outline:
In this course, students will learn series expansions of functions, partial derivatives of two-variable functions, and more.
Exercises will be conducted in parallel with lectures to acquire mathematical and computational techniques required in engineering and other subjects.
Style:
The content of the lectures gradually increases in quantity and quality. We recommend that students prepare for each lecture.
If there is anything you do not understand in the lectures, please review it immediately and try to understand it. Active questions are always recommended.
It is not enough to understand basic contents once. Please train repeatedly.
The lecture plan may be changed according to the student's level of understanding.
Notice:
Mini-exams will be carried out up to several times.
Evaluations will be made comprehensively by written exams (making up about 90% of the grade), by exercises and homework (making up about 10% of the grade).
Can take makeup exam in need aid up to maximum of 60 points.
Characteristics of Class / Division in Learning
Course Plan
|
|
|
Theme |
Goals |
| 1st Semester |
| 1st Quarter |
| 1st |
Expansion of functions (1) |
Review (mini-exam), guidance, 1st order approximation.
2nd order approximation, n-th order approximation.
|
| 2nd |
Expansion of functions (2) |
Sufficient conditions for extrema.
Properties for limits of sequences, limits of geometric sequences.
|
| 3rd |
Expansion of functions (3) |
Convergence and divergence of series.
Convergence and divergence of geometric series. (Optional: Radius of power series.)
|
| 4th |
Expansion of functions (4) |
Maclaurin expansion.
(Maclaurin's Theorem, Error bound.)
|
| 5th |
Expansion of functions (5) |
Taylor expansion．
Euler's formula, de Moivre's theorem.
|
| 6th |
Partial differentiation (1) |
Supplementary information up to this lecture, and exercise.
Two-variable functions and equations for surfaces.
|
| 7th |
Partial differentiation (2) |
Limits and continuity for two-variable functions.
Partial derivatives, exercise.
|
| 8th |
Mid term examination |
|
| 2nd Quarter |
| 9th |
Partial differentiation (3) |
Answers and explanations for the mid-term exam, total derivatives.
Equations of tangent planes.
|
| 10th |
Partial differentiation (4) |
Derivatives of composite functions.
(In the case of polar coordinates.)
|
| 11th |
Applications of partial derivatives (1) |
Higher order partial derivatives.
(In the case of polar coordinates, Taylor's theorem for two-variable functions.)
|
| 12th |
Applications of partial derivatives (2) |
Maxima and minima.
Methods for determining extremal values. (A proof for the general case.)
|
| 13th |
Applications of partial derivatives (3) |
Derivatives of implicit functions.
Constrained extremum problems.
|
| 14th |
Applications of partial derivatives (4) |
Envelopes.
Supplementary information up to this lecture, and exercise.
|
| 15th |
Final examination |
|
| 16th |
Answers and explanations for the final exam, and class questionnaires |
Answers and explanations for the final exam, and class questionnaires.
Summary of the 1st semester and advice for summer vacation and the 2nd semester.
|
Evaluation Method and Weight (%)
| Examination | Presentation | Mutual Evaluations between students | Behavior | Portfolio | Other | Total |
| Subtotal | 90 | 0 | 0 | 0 | 0 | 10 | 100 |
| Basic Ability | 90 | 0 | 0 | 0 | 0 | 10 | 100 |
| Technical Ability | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Interdisciplinary Ability | 0 | 0 | 0 | 0 | 0 | 0 | 0 |