MA3110: Mathematical Analysis II
AY2014/2015, Semester 2, Lecturer: Lee Soo Teck
Course Coverage:
1. Differentiable Functions
2. The Riemann Integral
3. Sequences & Series of Functions
4. Power Series
This module is the second of three parts to real analysis. In this part, the study on properties of functions continue to differentiable and integrable functions. The concept of sequences and series is extended from constant terms to functions. In the last section, properties of power series is examined.
The first half of the module is a continuation from where MA2108 left off. The topic on differentiable functions cover the theorems related to differentiation, such as the sum and difference of the derivatives, product rule, quotient rule and chain rule. Other important theorems that are derived from the differentiable property includes the Interior Extremum Theorem, Rolle's Theorem and Mean Value Theorem. A great portion of time is devoted to Mean Value Theorem and its applications. Some renowned results in applied mathematics that are covered in this chapter are the first and second derivative tests, Cauchy Mean Value Theorem, L'Hospital Rule and Taylor's Theorem.
The Riemann Integral is of equal if not greater importance in real analysis. From the integral, we derive not only the Riemann Integrability Criterion, but also the Fundamental Theorem of Calculus. Many important results follow these theorems, especially the latter. Familiar theorems on the different methods of integration (by substitution and by parts) are developed alongside other theoretical results such Mean Value Theorem of Integral Calculus and Taylor's Theorem with integral form of remainder.
The main topic of the module is on the sequences and series of functions. The first concept that were introduced right at the beginning of the lecture is the difference between pointwise and uniform convergence of function. Subsequently, the preservation of properties such as continuity, differentiability and integrability from the sequence functions to the limiting function is examined rigorously before being extended to the infinite series of functions.
In the last chapter, we studied power series, its convergence and relation to Taylor and Maclaurin series. Abel's Formula, Abel's Theorem, Merten's Theorem are some fundamental results derived in this chapter. These results, together with other properties of power series, are used to derive the exponential, logarithmic and trigonometric functions at the end of the module.
Like the preceding module, the focus of this module is on the preservation and implication of theorems. This module extends the topics of MA2108 in both breadth and depth, completing the main topics of univariate real analysis, thus providing the necessary background for the last of three part to real analysis, which generalizes the result to metric spaces, measure spaces, Banach spaces and Hilbert spaces.
Workload: Moderate
Difficulty: Moderate
Grade: B+
Course Coverage:
1. Differentiable Functions
2. The Riemann Integral
3. Sequences & Series of Functions
4. Power Series
This module is the second of three parts to real analysis. In this part, the study on properties of functions continue to differentiable and integrable functions. The concept of sequences and series is extended from constant terms to functions. In the last section, properties of power series is examined.
The first half of the module is a continuation from where MA2108 left off. The topic on differentiable functions cover the theorems related to differentiation, such as the sum and difference of the derivatives, product rule, quotient rule and chain rule. Other important theorems that are derived from the differentiable property includes the Interior Extremum Theorem, Rolle's Theorem and Mean Value Theorem. A great portion of time is devoted to Mean Value Theorem and its applications. Some renowned results in applied mathematics that are covered in this chapter are the first and second derivative tests, Cauchy Mean Value Theorem, L'Hospital Rule and Taylor's Theorem.
The Riemann Integral is of equal if not greater importance in real analysis. From the integral, we derive not only the Riemann Integrability Criterion, but also the Fundamental Theorem of Calculus. Many important results follow these theorems, especially the latter. Familiar theorems on the different methods of integration (by substitution and by parts) are developed alongside other theoretical results such Mean Value Theorem of Integral Calculus and Taylor's Theorem with integral form of remainder.
The main topic of the module is on the sequences and series of functions. The first concept that were introduced right at the beginning of the lecture is the difference between pointwise and uniform convergence of function. Subsequently, the preservation of properties such as continuity, differentiability and integrability from the sequence functions to the limiting function is examined rigorously before being extended to the infinite series of functions.
In the last chapter, we studied power series, its convergence and relation to Taylor and Maclaurin series. Abel's Formula, Abel's Theorem, Merten's Theorem are some fundamental results derived in this chapter. These results, together with other properties of power series, are used to derive the exponential, logarithmic and trigonometric functions at the end of the module.
Like the preceding module, the focus of this module is on the preservation and implication of theorems. This module extends the topics of MA2108 in both breadth and depth, completing the main topics of univariate real analysis, thus providing the necessary background for the last of three part to real analysis, which generalizes the result to metric spaces, measure spaces, Banach spaces and Hilbert spaces.
Workload: Moderate
Difficulty: Moderate
Grade: B+