Calculus I

This course delves into the fundamental concepts of Calculus, exploring the relationship between numbers and shapes that is central to mathematics. It covers the historical development of calculus from ancient practical needs to its formalization by Newton and Leibniz, and its subsequent applications in various scientific fields. The course aims to provide a comprehensive understanding of calculus as the study of continuous quantities, emphasizing its logical foundation and connection to intuitive ideas. It addresses key areas such as sequences and series convergence, the behavior of functions, and the rigorous definitions of limits and continuity. The curriculum is designed to build a solid logical basis for mathematical analysis, enabling students to grasp complex mathematical objects and concepts through a constructive approach. Compared to introductory calculus courses that focus more on intuitive geometric and algebraic reasoning, this course emphasizes clarifying the crucial points often bypassed in simpler treatments. It bridges the gap between rigorous logical methods and intuitive understanding, while still including practical examples and problem-solving.

Curriculum

The course covers a wide range of topics in Calculus, from foundational concepts of real numbers and sequences to advanced topics like differential equations and series. The lectures are structured to progressively build understanding, starting with basic definitions and properties and moving towards more complex theorems and applications.

1. Module Set
   • Introduction
   • Least Upper Bound, Greatest Lower Bound, Dedekind Cut, Definition and Properties of Sequence Limits
   • Monotonic Sequence Convergence, Interval Nesting Theorem, Concept of Cauchy Sequences
   • Cauchy Pointwise Convergence
   • Equivalent Statements for Series Convergence, Absolute and Conditional Convergence, Comparison Test, Ratio and Root Tests, Alternating Series and Leibniz's Test
   • Dirichlet Series Rearrangement Theorem (Rearranging terms does not affect the sum of absolutely convergent series), Riemann Series Rearrangement Theorem (Conditionally convergent series can be rearranged to converge to any number)
   • Term-by-Term Expansion of Products of Absolutely Convergent Series
   • Metric Spaces, Open and Closed Sets
   • Review of Previous Definitions: Basic Properties of Open and Closed Sets, Definition of Sequence Limits in Metric Spaces, Bolzano-Weierstrass Theorem, Open Covers, Compact Sets, Heine-Borel Theorem
   • Review of Compactness and Heine-Borel Theorem: Lebesgue Number of an Open Cover
   • Isolated Points, Limit Points, and Accumulation Points of a Subset in a Metric Space
   • Limits of Mappings Between Metric Spaces
   • Review: Isolated Points, Limit Points and Accumulation Points, Mapping Limits; Continuity of Mappings and Equivalent Statements, Concept of Metric Subspaces
   • Summary: Subspaces and Continuous Mappings; Continuous Mappings Preserve Compactness, Continuous Functions Have Maxima and Minima on Compact Sets, Intermediate Value Theorem
   • Review: Continuity of Mappings/Functions, Relationship Between Mapping Restriction to Subspaces and Continuity
   • Meaning of Rational Exponents
   • Meaning of Real Number Exponents
   • Construction and Properties of Exponential and Logarithmic Functions (Cont.)
   • Uniform Continuity of Mappings/Functions, Continuous Mappings on Compact Metric Spaces are Uniformly Continuous, (Supplement) Proving the Existence of Maxima and Minima of Continuous Functions Using Constructive Sequences
   • Continuation of Unit 18: Exponent Laws, Continuity of Exponential Functions
   • Various Upper and Lower Limits of Real-Valued Functions, Left and Right Limits of Single-Variable Functions
   • Examples of Non-existent Limits
   • Derivatives and Differentiability of Functions
   • Relationship Between Convexity and Derivatives of Functions
   • Derivatives of Exponential and Trigonometric Functions
   • Rules for Differentiation of Arithmetic Operations
   • Derivative of Inverse Functions, Derivative of Composite Functions (Chain Rule)
   • Rolle's Theorem, Mean Value Theorem, L'Hôpital's Rule
   • Continuation of Unit 26 Chain Rule: Concept of Approximating a Given Function to a Certain Order at a Point
   • Sign of Derivatives and Monotonicity of Functions, Partial Derivatives, Finding Maxima and Minima
   • Continuation of 11/03B: Approximating a Given Function to a Certain Order at a Point Using Polynomials - Taylor Polynomials
   • Properties of k-th Order Approximation of the k-th Taylor Polynomial, Remainder Term of Taylor Approximation, Chain Rule for Taylor Approximation of Composite Functions to the k-th Order
   • Correction to Unit 30: Extreme Values of Single-Variable Functions at Endpoints Do Not Necessarily Imply Zero Derivative
   • Continuation of Unit 30: Examples of Finding Maxima and Minima
   • Continuation of Unit 32: Taylor Polynomials of Products and Quotients
   • Review of Midterm Exam Problems
   • Continuation of Unit 35: k-th Order Polynomial Approximation of Products
   • Differentiability of Multivariable Vector-Valued Functions, Continuously Differentiable Functions, Chain Rule for Multivariable Functions
   • Uniform Convergence of Function Sequences, Uniform Convergence Preserves Continuity
   • Complete Metric Spaces, Uniform Cauchy Criterion for Function Sequences is Equivalent to Uniform Convergence When the Codomain is Complete, Weierstrass M-test for Function Series
   • Continuous Curves that Fill a Triangle
   • Weierstrass's Continuous Function with No Derivative
   • Continuation of the Previous Unit: Weierstrass's Continuous Function with No Derivative
   • Fundamental Theorem of Signed Area; Antiderivatives/Indefinite Integrals; Integration by Parts and Substitution Method for Indefinite Integrals
   • Upper and Lower Sums of Bounded Functions for Interval Partitions; Upper and Lower Integrals; Basic Properties of Darboux Integrable Functions; Continuous Functions and Monotonic Functions are Darboux Integrable
   • Darboux's Theorem: Darboux Integral Equals the Limit of Riemann Sums
   • Discussion: Example 1 from Unit 45, 37:08
   • Countable Sets; Concept of Measure Zero; Lebesgue's Criterion for Darboux-Riemann Integrability
   • Review of Darboux Integral; The 'Constant' Arising in Indefinite Integration
   • Examples of Indefinite Integral Calculation - Powers of Cosine
   • Integrals of Rational Functions and Functions Involving Square Roots of Quadratic Polynomials - Partial Fraction Decomposition and Various Trigonometric/Hyperbolic Substitutions
   • Indefinite Integrals of Rational Combinations of Trigonometric/Hyperbolic Functions
   • Integral Expression for the Remainder Term of Taylor Expansion
   • Improper Integrals
   • Continuation of Unit 54: Absolute Convergence of Improper Integrals; Gamma Function
   • Transcendence of e
   • Discussion: [0,1] is not of measure 0
   • Discussion: Limits of Integration in Variable Substitution; Indefinite Integral of 1/(3 + sin^2 x)
   • Lebesgue's Criterion for Darboux-Riemann Integrability
   • Uniform Convergence and Limit Under the Integral Sign
   • Definite Integrals of Powers of Sine Function and Their Corollaries
   • Estimation of n!: Stirling's Formula
   • Differentiation and Integration of Integrals with Respect to a Parameter
   • Differentiation and Integration of Integrals with Respect to a Parameter: Improper Integrals
   • Differentiation and Integration of Integrals with Respect to a Parameter: Calculation Examples

Careers

While this course focuses on foundational calculus, the skills developed are crucial for a wide range of scientific and quantitative fields. Graduates often pursue careers in research, engineering, data analysis, finance, and technology.

• Mathematician
• Data Scientist
• Engineer
• Financial Analyst
• Researcher

Frequently asked questions

What are the tuition fees for the Calculus I programme?

Tuition rates for the Calculus I programme are available on the NTU Finances page. The estimated cost of living per semester in Taipei is TWD 84,000 – 113,000 (USD 2,800 – 3,767), which includes books, housing, insurance, and general living expenses but excludes tuition fees.

What is the duration of the Calculus I programme?

The provided data does not specify the duration of the Calculus I programme.

What are the main topics covered in the Calculus I programme?

The Calculus I programme covers fundamental concepts of calculus, including sequences and series convergence, the behavior of functions, limits, continuity, real and complex number exponents, derivatives, and integration, with an emphasis on rigorous logical methods.

What are the potential career paths after completing Calculus I?

Skills developed in this course are crucial for fields such as research, engineering, data analysis, finance, and technology. Potential roles include Mathematician, Data Scientist, Engineer, Financial Analyst, and Researcher.

In what language is the Calculus I programme taught?

The provided data does not specify the language of instruction for the Calculus I programme.