Course Introduction

• Prerequisite：high school mathematics
• Recommended for： freshmen
• Introduction：

This course provides a first introduction into the theory of differentiation and integration. The course mainly serves as a bridge between high school mathematics and university mathematics. Its main goal is to make students acquainted with rigorous mathematical thinking. This is done via learning basic concepts such as limits, continuity, differentiability, etc. on the one hand and fundamental theorems such as the intermediate value theorem, the extreme value theorem, the mean value theorem, etc. on the other hand.Moreover, the course is intended to train students problem solving skills as well as writing and oral skills. Finally, the course equips students with the basic tools needed in the more applied sciences and is the entrance door to more advanced courses on mathematics.

• Syllabus：

1. Limits and Continuity:
   The Basic (Intuitive) Concepts of Limit, The Definition of Limit, Evaluating Limits and Rules for Finding Limits, Some Limit Theorems (with proofs), One-Sided Limits, Continuity.
2. Differentiation:
   Definition of Derivative, Basic Differentiation Rules, and Higher-Order Derivatives, The Chain Rule, Differentiatingthe Trigonometric Functions, Implicit Differentiation.
3. Applications of Differentiation:
   Extreme Value Theorem, The Role's Theorem and the Mean-Value Theorem (with proofs), Increasing and Decreasing Functions and the First Derivative Test for Local Extreme Values, Concavity and the Second Derivative Test, Limits at Infinity, Asymptotes and Summary of Curve Sketching, Linearization and Differentials, Optimization Problems.
4. Integration:
   AntiderivativesandIndefinite Integrals,Riemann Sums and Definite Integrals, Properties and the Mean-Value Theorem for Integrals, The Fun- damental Theorem of Calculus (with proof), Integral by Substitution: Change of Variables.
5. Transcendental Functions:
   Inverse Functions and Their Derivatives, The Natural Logarithm Function: Differentiation and Integration, The Exponential Functions: Differ- entiation and Integration, Logarithm and Exponential Functions with Bases Other Than e , Indeterminate Forms and Rule, Inverse Trigonometric Functions: Differentiation and Integration, (Optional) Hyperbolic Functions: Differentiation and Integration.
6. Techniques of Integration:
   Basic Integration Formulas, Integration by Parts, Partial Fractions, (Optional) Trigonometric Integrals, Trigonometric Substitutions, Improper Integrals.
7. Applications of Integrals:
   Area Problems: Area of a Region Between Two Curves, Volumes of Solids of Revolution, Area of Surface of Revolution, Lengths of Plane Curves, Solutions of Differentiation Equations, and Growth/Decay Problems.
8. Infinite Series:
   Sequences, Theorems for Calculating Limits of Sequences, Infinite Series and Convergence, The Integral Test, The Comparison Test, Alter- nating Series, Absolute and Conditional Convergence, The Ratio and Root Tests, Power Series, Taylor and Maclaurin Series Approxima- tions of Functions and Error Estimates.
9. Parametrized Curves and Analytic Geometric in Space:
   Parametrizations of Plane Curves, Calculus with Parametrized Curves, (Optional) Polar Coordinates and Polar Graphs, (Optional) Cylindrical and Spherical Coordinates.
10. Vector-Valued Functions:
    Review of Vectors in Space, Dot Products and Cross Products, Vector-Valued Functions and Space Curves, Differentiation and Integration of Vector-Valued Functions, Tangent Vectors and Normal Vectors.
11. Multivariable Functions and Partial Derivatives:
    Introduction to Functions of Several Variables, Limits and Continuity, Partial Derivatives, Directional Derivatives and Gradients, Differentiability and Differentials, The Mean-Value Theorem and Chain Rules for Multivariable Functions, Tangent Plane and Normal Lines, Extreme Values and Saddle Points, Lagrange Multipliers, Taylor s Formula.
12. Multiple Integrals:
    Double Integrals as the Limit of Riemann Sums, Evaluation of Double Integrals, Some Applications of Double Integrals: Volumes and Surface Areas, Triple Integrals, Change of Variables: Jacobians, (Optional) Double Integrals in Polar Coordinates, and Triple Integrals in Cylindrical and Spherical Coordinates.
13. Integration in Vector Field:
    Vector Field, Line Integrals, Conservative Vector Field and Path Independence, Green's Theorem, Parametrized Surfaces, Surface Integrals, The Divergence Theorem, The Stokes's Theorem and a Unified Theory.

• Reference：

1. Calculus (Early Transcendentals), James Stewart, 6th Edition

• Summary：

1. Homepage of NYCU's Calculus Education