Learn fundamental calculus concepts in this comprehensive mathematics course covering functions, limits, derivatives, and integrals. Master essential topics including function modeling, exponential and logarithmic functions, differentiation rules, applications of derivatives, integration techniques, and differential equations. Explore practical applications through physics and engineering problems while developing skills in calculating areas, volumes, and surface areas. Progress from basic limit concepts through advanced topics like parametric equations, polar coordinates, and conic sections. Apply mathematical principles to real-world scenarios through optimization problems, Newton's method, and modeling with differential equations. Practice integration techniques including substitution, integration by parts, and trigonometric integrals while learning to solve both definite and improper integrals.
Overview
Syllabus
- Functions and Models
- Four Ways to Represent a Function
- Mathematical Models:A Catalog of Essential Functions. New Functions from Old Functions
- Exponential Functions, Inverse Functions and Logarithms
- Limits and Derivatives
- The Limit of a Function. Calculating Limits Using the Limits Laws
- The Precise Definition of a Limit. Continuity
- Limits at Infinity;Horizontal Asymptotes
- Derivatives and Rates of Change. The Derivative as a Function
- Differentiation Rules
- Derivatives of Polynomials and Exponential Functions
- The Product and Quotient Rules
- Derivatives of Trigonometric Functions
- The Chain Rule，Implicit Differentiation
- Derivatives of Logarithmic Functions
- Exponential Growth and Decay
- Related Rates
- Linear Approximations and Differentials
- Hyperbolic Functions
- Applications of Differentiation
- Maximum and Minimum Values
- The Mean Value Theorem
- How Derivatives Affect the Shape of a Graph
- Indeterminate Froms and 1'Hospital's Rule
- Optimization Problems
- Newton's Method
- Antidervatives
- Integrals
- Areas and Distances，The Definite Integral
- The Fundamental Theorem of Calculus
- Indefinite Integrals and the Net Change Theorem
- The Substitution Rule
- Applications of Integration
- Areas Between Curves
- Volumes
- Volumes by Cylindrical Shells
- Work
- Average Value of a Function
- Techniques of Integration
- Integration by Patrs
- Trigonometric Integral
- Trigonometric Substitution
- Integration of Rational Functions by Parital Fractions
- Approximate Integration
- Improper Integrals
- Further Applications of Integration
- Arc Length
- Area of a Surface of Revolution
- Applications to Physics and Engineering
- Differential Equations
- Modeling with Differential Equations
- Separable Equations
- Linear Equations
- Parametric Equations and Polar Coordinates
- Curves Defined by Parametric Equations
- Calculus with Parametric Curves
- Polar Coordinates
- Areas and Lengths in Polar Coordinates
- ConicSections in Polar Coordinates
Taught by
Faen Wu
