A Rapid Introduction To Calculus

Master Essential Concepts in Calculus Quickly

What you'll learn

Learn fundamental concepts of Pre-Calculus and Calculus

Learn how to present formal proofs in Calculus

Gain an intuitive and formal understanding of limits, continuous functions, derivatives and integrals

Understand why is Calculus important

Requirements

You should know how to solve an equation such as 2x + 3 = 0 before taking the course.

The most important requirement is that you feel free to ask questions in the Q&A in case anything is unclear. I'm always happy to help and am open to interesting discussions or even just clarifying a topic.

Description

"A Rapid Introduction to Calculus" is a concise and focused calculus course designed to provide students with a foundational understanding of the key concepts and techniques in calculus. This course is aimed at students who need a quick but solid introduction to calculus and may not require an in-depth exploration of the subject. Here's a breakdown of the curriculum for this course:1. Mathematical Preliminaries:   - Review of basic algebraic concepts and techniques.   - Introduction to mathematical notation and terminology.2. Functions:   - Definition and properties of functions.   - Types of functions, including linear, quadratic, exponential, and trigonometric functions.   - Function notation and basic operations with functions.3. Limits:   - Introduction to the concept of a limit.   - Calculating limits of functions algebraically and graphically.   - Understanding the intuitive notion of limits.4. Continuous Functions:   - Definition of continuity.   - Identifying and analyzing points of discontinuity.   - Intermediate Value Theorem.5. Differentiable Functions:   - Introduction to the derivative of a function.   - Calculation of derivatives using basic rules.   - The concept of instantaneous rate of change.6. Main Theorems Involving the Derivative:   - Rolle's Theorem and the Mean Value Theorem.   - Applications of the Mean Value Theorem.   - Connection between the derivative and the behavior of functions.7. Graphing Functions:   - Analysis of functions through their first and second derivatives.   - Sketching graphs of functions.   - Identifying critical points and inflection points.8. Integration:   - Introduction to the concept of integration.   - Calculation of definite and indefinite integrals.   - Basic techniques of integration, including substitution and integration by parts.This course is structured to provide students with a solid foundation in calculus, covering the essential topics necessary for understanding and working with calculus concepts. The focus on key concepts and practical applications makes it suitable for those who need to quickly grasp the fundamentals of calculus for their academic or professional needs.

Overview

Section 1: Introduction

Lecture 1 Introduction

Lecture 2 History and Motivation

Section 2: Mathematical Preliminaries

Lecture 3 Set Theory Part 1

Lecture 4 Set Theory Part 2

Lecture 5 Linear Functions and Linear Equations

Lecture 6 Quadratic Functions and Quadratic Equations

Lecture 7 Important Algebraic Identities

Lecture 8 Factoring Trinomials

Lecture 9 Inequalities

Lecture 10 Absolute Value and Distance

Section 3: Functions

Lecture 11 What is a Function ?

Lecture 12 Domain and Range of a function + Examples

Lecture 13 Domain Solution to Homework

Lecture 14 Monotone Functions

Lecture 15 Monotone Functions Exercise

Lecture 16 Injections (one-to-one functions)

Lecture 17 Image of a function

Lecture 18 Surjections (onto function)

Lecture 19 Function Composition

Lecture 20 Inverse Functions

Lecture 21 Bounded Functions

Lecture 22 Even and Odd Functions

Lecture 23 Polynomials and Rational Functions

Lecture 24 Exponents and Logarithms

Lecture 25 Trigonometric Functions

Lecture 26 Elementary Functions

Section 4: Limits

Lecture 27 What is a Limit ?

Lecture 28 Examples of Limits

Lecture 29 One Sided Limits

Lecture 30 Infinite Limits

Lecture 31 Limit Arithmetic

Section 5: Continuous Functions

Lecture 32 Continuous Functions

Lecture 33 Roots of Polynomials of Odd Degree

Anyone who wants to learn Calculus,Anyone who wants to get a rapid overview of the main results in Calculus,People with a minimal mathematical background who want to learn Calculus