Calculus I : Keypoints And Techniques

A concise and review course for Calculus I

What you'll learn

Methods for finding limits: limit laws, l'Hospital's rule, factoring, rationalization,order o infinity

Continuity and types of discontinuous points: removable,jump,infinity and oscillating discontinuous points

Derivatives: product, quotient, chain rule, implicit differentiation, logarithm differentiation,tangent and normal line

Derivatives and the shape of a curve: increasing, decreasing,maximum, minimum, concave up, concave down, inflection points, asymptotes

Applicatin of derivatives: optimization, related rates,Newton's method

Requirements

You should have completed high school mathematics course.

You should familiar with power functions, exponential functions, logarithm functions, trigonometric functions

Description

This course is designed to emphasize the core concepts, key computational methods, and essential techniques of Calculus I. We will streamline our focus by skipping trivial details, overly elementary topics, and non-essential theorem proofs.By the end of this course, you will have a solid grasp of all the fundamental topics in Calculus I, establishing a strong foundation for future studies and ensuring you are well-prepared for the final exam.Practice exercises are assigned at the end of each lesson as an essential part of the course. They are designed to help you better understand and master the material. The exercises are concise and won’t take much time to complete, so please make an effort to work through them.The course content is organized as follows:1. Methods to evaluate limits: limit laws; l'Hospital's rule; factoring; compare the order of infinity; rationalization;squeeze theorem; limits with trigonometric functions; one-sided limits.2. Continuity and discontinuous points: definition of continuity; removable discontinuous points; step discontinuous points; infinity discontinuous points; oscillating discontinuous point; intermediate value theorem; horizongtal , vertical and slant asymptotes.3. Derivative and defferential rules: definition of derivative; basic differential formulas; summation and subtraction rule; product and quotient rule; chain rule; implicit differentiation; logarithm differentiation; derivative for inverse functions; tangent and normal line; higher order derivatives; linear approximation and differential.4. Applications of derivative: increasing and decreasing; concave up and concave down; local and global maximum and minimum; inflection points; curve sketching; related rates; optimization; Newton's method; mean value theorem.

Overview

Section 1: Introduction

Lecture 1 Limit laws

Lecture 2 L'Hospital's Rule I

Lecture 3 L'Hospital's Rule II

Lecture 4 L'Hospital's Rule III

Lecture 5 Factoring

Lecture 6 Compare the Order of Infinity

Lecture 7 Rationalization

Lecture 8 Squeeze Theorem

Lecture 9 Limits Involve Trigonometric Functions

Lecture 10 Left and Ritht Limits

Section 2: Continuity and Discontinuous Points

Lecture 11 Continuity

Lecture 12 Removable Discontinuous Points

Lecture 13 Jump Discontinuous Points

Lecture 14 Infinity Discontinuous Points

Lecture 15 Oscillating Discontinuous Points

Lecture 16 Intermediate Value Theorem

Section 3: Asymptotes

Lecture 17 Horizontal and Vertical Asymptotes

Lecture 18 Slant Asymptotes

Section 4: Derivative and Derivative Rules

Lecture 19 Definition of Derivative

Lecture 20 Basic Formulas of Derivative and Summation & Subtraction Rule

Lecture 21 Product Rule

Lecture 22 Quotient Rule

Lecture 23 Tangent and Normal Line

Lecture 24 Chain Rule

Lecture 25 Chain Rule Mixed with Product and Quotient Rule

Lecture 26 Chain Rule Mixed with Summation and Subtraction Rule

Lecture 27 Implicit Differentiation

Lecture 28 Logarithm Differentiation

Lecture 29 Derivative of Inverse Functions

Lecture 30 Higher Order Dirivatives

Lecture 31 Linear Approximation

Section 5: Derivative and Shape of Function

Lecture 32 Increasing and Decreasing

Lecture 33 Concave Up and Concave Down

Lecture 34 Local Maximum and Minimum

Lecture 35 Global Maximum and Minimum

Lecture 36 Inflection Points

Lecture 37 Curve Sketching

Lecture 38 More Examples on Curve Sketching

Section 6: Other Applications of Derivatives

Lecture 39 Related Rates

Lecture 40 Optimization

Lecture 41 Newton's Method

Lecture 42 Mean Value Theorem

For undergraduate students who want to prepare for final exam. For people who want to quick review the key material of calculus I. For people who want to study Calculus I in a concise form.