Overview

This course covers the fundamentals of Calculus, including topics such as limits, continuity, derivatives, integration, and applications of differentiation and integration. By the end of the course, students will be able to evaluate limits, find derivatives, determine extrema, sketch curves, calculate integrals, and work with exponential and logarithmic functions. The course uses a combination of theoretical explanations, problem-solving exercises, and real-world applications to help learners grasp the concepts effectively. This course is intended for students who want to build a strong foundation in Calculus, including high school and college students studying mathematics, physics, engineering, or related fields.

Syllabus

Calculus 1.1 A Preview of Calculus.
Calculus 1.2.1 Find Limits Graphically and Numerically: Estimate a Limit Numerically or Graphically.
Calculus 1.2.2 Find Limits Graphically and Numerically: When Limits Fail to Exist.
Calculus 1.2.3 Find Limits Graphically and Numerically: The Formal Definition of A Limit.
Calculus 1.3.1 Evaluating Limits Using Properties of Limits.
Calculus 1.3.2 Evaluating Limits By Dividing Out or Rationalizing.
Calculus 1.3.3 Evaluating Limits Using the Squeeze Theorem.
Calculus 1.4.1 Continuity on Open Intervals.
Calculus 1.4.2 Continuity on Closed Intervals.
Calculus 1.4.3 Properties of Continuity.
Calculus 1.4.4 The Intermediate Value Theorem.
Calculus 1.5.1 Determine Infinite Limits.
Calculus 1.5.2 Determine Vertical Asymptotes.
Calculus 2.1.1 Find the Slope of a Tangent Line.
Calculus 2.1.2 Derivatives Using the Limit Definition.
Calculus 2.1.3 Differentiability and Continuity.
Calculus 2.2.1 Basic Differentiation Rules.
Calculus 2.2.2 Rates of Change.
Calculus 2.3.1 The Product and Quotient Rules.
Calculus 2.3.2 Derivatives of Trigonometric Functions.
Calculus 2.3.3 Higher Order Derivatives.
Calculus 2.4.1 The Chain Rule.
Calculus 2.4.2 The General Power Rule.
Calculus 2.4.3 Simplifying Derivatives.
Calculus 2.4.4 Trigonometric Functions and the Chain Rule.
Calculus 2.5.1 Implicit and Explicit Functions.
Calculus 2.5.2 Implicit Differentiation.
Calculus I - 2.6.1 Related Rates - Water Ripples (2D Circle).
Calculus I - 2.6.2 Related Rates - Balloon Inflation (Sphere).
Calculus I - 2.6.3 Related Rates - Modeling with Triangles.
Calculus 3.1.1 Extrema of a Function on an Interval.
Calculus 3.1.2 Relative Extrema of a Function on an Open Interval.
Calculus 3.1.3 Find Extrema on a Closed Interval.
Calculus 3.2.1 Rolle’s Theorem.
Calculus 3.2.2 The Mean Value Theorem.
Calculus 3.3.1 Increasing and Decreasing Intervals.
Calculus 3.3.2 The First Derivative Test.
Calculus 3.4.1 Intervals of Concavity.
Calculus 3.4.2 Points of Inflection.
Calculus 3.4.3 The Second Derivative Test.
Calculus 3.4.4 Putting It All Together.
Calculus 3.5.1 Determine Finite Limits at Infinity.
Calculus 3.5.2 Determine Horizontal Asymptotes of a Function.
Calculus 3.5.3 Horizontal Asymptotes - Tricky Examples.
Calculus 3.5.4 Determine Infinite Limits at Infinity.
Calculus 3.6.1 A Summary of Curve Sketching.
Calculus 3.6.2 Curve Sketching - Full Practice.
Calculus 3.7.1 Optimization Problems.
Calculus 3.7.2 Optimization Practice.
Calculus 4.1.1 Antiderivatives.
Calculus 4.1.2 Basic Integration Rules.
Calculus 4.1.3 Find Particular Solutions to Differential Equations.
Calculus 4.2.1 Sigma Notation.
Calculus 4.2.2 The Concept of Area.
Calculus 4.2.3 The Approximate Area of a Plane Region.
Calculus 4.2.4 Finding Area By The Limit Definition.
Calculus 4.3.1 Riemann Sums.
Calculus 4.3.2 Definite Integrals.
Calculus 4.3.3 Properties of Definite Integrals.
Calculus 4.4.1 The Fundamental Theorem of Calculus.
Calculus 4.4.2 The Mean Value Theorem for Integrals.
Calculus 4.4.3 The Average Value of a Function.
Calculus 4.4.4 The Second Fundamental Theorem of Calculus.
Calculus 4.5.1 Use Pattern Recognition in Indefinite Integrals.
Calculus 4.5.2 Change of Variables for Indefinite Integrals.
Calculus 5.1.1 Properties of the Natural Logarithmic Function.
Calculus 5.1.2 The Number e.
Calculus 5.1.3 The Derivative of the Natural Logarithmic Function.
Calculus 5.2.1 The Log Rule for Integration.
Calculus 5.2.2 Integrals of Trigonometric Functions.
Calculus 5.3.1 Verify Functions are Inverses of One Another.
Calculus 5.3.2 Determine Whether a Function Has An Inverse.
Calculus 5.3.3 Find the Inverse of a Function.
Calculus 5.3.4 Find the Derivative of an Inverse of a Function.
Calculus 5.4.1 The Natural Exponential Function.
Calculus 5.4.2 Derivatives of the Natural Exponential Function.
Calculus 5.4.3 Integrals of the Natural Exponential Function.
Calculus 5.5.1 Exponential Functions with Bases Other than e.
Calculus 5.5.2 Differentiate and Integrate with Bases Other than e.
Calculus 5.5.3 Applications of Bases Other than e.
Calculus 5.6.1 Indeterminate Forms.
Calculus 5.6.2 L’Hôpital’s Rule.
Calculus 5.7.1 Inverse Trigonometric Functions.
Calculus 5.7.2 Derivatives of Inverse Trigonometric Functions.
Calculus 5.8.1 Integrate Inverse Trigonometric Functions.
Calculus 5.8.2 Integrate Using the Completing the Square Technique.