Become A Calculus 1 Master

Calculus Demystified: Visual, Intuitive, and Application-Driven Learning

What you'll learn

How to represent, evaluate, and analyze functions graphically and algebraically

echniques for identifying domain, range, symmetry, and monotonicity of functions

Composition and decomposition of functions, including transformations and piece-wise definitions

Core trigonometric concepts including radian measure, unit circle, and trigonometric graphs

Properties and applications of exponential and logarithmic functions

How to compute and interpret limits, including limits at infinity and discontinuities

The formal definition of a limit and its role in defining continuity and derivatives

How to compute derivatives using first principles and differentiation rules (power, product, quotient, chain)

Applications of derivatives in velocity, acceleration, optimization, and curve sketching

Implicit differentiation and derivatives of inverse, logarithmic, and trigonometric functions

Real-world applications of derivatives in science, economics, and engineering

Techniques for solving related rates problems and optimization scenarios

Introduction to antiderivatives and indefinite integrals

How to approximate area under curves using Riemann sums and sigma notation

The Fundamental Theorem of Calculus and its use in computing definite integrals

Integration techniques including u-substitution and symmetry-based strategies

How to apply integrals to model displacement, cost, growth, and net change

Requirements

A solid understanding of Algebra (including solving equations, manipulating expressions, and working with functions)

Familiarity with Trigonometry is helpful but not required—key concepts are reviewed in the course

Commitment to practice regularly and review lessons as needed

Description

Are you ready to conquer calculus with confidence? Whether you're a high school student, college learner, or lifelong math enthusiast, this course is your ultimate guide to mastering Calculus I—from foundational functions to advanced integration techniques.This course is meticulously structured to take you on a journey through the essential concepts of calculus, with over 200+ bite-sized lessons, real-world applications, and step-by-step walkthroughs that make even the most complex topics feel intuitive.What You’ll ExperienceYou’ll begin with the language of functions, learning how to interpret, manipulate, and visualize them in multiple forms. You’ll explore how functions behave—how they grow, shrink, reflect, and transform—and how they form the backbone of all calculus concepts.From there, you’ll dive into trigonometry, exponentials, and logarithms, building the tools needed to understand more complex relationships. You’ll learn how to work with inverse functions, how to graph them, and how they relate to real-world phenomena like sound waves and population growth.Then comes the heart of calculus: limits. You’ll explore both the intuitive and rigorous definitions, learning how to compute limits graphically and algebraically. You’ll understand why limits matter, how they define continuity, and how they lead to the concept of the derivative.With derivatives, you’ll unlock the ability to measure instantaneous change—velocity, acceleration, slope, and more. You’ll master the power rule, product rule, quotient rule, and the chain rule, and apply them to a wide variety of functions, including trigonometric, exponential, and logarithmic ones.But this course doesn’t stop at computation. You’ll explore real-world applications: how derivatives help us solve optimization problems, model physical systems, and understand rates of change in science and economics. You’ll sketch curves, analyze critical points, and use the Mean Value Theorem to make powerful inferences about function behavior.Finally, you’ll enter the world of integration—the reverse process of differentiation. You’ll learn how to compute antiderivatives, use Riemann sums to approximate area, and apply the Fundamental Theorem of Calculus to connect everything you’ve learned. You’ll explore definite integrals, u-substitution, and how integrals are used to calculate displacement, cost, and growth.What Makes This Course Different?Comprehensive Coverage: Every major topic in Calculus I is covered in depth—from functions and trigonometry to derivatives, optimization, and integrals.Visual Learning: Graphs, diagrams, and animations help you see the math and build intuition.Real-World Applications: Learn how calculus is used in science, engineering, and everyday problem-solving.Practice-Driven: Includes guided examples, exercises, and walkthroughs to reinforce every concept.Modular Design: Learn at your own pace, revisit topics anytime, and build your understanding layer by layer.

Overview

Section 1: Functions

Lecture 1 Four Ways to Represent a Function

Lecture 2 Evaluation of Functions (Graphical)

Lecture 3 The Monotonicity of the Function (Graphical)

Lecture 4 A Survey of Computing Domains of Functions

Lecture 5 Difference Quotient of a Quadratic

Lecture 6 Symmetry of Functions

Section 2: Library of Functions

Lecture 7 Power Functions

Lecture 8 Reciprocal Functions and their Graphs

Lecture 9 Radical Functions

Lecture 10 Piece-wise Functions

Section 3: Composition

Lecture 11 Graph Transformations

Lecture 12 Algebra of Functions (Algebraic)

Lecture 13 Function Composition

Lecture 14 Composition of Square Root Functions

Lecture 15 Composition of Rational Functions

Lecture 16 Function Decomposition

Section 4: Trigonometry

Lecture 17 Radian Measure

Lecture 18 Arc Length

Lecture 19 Definitions of the Six Trigonometric Ratios

Lecture 20 Right Triangle Trigonometry

Lecture 21 The Unit Circle Diagram

Lecture 22 The Graph of Sine

Section 5: Exponentials

Lecture 23 Exponential Laws (College Algebra)

Lecture 24 Graphs of Exponential Functions

Lecture 25 Curve Fitting Exponential Functions

Lecture 26 Exponential Growth

Section 6: Logarithms and Inverse Trigonometry

Lecture 27 One-to-One Functions

Lecture 28 Inverse Functions

Lecture 29 The Inverse Function Property

Lecture 30 Computing Inverse Functions Algebraically

Lecture 31 Finding Inverse Functions of Square Root Functions

Lecture 32 Inverses of Linear Fractionals

Lecture 33 An Introduction to Logarithms

Lecture 34 Logarithms ARE the Exponents

Lecture 35 Graphs of Logarithms

Lecture 36 Laws of Logarithms

Lecture 37 The Change of Base Formula (Logarithms)

Lecture 38 Solving Logarithmic Equations

Lecture 39 The Inverse Trigonometric Functions

Lecture 40 Computing Inverse Trigonometric Functions

Lecture 41 Inverse Trigonometric Expressions and Triangle Diagrams

Section 7: Error

Lecture 42 Error and Allowance (A Precursor to Limits)

Lecture 43 An Example of Computing Delta for a Function Given an Epsilon

Lecture 44 The Precise Definition of the Limit

Section 8: Limits

Lecture 45 The Intuitive Definition of a Limit

Lecture 46 Computing Limits from the Graph of a Function

Lecture 47 Why Do We Need a Precise Definition of a Limit?

Section 9: Limit Laws

Lecture 48 Using Limit Laws to Compute Limits

Lecture 49 Computing Limits of a Function using a Simplified Form

Lecture 50 Limits of Piece-wise Functions

Lecture 51 Simplifying a Limit of a Difference Quotient (Polynomial)

Lecture 52 Simplifying a Limit of a Difference Quotient (Radical)

Lecture 53 Simplifying a Limit of a Difference Quotient (Rational)

Section 10: The Squeeze Theorem

Lecture 54 The Squeeze Theorem

Lecture 55 Simplifying a Limit of a Difference Quotient (Exponential)

Lecture 56 Simplifying a Limit of a Difference Quotient (Trigonometric)

Section 11: Discontinuities

Lecture 57 Continuous Functions

Lecture 58 Discontinuities

Lecture 59 Continuity of Piece-wise Functions

Section 12: Continuity Laws

Lecture 60 Finding Values to Make Piece-wise Functions Continuous

Lecture 61 Combining Continuous Functions

Lecture 62 Composition of Continuous Functions

Lecture 63 The Intermediate Value Theorem (Calculus I)

Section 13: Limits at Infinity

Lecture 64 Vertical Asymptotes

Lecture 65 Limits at Infinity

Lecture 66 Arithmetic at Infinity

Lecture 67 Horizontal Asymptotes

Lecture 68 Vertical and Horizontal Asymptotes

Lecture 69 Limits at Infinity and the Squeeze Theorem

Lecture 70 Limits at Infinity Involving Radicals

Lecture 71 Limits at Infinity Involving Exponentials

Lecture 72 The End Behavior of Dampened Harmonic Motion

Lecture 73 Limits at Infinity Involving Arctangent

Section 14: Tangent Lines

Lecture 74 Tangent Lines

Lecture 75 Instantaneous Rate of Change and Velocity

Section 15: Instantaneous Rates of Change

Lecture 76 The Derivative of a Function

Lecture 77 Computing Derivatives from the Definition (Tangent Lines)

Lecture 78 The Reverse-FOIL Method

Lecture 79 Computing Derivatives from the Definition (Rational)

Lecture 80 Computing Derivatives from the Definition (Velocity)

Lecture 81 Derivatives

Lecture 82 Criteria for a Function Being Differentiable

Lecture 83 Graphing the Derivative of a Function from Its (Sometimes Non-Differentiable)

Section 16: Power Rule

Lecture 84 The Power Rule

Lecture 85 The Linearity of the Derivative

Lecture 86 Finding Acceleration of a Motion Function

Lecture 87 The Derivative of e^x

Section 17: Product Rule

Lecture 88 The Product Rule

Lecture 89 The Quotient Rule

Lecture 90 Combining the Quotient and Product Rules

Lecture 91 We Don't Always Need the Quotient Rule

Section 18: Trigonometric Derivatives

Lecture 92 Trigonometric Limits

Lecture 93 The Derivatives of Sine and Cosine

Lecture 94 The Derivatives of Tangent and Other Trigonometric Functions

Lecture 95 Higher Derivatives of Sine

Section 19: Chain Rule

Lecture 96 The Chain Rule

Lecture 97 Trigonometric Derivatives and the Chain Rule

Lecture 98 Exponential Derivatives and the Chain Rule

Lecture 99 Combining the Product Rule and the Chain Rule

Lecture 100 The Chain Rule and the Quotient Rule

Lecture 101 Examples of the Chain Rule

Lecture 102 Using the Chain Rule on the Composition of Three Functions

Lecture 103 Finding the Equation of a Tangent Line using the Chain Rule

Lecture 104 Using the Chain Rule Graphically

Lecture 105 Derivatives of Exponential Functions

Lecture 106 The Chain Rule and a Story Problem

Section 20: Implicit Differentiation

Lecture 107 Implicit Differentiation

Lecture 108 Implicit Differentiation vs. Explicit Differentiation

Lecture 109 Implicit Differentiation (Polynomial Relation)

Lecture 110 Implicit Differentiation (Radical Relation)

Lecture 111 Implicit Differentiation (Trigonometric Relation)

Lecture 112 Implicit Differentiation (Folium of Descartes)

Lecture 113 Second Derivatives with Implicit Differentiation

Lecture 114 Derivatives of Inverse Trigonometric Functions

Section 21: Logarithmic Differentiation

Lecture 115 Derivatives of Logarithms

Lecture 116 Derivatives of Logarithms with Absolute Value

Lecture 117 Logarithmic Differentiation

Lecture 118 The Proof of the Power Rule by Logarithmic Differentiation

Lecture 119 Taking Derivatives of Functions involving Exponents and Bases

Lecture 120 Taking Derivatives of Functions involving Absolute Values

Section 22: Derivatives in Science

Lecture 121 Rates of Change in Science

Lecture 122 Derivatives and Linear Density

Lecture 123 Derivatives and Isothermal Compressibility

Lecture 124 Derivatives and Population Growth

Lecture 125 Derivatives and Economics

Section 23: Related Rates

Lecture 126 Related Rates

Lecture 127 Related Rates and a Falling Ladder

Lecture 128 Related Rates and an Inverted Conical Tank of Water

Lecture 129 Related Rates and Two Approaching Cars

Lecture 130 Strategies for Solving Related Rates Problems

Lecture 131 Related Rates and a Trapezoidal Trough

Lecture 132 Related Rates and Expanding Gases

Lecture 133 Related Rates and Rotating Searchlight

Section 24: Hyperbolic Derivatives

Lecture 134 The Hyperbolic Functions

Lecture 135 Derivatives of the Hyperbolic Functions

Lecture 136 The Inverse Hyperbolic Functions

Lecture 137 Derivatives of the Inverse Hyperbolic Functions

Section 25: Extreme Value Theorem

Lecture 138 Local Extrema

Lecture 139 Critical Numbers

Lecture 140 Absolute Extrema

Lecture 141 The Extreme Value Theorem

Lecture 142 The Extreme Value Problem

Section 26: Mean Value Theorem

Lecture 143 Rolle's Theorem

Lecture 144 The Mean Value Theorem

Lecture 145 Proving that an Equation has Exactly One Solution

Lecture 146 The Assumptions of the Mean Value Theorem

Lecture 147 Inferences of the Mean Value Theorem

Lecture 148 Two Functions with the Same Derivative Differ by a Constant

Section 27: First and Second Derivative Tests

Lecture 149 The First Derivative Test

Lecture 150 Determining Local Extrema using the First Derivative Test

Lecture 151 A Remark about Critical Numbers

Lecture 152 The Test for Concavity

Lecture 153 The Second Derivative Test

Section 28: l'Hospital's Rule

Lecture 154 l'Hospital's Rule

Lecture 155 More Practice on l'Hospital's Rule

Lecture 156 l'Hospital's Rule and Product Indeterminants

Lecture 157 l'Hospital's Rule and Exponential Indeterminants

Lecture 158 l'Hospital's Rule and Difference Indeterminants

Section 29: Curve Sketching

Lecture 159 Curve Sketching (Calculus I)

Lecture 160 Curve Sketching (Polynomial Function)

Lecture 161 Curve Sketching (Rational Function with Oblique Asymptote)

Lecture 162 Curve Sketching (Rational Function with Horizontal Asymptote)

Lecture 163 Curve Sketching (Radical Ratio)

Lecture 164 Curve Sketching (Logarithmic Ratio)

Lecture 165 Curve Sketching (Trigonometric Ratio)

Section 30: Optimization

Lecture 166 Optimization (Calculus I)

Lecture 167 (Optimization) Maximizing the Product of Two Points on a Line

Lecture 168 (Optimization) - Finding the Minimal Distance between a Point and a Parabola

Lecture 169 (Optimization) - Finding a Maximum Rectangle in a Semicircle

Lecture 170 (Optimization) - Finding Minimum Distance of a Path

Lecture 171 (Optimization) - Finding Minimum Distance of a Path Reprise

Lecture 172 (Optimization) - Finding the Maximum Volume of a Box

Section 31: Newton's Method

Lecture 173 Tangent Line Approximation

Lecture 174 Newton's Method

Section 32: Antiderivatives

Lecture 175 What is an Antiderivative?

Lecture 176 The Power Rule for Antiderivatives

Lecture 177 Linearity Property of Antiderivatives

Lecture 178 Basic Antiderivatives

Lecture 179 Initial Value Problem for Antiderivatives

Section 33: Summation Notation

Lecture 180 Sigma Notation

Lecture 181 Properties of Sigma

Lecture 182 Examples of Sigma Notation

Lecture 183 Geometric Sums

Section 34: Area Under the Curve

Lecture 184 Approximating π using Rectangles

Lecture 185 Area under the Curve

Lecture 186 Riemann Sum Calculators

Lecture 187 Upper and Lower Sums

Lecture 188 Velocity, Displacement, and Area under the Curve

Section 35: Definite Integrals

Lecture 189 The Definite Integral

Lecture 190 The Definition of Definite Integrals

Lecture 191 Computing Definite Integrals by the Definition

Lecture 192 Computing Definite Integrals by the Definition involving a Geometric Sum

Lecture 193 Properties of Definite Integrals

Lecture 194 Comparison Test of Definite Integrals

Section 36: The Fundamental Theorem of Calculus

Lecture 195 Integral Functions

Lecture 196 The Fundamental Theorem of Calculus - Part 1

Lecture 197 Computing Derivatives using the Fundamental Theorem of Calculus - Part 1

Lecture 198 Computing Derivatives using the Fundamental Theorem of Calculus, where the limit

Lecture 199 Computing Derivatives using the Fundamental Theorem of Calculus, where the limit

Lecture 200 The Fundamental Theorem of Calculus - Part 2

Lecture 201 Computing Integrals using the Fundamental Theorem of Calculus

Lecture 202 Finding Areas using the Fundamental Theorem of Calculus

Lecture 203 The Limitations of the Fundamental Theorem of Calculus

Lecture 204 Integrals in Science

Lecture 205 Integrals and Displacement

Lecture 206 The Net Change Theorem and Cost

Lecture 207 The Net Change Theorem and Growth

Section 37: u-Substitution

Lecture 208 What is u-Substitution?

Lecture 209 u-Substitution and Indefinite Integrals

Lecture 210 Examples of Finding Antiderivatives Using u-Substitution

Lecture 211 u-Substitution When the Inner Derivative Isn't Quite Right

Lecture 212 The Antiderivative of Tangent

Lecture 213 u-Substitution and Definite Integrals

Lecture 214 Definite Integrals and Symmetry

High school or college students taking Calculus I,STEM professionals needing a refresher,Self-learners passionate about mathematics,Anyone who’s ever said, “I wish someone had explained calculus this way”