Learn Calculus 1:
Free Course
with Videos
and Guided Notes
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Welcome to Our Calculus 1 Videos and Guided Notes Page!
This free resource is designed to help you master the core concepts of Calculus 1 using video lessons and guided notes PDFs. Our series follows Stewart’s Calculus: Early Transcendentals and covers Chapters 1–6. Whether you’re learning calculus for the first time or reviewing before Calculus 2, these materials provide step-by-step explanations and practice.
What’s Included in This Free Calculus 1 Course?
Video Lessons
Every section from Chapters 1–6 is explained with clear examples and detailed walkthroughs. Chapter 1 contains shorter review videos on algebra, trigonometry, and functions to refresh your background skills. Our lessons and guided notes follow the structure of Stewart’s textbook.
Chapter 1: Functions and Models (Review)
– Algebra and trigonometry review
– Graphs of functions
– Types of functions and transformationsChapter 2: Limits and Derivatives
– Understanding limits
– Limit laws
– Continuity and the Intermediate Value Theorem
– Derivatives and Rates of ChangeChapter 3: Differentiation Rules
– Definition of the derivative
– Rules of differentiation
– Derivatives of trigonometric, exponential, and logarithmic functionsChapter 4: Applications of Differentiation
– Curve sketching and optimization
– Related rates
– Mean Value TheoremChapter 5: Integrals
– Definite and indefinite integrals
– The Fundamental Theorem of Calculus
– Techniques of integrationChapter 6: Applications of Integration
– Area under curves
– Volumes of solids of revolution
– Average value of a function
Free Guided Notes (Downloadable PDFs)
Our Calculus 1 guided notes PDFs are designed to be completed while watching the videos. They help you:
Reinforce understanding by filling in definitions, formulas, and key steps.
Improve retention through active engagement.
Stay organized with a structured study guide for quizzes, midterms, and finals.
Course Schedule and Video Updates
All Chapter 1 Review Videos are Available Now (functions, algebra, trigonometry, and graphs review).
New videos for Chapters 2–6 (limits, derivatives, applications, and integrals) will be posted every Tuesday, Thursday, and Sunday, at 3:00 p.m. ET beginning Tuesday, September 2nd. Each new video will be paired with a guided notes PDF to help you follow along.
Stay on track with this consistent release schedule and use the guided notes to reinforce your learning.
How to Learn Calculus 1 Effectively
Watch the video lessons for each section.
Download the guided notes PDF to follow along as you watch.
Engage actively by completing the notes during the lesson.
Review your completed notes as a study resource for exams.
Jump to Example Problems for Extra Practice
All of the worked examples from our course videos are organized for you in one place! Visit our Calculus 1 Example Problems page to quickly find problems by topic—perfect for reviewing before quizzes, midterms, or finals. 🔗 Go to the Calculus 1 Example Problems
Start Learning Calculus 1 Now!
From reviewing algebra and trigonometry in Chapter 1 to mastering limits, derivatives, and integrals in later chapters, this Calculus 1 course with guided notes and videos is built to help you succeed. Get started today with our free lessons and interactive notes that make calculus concepts clear, practical, and approachable..
Videos and Notes
Chapter 1: Functions and Models
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
1.1 Four Ways to Represent a Function
Functions and the Vertical Line Test
Evaluating Functions from a Formula, Table, and Graph
Average Rate of Change of a Function
Domain of a Function
Domain of a Composite Function
Piecewise Functions
Graphing Absolute Value Functions
Even and Odd Functions
Identify Increasing, Decreasing, and Constant Intervals on a Graph
1.2 Mathematical Models: A Catalog of Essential Functions
Linear Functions and Models
Polynomial Basics
Parent Functions
1.3. New Functions from Old Functions
Transformations of Functions
Graphing Transformations of Functions
Graphing Quadratic Functions
How to Graph Sine, Cosine, and Sinusoidal Functions
How to Graph the Tangent Function
Composite Functions
1.4. Exponential Functions
Introduction to Exponential Functions
Laws of Exponents
Graphing Exponential Functions
The Natural Exponential Function and Natural Logarithm
Exponential Growth and Decay Word Problems
1.5. Inverse Functions and Logarithms
Introduction to Inverse Functions
How to Find the Inverse of a Function
Horizontal Line Test and One-to-One Functions
How to Evaluate Logarithms
How to Solve Exponential Equations
How to Solve Logarithmic Equations
Change of Base Formula for Logarithms
Inverse Trigonometric Functions: Domain, Range, and Graphs
How to Evaluate Inverse Trigonometric Functions
How to Evaluate Compositions of Inverse Trigonometric Functions
Chapter 2: Limits and Derivatives
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
2.1 The Tangent and Velocity Problems
In this video, you’ll be introduced to two of the fundamental problems that led to the development of calculus: the tangent problem and the velocity problem. We’ll explore what each problem is asking, why it matters, and how these ideas connect to the concept of limits — the foundation of differential calculus.
2.2 The Limit of a Function
In this video, you’ll learn the foundational concept of limits — the building block of calculus. We’ll cover intuitive and formal definitions, explore one-sided limits, and discuss infinite limits and asymptotes. Through examples, you’ll see how limits describe the behavior of functions near a point, even when the function value at that point may not be defined.
2.3 Calculating Limits Using the Limit Laws
In this video, you’ll discover how to evaluate limits efficiently by applying fundamental limit laws. We’ll also cover strategies for handling indeterminate forms and introduce the squeeze theorem, a powerful tool for solving tricky limits. With worked-out examples, you’ll see how these techniques build the foundation for more advanced topics in calculus.
2.4 The Precise Definition of a Limit
In this video, you’ll explore the epsilon–delta definition of a limit, the rigorous foundation behind calculus. We’ll discuss why an informal idea of “getting close” isn’t enough, then carefully build the precise definition step by step. You’ll also see how to use graphs to find δ for a given ε and how to prove a limit statement formally with worked-out examples.
2.5 Continuity
In this video, you’ll explore one of the most important ideas in calculus—continuity. We’ll define what it means for a function to be continuous at a point, look at continuity on an interval, and discuss different types of discontinuities with clear examples. Finally, we’ll see how the Intermediate Value Theorem applies when working with continuous functions.
2.6 Limits at Infinity; Horizontal Asymptotes
In this video, you’ll explore how functions behave as x grows larger and larger (positively or negatively) without bound. We’ll define what a limit at infinity means, work through examples, and see how these limits reveal horizontal asymptotes. You’ll also learn algebraic strategies for handling indeterminate forms like ∞/∞ and ∞ – ∞.
2.7 Derivatives and Rates of Change
In this video, you’ll discover one of the most fundamental ideas in calculus — the derivative. We’ll connect derivatives to slopes of tangent lines, equations of tangent lines, and velocity, giving you both geometric and real-world perspectives. Step-by-step examples will show how limits are used to define and calculate derivatives.
2.8 The Derivative as a Function
In this video, you’ll go beyond finding a single derivative value and learn how to think of the derivative as its own function. We’ll explore how to compute derivatives using the limit definition, how to graph a derivative based on the original function, and what it means for a function to be differentiable. We’ll also introduce higher-order derivatives and what they represent.
Chapter 3: Differentiation Rules
© Understand The Math LLC. All Rights Reserved. These materials are freely provided for non-commercial educational use. Redistribution is permitted with proper attribution to UnderstandTheMath.com.
3.1 Derivatives of Polynomials and Exponential Functions
In this video, you’ll master the core differentiation rules for polynomials and exponential functions. We’ll start with the power rule — one of the most important tools for finding derivatives quickly — and then cover the constant multiple, sum, and difference rules. Finally, we’ll learn how to differentiate the natural exponential function and apply these rules to solve example problems.
3.2 The Product and Quotient Rules
In this video, you’ll learn two powerful differentiation rules for handling products and quotients of functions. We’ll break down each rule, go through worked-out examples, and practice applying them step by step.
3.3 Derivatives of Trigonometric Functions
In this video, you’ll learn how to find the derivatives of all six trigonometric functions. We’ll start with the fundamental trigonometric limits that form the basis for deriving the derivatives of sine and cosine, then work through the derivations step by step. Finally, we’ll summarize the derivatives of all six trig functions and solve practice problems.
3.4 The Chain Rule
In this video, you’ll learn how to use the chain rule to differentiate composite functions. We’ll review how to identify inner and outer functions, introduce the chain rule formulas, and then apply them to a series of step-by-step derivative examples, including functions with multiple layers.
3.5 Implicit Differentiation and Derivatives of Inverse Trigonometric Functions
In this video, you’ll learn how to find derivatives of equations where y is not given explicitly as a function of x. We’ll compare explicit and implicit functions, outline the method for implicit differentiation, and solve multiple examples, including tangent lines and higher-order derivatives.
In this video, you’ll learn the derivative formulas for all six inverse trigonometric functions. We’ll derive them step by step, then apply them in worked-out examples, including problems that combine inverse trig functions with the chain rule.
3.6 Derivatives of Logarithmic Functions
In this video, you’ll learn the derivative formulas for logarithmic functions and how to apply them step by step. We’ll also introduce logarithmic differentiation — a powerful technique for differentiating products, quotients, and expressions with both the base and exponent as functions of x.
3.7 Rates of Change in the Natural and Social Sciences
In this video, you’ll explore one of the most important applications of calculus — exponential growth and decay. We’ll derive the model, explain what the constant k represents, and work through examples from biology, physics, and everyday life.
3.8 Exponential Growth and Decay
In this video, you’ll explore one of the most important applications of calculus — exponential growth and decay. We’ll derive the model, explain what the constant k represents, and work through examples from biology, physics, and everyday life.
3.9 Related Rates
In this video, you’ll learn how to approach related rates problems step by step. We’ll review the main strategy, go over the standard formulas that appear most often, and solve eight classic related rates examples covering geometry, motion, and trigonometry.
3.10 Linear Approximations and Differentials
In this video, you’ll learn how to use tangent lines to approximate complicated functions and apply differentials to estimate error. We’ll cover the linearization formula, work through examples, and see how to apply these techniques to real-world problems.3.11 Hyperbolic Functions
3.11 Hyperbolic Functions
In this video, you’ll discover hyperbolic functions — the counterparts to trigonometric functions, but based on the hyperbola instead of the circle. We’ll define the six hyperbolic functions, prove key identities, find their derivatives, and introduce inverse hyperbolic functions.
Chapter 4: Applications of Differentiation
4.1 Maximum and Minimum Values
In this video, you’ll learn how to determine where a function reaches its highest and lowest points — both locally and absolutely. We’ll define local and absolute extrema, introduce critical numbers, and apply the closed interval method to find maximum and minimum values step by step, with multiple worked examples.
4.2 The Mean Value Theorem
In this video, you’ll learn about the Mean Value Theorem — one of the most important results connecting derivatives and rates of change. We’ll start with Rolle’s Theorem as a special case, explain how both theorems relate geometrically, and work through multiple examples step by step.
4.3 How Derivatives Affect the Shape of a Graph
In this video, you’ll learn how the first and second derivatives of a function reveal key features of its graph — including where it increases or decreases, where it has local maxima and minima, and where its concavity changes. We’ll use the increasing/decreasing test, first and second derivative tests, and the concavity test with worked-out examples.
4.4 Indeterminant Forms and L’Hôpital’s Rule
n this video, you’ll learn how to use L’Hôpital’s Rule to evaluate limits that lead to indeterminate forms such as 0/0 and ∞/∞. We’ll also cover how to handle other indeterminate forms, including 0·∞, ∞ – ∞, and indeterminate powers like 0⁰, ∞⁰, and 1^∞. Each concept includes clear, step-by-step examples.
4.5 Summary of Curve Sketching
In this video, you’ll learn a systematic approach to curve sketching using derivatives. We’ll review all the key concepts — domain, intercepts, symmetry, asymptotes, increasing/decreasing intervals, extrema, and concavity — and apply them step by step to four complete examples.
4.6 Graphing with Calculus and Calculators
In this video, you’ll learn how to use technology together with calculus concepts to analyze and sketch a detailed graph of a function. We’ll apply first and second derivatives to identify all key features — intervals of increase and decrease, local extrema, concavity, and inflection points — and use graphing tools to visualize them.
4.7 Optimization Problems
In this video, you’ll learn how to solve optimization problems — finding maximum or minimum values of functions subject to given conditions. We’ll go step by step through each stage: defining variables, creating equations, reducing to a single variable, differentiating, and identifying maximum or minimum values.
4.8 Newton's Method
In this video, you’ll learn how Newton’s Method (also called the Newton–Raphson Method) uses iteration to approximate solutions to equations of the form f(x) = 0. You’ll see how the method combines function values and their derivatives to refine guesses and move closer to the true root with each step.
4.9 Antiderivatives
In this video, you’ll learn what an antiderivative is, how it relates to derivatives, and how to find the most general antiderivative of a given function. We’ll review the basic rules and formulas, work through several examples, and apply them to problems involving boundary and initial conditions.
Chapter 5: Integrals
Videos and guided notes will be posted three times each week—Tuesday, Thursday, and Sunday at 3:00 p.m. ET —starting September 2nd.
5.1 Areas and Distances
In this video, you’ll learn how the concept of area leads to the definition of the definite integral. We’ll start by approximating area under a curve using rectangles, then use limits to find the exact area. You’ll also learn how to write an expression for area as a limit and identify a function and interval from a given limit expression.
5.2 The Definite Integral
In this video, you’ll learn the definition of the definite integral as the limit of a Riemann sum and how it represents the exact area under a curve. We’ll explore how to evaluate definite integrals using limits, interpret them geometrically as areas, and apply the comparison property to estimate their values.
5.3 The Fundamental Theorem of Calculus
In this video, you’ll learn how the Fundamental Theorem of Calculus connects two major ideas — differentiation and integration. You’ll see how it provides a powerful bridge between finding rates of change and calculating total accumulation, and how it allows us to evaluate definite integrals efficiently using antiderivatives.
5.4 Indefinite Integrals and the Net Change Theorem
In this video, you’ll learn how indefinite integrals are connected to antiderivatives and how the Net Change Theorem links rates of change to total change. We’ll review integral formulas, work through example problems, and apply these ideas to real-world motion and rate problems.
5.5 The Substitution Rule
In this video, you’ll learn how to apply the Substitution Rule, also known as u-substitution, to evaluate integrals that can’t be solved directly with basic formulas. We’ll cover the main idea behind substitution, outline the step-by-step process, and work through both indefinite and definite integral examples.
Chapter 6: Applications of Integration
6.1 Areas Between Curves
In this video, you’ll learn how to find the area enclosed between two curves using integration. We’ll go through the step-by-step method, explain when to integrate with respect to x or y, and work through several complete examples.
6.2 Volumes
6.3 Volumes by Cylindrical Shells
6.4 Work
6.5 Average Value of a Function