Learn Calculus 1 - Free Course with Videos and Notes
Master limits, derivatives, integrals, and their applications with structured video lessons, guided notes, example problems, and exam review resources. This free course builds understanding and confidence through clear, step-by-step instruction.
This series follows the textbook Calculus: Early Transcendentals by James Stewart and covers Chapters 1–6.
Calculus 1 Video Lessons with Guided Notes
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.
Chapter 1 - Functions and Models
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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
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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
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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.
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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.
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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.
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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.
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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.
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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 ∞ – ∞.
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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.
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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
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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.
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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.
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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.
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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.
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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.
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Chapter 4 - Applications of Differentiation
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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.
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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.
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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.
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In 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.
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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.
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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.
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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.
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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
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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.
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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.
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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.
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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
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In this video, you’ll learn how to compute volumes of solids of revolution using the disk and washer methods. We’ll walk through how to set up each integral, decide whether to use disks or washers, and interpret the geometry from the graph. Then we’ll apply these ideas to eight fully worked examples, including rotations about the x-axis, y-axis, and horizontal/vertical lines.
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In this video, you’ll learn how to use the cylindrical shells method to find the volume of a solid of revolution. We’ll walk through the main formulas, explain how to identify the radius and height of each shell, and then apply the method to four complete examples involving rotation about vertical and horizontal lines.
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In this video, you’ll learn how to compute work when a constant force or a variable force moves an object. We’ll review the formula for work, discuss when to use a definite integral, and work through examples involving lifting objects and stretching/compressing springs using Hooke’s Law.
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In this video, you’ll learn how to compute the average value of a function on a closed interval. We’ll start with the definition using Riemann sums, develop the formula using a definite integral, and interpret the average value as the height of a rectangle with the same area as the region under the curve. Then we’ll work through several examples step by step.
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Calculus 1 Example Problems
Select a chapter below to access our complete step-by-step example problem videos.
Calculus 1 Exam Reviews
Master exam-style problems and build your test-day confidence with structured exam reviews, downloadable practice problems, and extra study support.
Exam Reviews
Part 1 – Limits and Continuity
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Part 2 – Derivative Basics
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Part 3 – Advanced Differentiation Techniques
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Part 4 – Derivative Applications
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Part 5 – Integration Basics
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Part 6 – Applications of Integration
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About Dr. Cheryl Hile
Understand The Math was created by Dr. Cheryl Hile, who earned a Ph.D. in Engineering Science and Applied Mathematics from Northwestern University. She has more than 25 years of experience teaching mathematics at the university level, including 20 years at Penn State. The lessons and guided notes on this site are based on materials she developed for her university courses.