Duration of the online course: 7 hours and 37 minutes
New
Build real calculus skills fast: master limits, derivatives, and integrals with a free online course plus practice for exams and problem-solving confidence.
In this free course, learn about
Compute average vs instantaneous velocity; interpret slope numerically and graphically
Understand limits: definition, infinite limits, limits at infinity, and squeeze theorem
Apply limit laws and algebraic tricks to evaluate polynomial/rational/trig limits
Know key limits like sin(x)/x and use them to simplify related expressions
Analyze continuity, piecewise continuity, and apply the Intermediate Value Theorem
Define derivative via difference quotient; tangent line slope and linear approximation
Differentiate using rules: power, product, quotient, chain; plus trig, exp, ln
Use implicit and logarithmic differentiation; derivatives of inverse trig (e.g., arctan)
Solve related rates problems by differentiating geometric relationships in time
Optimize with derivatives: maxima/minima, concavity, 2nd derivative test, MVT ideas
Evaluate indeterminate limits with L'Hopital’s Rule; compare exponential vs polynomial growth
Find antiderivatives and constants; power antiderivative and general families
Define definite integrals via Riemann sums; compute from definition and reverse sums
Use FTC I/II, substitution (u-sub) for indefinite/definite integrals, and average value formula
About the free online course
This free online course helps you move from memorizing formulas to actually understanding calculus. Instead of treating ideas like limits, derivatives, and integrals as separate chapters, you learn how they connect and why they work. You will build intuition from the kinds of questions that show up in real problems: how velocity changes in time, what a function is doing near a point, and how a small change in x affects the output. That big-picture understanding makes later techniques feel natural rather than mysterious.
You start by exploring change through average and instantaneous velocity, then develop the language of limits using numerical and graphical thinking. As the course progresses, you get comfortable with key tools that make challenging limits manageable, including limit laws, algebraic strategies, and methods for tricky behaviors like oscillation, infinite limits, and limits at infinity. Along the way, you strengthen your reasoning with important ideas about continuity and guaranteed values, giving you a solid foundation for everything that follows.
From there, the course builds the derivative from the ground up using the definition, so the rules are not just facts to memorize. You learn to compute derivatives efficiently and interpret them as slopes, rates of change, and local behavior. Techniques such as product and quotient rules, chain rule, implicit differentiation, and logarithmic differentiation become practical tools for modeling and solving problems. Applications are emphasized so the mathematics feels useful, including tangent-line approximation, related rates, and optimization that trains you to translate a situation into equations and make decisions with calculus.
The final arc ties the story together through antiderivatives and definite integrals, connecting area and accumulation to derivatives via the Fundamental Theorem of Calculus. You gain confidence with Riemann-sum definitions, evaluating integrals, and substitution as the reverse of the chain rule, including both indefinite and definite cases. With practice questions and exam-style walkthroughs, you will be prepared for coursework, placement tests, and future STEM classes, with clearer intuition and stronger problem-solving habits.
Course content
Video class: The Velocity Problem | Part I: Numerically07m
Exercise: What is the difference between average velocity and instantaneous velocity as described in the context of the velocity problem?
Video class: The Velocity Problem | Part II: Graphically07m
Exercise: What is the average velocity of a car traveling from 100 miles to 110 miles over 15 minutes?
Video class: A Tale of Three Functions | Intro to Limits Part I04m
Exercise: Which function has a different behavior at x=1 compared to others?
Video class: A Tale of Three Functions | Intro to Limits Part II08m
Exercise: What is the definition of a limit in calculus as it relates to the behavior of a function near a designated point, regardless of the function's behavior at that point?
Video class: What is an infinite limit?04m
Video class: Limit Laws | Breaking Up Complicated Limits Into Simpler Ones06m
Exercise: Which scenario demonstrates the proper application of limit laws?
Video class: Building up to computing limits of rational functions03m
Exercise: What is the limit of the function f(x) = 3x^3 + x - 1 as x approaches the value a?
Video class: Limits of Oscillating Functions and the Squeeze Theorem06m
Exercise: What is the limit of x sin(1/x) as x approaches 0 from the right?
Video class: Top 4 Algebraic Tricks for Computing Limits07m
Video class: A Limit Example Combining Multiple Algebraic Tricks07m
Exercise: Which of the following statements correctly describes a characteristic of the function presented in the limit problem?
Video class: Limits are simple for continuous functions07m
Video class: Were you ever exactly 3 feet tall? The Intermediate Value Theorem04m
Exercise: What does the Intermediate Value Theorem state?
Video class: Example: When is a Piecewise Function Continuous?03m
Exercise: What is the value of 'C' that makes the piecewise function continuous at x=1, given the piecewise function includes 'Cx^2 + 1' for x values less than one, and '2x - C' for x values equal or greater than one?
Video class: Limits at infinity06m
Exercise: What is the limit of arctan(X) as X approaches infinity?
Video class: Computing Limits at Infinity for Rational Functions07m
Exercise: What is the limit at infinity of a rational function with higher degree in the numerator?
Video class: Infinite Limit vs Limits at Infinity of a Composite Function09m
Exercise: What does the function f(x) = e^(x-3)/(x-2) tend towards as x approaches 2 from the left?
Video class: The most important limit in Calculus // Geometric Proof11m
Exercise: What is the limit as x approaches zero for sin(x)/x?
Video class: Definition of the Derivative | Part I06m
Exercise: What is the slope of the tangent line approximated by?
Video class: Applying the Definition of the Derivative to 1/x05m
Exercise: Consider the function f(x) = 1/x. What is the derivative f'(x), found by using the definition of the derivative as a limit of the difference quotient?
Video class: Definition of Derivative Example: f(x) = x 1/(x 1)06m
Video class: The derivative of a constant and of x^2 from the definition05m
Video class: Derivative Rules: Power Rule, Additivity, and Scalar Multiplication07m
Video class: How to Find the Equation of a Tangent Line05m
Video class: The derivative of e^x.02m
Video class: The product and quotient rules05m
Video class: The derivative of Trigonometric Functions05m
Video class: Chain Rule: the Derivative of a Composition05m
Video class: Interpreting the Chain Rule Graphically05m
Exercise: When applying the chain rule to find the derivative H'(x) of the composition function H(x) = F(2x), what are the two steps involved?
Video class: The Chain Rule using Leibniz notation05m
Video class: Implicit Differentiation | Differentiation when you only have an equation, not an explicit function07m
Video class: Derivative of Inverse Trig Functions via Implicit Differentiation04m
Exercise: What is the derivative of arctangent of x?
Video class: The Derivative of ln(x) via Implicit Differentiation04m
Video class: Logarithmic Differentiation | Example: x^sinx03m
Video class: Intro to Related Rates06m
Exercise: If Car A is traveling east at 60 miles per hour, and Car B is traveling north at 50 miles per hour, and at a certain moment Car A has driven 0.4 miles and Car B has driven 0.3 miles, what is the rate at which the distance between the two cars is changing at that moment?
Video class: Linear Approximations | Using Tangent Lines to Approximate Functions09m
Video class: The MEAN Value Theorem is Actually Very Nice07m
Video class: Relative and Absolute Maximums and Minimums | Part I04m
Exercise: Which of the following statements describes the correct mathematical definition of a local maximum for a function f(x) at a point c?
Video class: Relative and Absolute Maximums and Minimums | Part II07m
Video class: Concavity and the 2nd Derivative Test09m
Video class: Using L'Hopital's Rule to show that exponentials dominate polynomials09m
Exercise: When evaluating the limit of (a * x) / (e^x) as x approaches infinity, why is L'Hopital's Rule applicable?
Video class: Applying L'Hopital's Rule to Exponential Indeterminate Forms07m
Video class: Ex: Optimizing the Volume of a Box With Fixed Surface Area11m
Video class: Folding a wire into the largest rectangle | Optimization example06m
Exercise: In an optimization problem involving folding a wire of length L into a rectangle, what length should each side of the width (X) be to maximize the area of the resulting rectangle?
Video class: Optimization Example: Minimizing Surface Area Given a Fixed Volume09m
Video class: Tips for Success in Flipped Classrooms OMG BABY!!!08m
Video class: What's an anti-derivative?06m
Exercise: What is the antiderivative of x^n, where n is a positive integer?
Video class: Solving for the constant in the general anti-derivative04m
Video class: The Definite Integral Part I: Approximating Areas with rectangles05m
Video class: The Definite Integral Part II: Using Summation Notation to Define the Definite Integral09m
Exercise: Which of the following statements correctly describes the purpose of summation notation in calculus?
Video class: The Definite Integral Part III: Evaluating From The Definition06m
Video class: Reverse Riemann Sums | Finding the Definite Integral Given a Sum10m
Video class: Fundamental Theorem of Calculus 1 | Geometric Idea Chain Rule Example11m
Exercise: What does the Fundamental Theorem of Calculus Part 1 primarily establish about the relationship between derivatives and integrals?
Video class: Fundamental Theorem of Calculus II05m
Video class: Intro to Substitution - Undoing the Chain Rule06m
Video class: Adjusting the Constant in Integration by Substitution03m
Exercise: Given the integral of e^(2x) / (1 + e^(2x)), what substitution should be made to simplify the integral?
Video class: Substitution Method for Definite Integrals **careful!**04m
Video class: Back Substitution - When a u-sub doesn't match cleanly!08m
Video class: Average Value of a Continuous Function on an Interval08m
Exercise: What is the general formula for the average value of a continuous function f(x) on an interval [a, b]?
Video class: Exam Walkthrough | Calc 1, Test 3 | Integration, FTC I/II, Optimization, u-subs, Graphing34m
Video class: Thank you Calc Students: Some final thoughts04m
Video class: CALCULUS SPEEDRUN || Limits || Episode 112m
Exercise: What is the correct approach to find the limit as x goes to 0 of the expression sin(3x) * cot(4x)?
Video class: 5 counterexamples every calculus student should know15m
This free course includes:
7 hours and 37 minutes of online video course
Digital certificate of course completion (Free)
Exercises to train your knowledge
100% free, from content to certificate
What is the difference between average velocity and instantaneous velocity in Calculus 1?
Average velocity measures total change in position over a time interval, while instantaneous velocity is the velocity at one moment, found using a limit or derivative.
How do you use the Fundamental Theorem of Calculus to evaluate a definite integral?
Find an antiderivative F of f, then compute F(b) minus F(a) for the integral from a to b.
When should L'Hopital's Rule be used for a limit?
Use it when direct substitution produces an indeterminate form such as 0/0 or infinity/infinity, and the required derivatives exist.
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