What Calculus 1 topics are covered in this free online course?

The course covers functions, limits, continuity, derivatives, curve analysis, optimization, integrals, area, volumes of revolution, and arc length.

How does this course explain the relationship between derivatives and integrals?

It introduces derivatives as rates of change and integrals as accumulation, then connects them through the Fundamental Theorem of Calculus.

Does the course include applications such as optimization, related rates, and area between curves?

Yes. It includes related rates, maximum and minimum optimization problems, area between curves, and volumes using disks, washers, and shells.

Free online course Calculus 1

Duration of the online course: 44 hours and 56 minutes

4.6

(60)

Build calculus confidence fast with a free online course: master limits, derivatives, and integrals to boost STEM grades and prep for exams.

In this free course, learn about

Compute slope, angle of inclination, and distance formula for lines
Define functions vs relations; domain/range; combine and compose functions
Use radians/degrees; graph trig functions; understand key trig limits
Understand limits, limit laws, and techniques for evaluating limits
Determine continuity and interpret it as “no breaks/jumps” in graphs
Define derivative via limits; interpret as slope, velocity, rate of change
Differentiate using rules: constant, product/quotient, chain, trig derivatives
Do implicit differentiation; distinguish explicit vs implicit functions
Solve related rates by linking changing quantities with derivatives
Analyze graphs with derivatives: increase/decrease, concavity, asymptotes
Apply Rolle’s Theorem, Mean Value Theorem, and 1st/2nd derivative tests
Solve optimization problems using critical points and constraints
Understand integrals via Riemann sums; compute definite/indefinite integrals
Apply FTC; compute areas, volumes (disks/washers/shells), and arc length

About the free online course

Calculus often feels like a sudden leap from familiar algebra into a world of symbols and new rules. This free online course is designed to make that leap manageable by building intuition first and then turning it into reliable problem-solving skills. You will start from the geometry of lines and slopes, connect that foundation to functions and graphs, and quickly move into the big ideas that power calculus: limits, continuity, derivatives, and integrals.

As you progress, you will learn how calculus explains change in a precise way. Limits provide the language for describing behavior near a point, and continuity clarifies when a function behaves smoothly enough to work with confidently. From there, the derivative becomes a practical tool for understanding instantaneous rate of change, slope of a curve, motion, and sensitivity in real situations. Step by step, you will strengthen your fluency with essential differentiation techniques, including product and quotient rules, the chain rule, and implicit differentiation, so that common expressions stop feeling intimidating and start feeling routine.

Beyond computation, the course emphasizes interpretation. You will see how derivatives reveal where a function increases or decreases, how concavity shapes a graph, and how the first and second derivative tests support clear conclusions. You will also practice the kind of thinking used in optimization, where calculus helps choose the best outcome under constraints, a skill that appears across science, engineering, and economics.

The second half of the journey focuses on integration as the natural partner of differentiation. You will build an understanding of area and accumulation from Riemann sums and limits, then learn to evaluate definite integrals efficiently and connect everything through the Fundamental Theorem of Calculus. With this perspective, topics like substitution, area between curves, and volumes of solids of revolution become logical extensions rather than disconnected tricks. By the end, you will be prepared to approach typical Calculus I assignments and exams with more confidence, accuracy, and insight.

Course content

Video class: Calculus 1 Lecture 0.1: Lines, Angle of Inclination, and the Distance Formula 48m
Exercise: _What is the slope of a line and how is it defined?
Video class: Calculus 1 Lecture 0.2: Introduction to Functions. 1h37m
Exercise: What is required for a mathematical relation to be a function?
Video class: Calculus 1 Lecture 0.3: Review of Trigonometry and Graphing Trigonometric Functions 1h20m
Exercise: _What is the relationship between radians and degrees?
Video class: Calculus 1 Lecture 0.4: Combining and Composition of Functions 15m
Exercise: What does F + G(x) represent when combining functions?

Video class: Calculus 1 Lecture 1.1: An Introduction to Limits 1h27m
Exercise: What is the foundational concept in calculus discussed?
Video class: Calculus 1 Lecture 1.2: Properties of Limits. Techniques of Limit Computation 3h00m
Exercise: What is the limit of tan(X) as X approaches a?
Video class: Calculus 1 Lecture 1.4: Continuity of Functions 1h26m
Exercise: _What is the layman's term definition of continuity for a function?

Video class: Calculus 1 Lecture 1.5: Slope of a Curve, Velocity, and Rates of Change 1h50m
Exercise: What is the slope of a curve at a point using limits?
Video class: Calculus 1 Lecture 2.1: Introduction to the Derivative of a Function 1h16m
Exercise: _What is the derivative?
Video class: Calculus 1 Lecture 2.2: Techniques of Differentiation (Finding Derivatives of Functions Easily) 1h12m
Exercise: What is the derivative of a constant?
Video class: Calculus 1 Lecture 2.3: The Product and Quotient Rules for Derivatives of Functions 1h02m
Exercise: _What is the product rule in calculus?
Video class: Calculus 1 Lecture 2.4: Applications of the Derivative 40m
Video class: Calculus 1 Lecture 2.5: Finding Derivatives of Trigonometric Functions 48m
Exercise: What is the derivative of sin(x)?
Video class: Calculus 1 Lecture 2.6: Discussion of the Chain Rule for Derivatives of Functions 1h34m
Exercise: Understanding the Chain Rule in Calculus
Video class: Calculus 1 Lecture 2.7: Implicit Differentiation 1h08m
Exercise: _What is the difference between an explicit and implicit function?
Video class: Calculus 1 Lecture 2.8: Related Rates 53m
Exercise: What is the key concept in related rates in calculus?

Video class: Calculus 1 Lecture 3.1: Increasing/Decreasing and Concavity of Functions 1h34m
Exercise: Identify the interval of increase for a function based on its slope
Video class: Calculus 1 Lecture 3.2: A BRIEF Discussion of Rolle's Theorem and Mean-Value Theorem. 06m
Exercise: What does Rolle’s Theorem guarantee about a differentiable function crossing the x-axis?
Video class: Calculus 1 Lecture 3.3: The First Derivative Test for Increasing and Decreasing 26m
Exercise: _What does the first derivative test tell us about a curve?
Video class: Calculus 1 Lecture 3.4: The Second Derivative Test for Concavity of Functions 36m
Exercise: What does the second derivative tell us about a function?
Video class: Calculus 1 Lecture 3.5: Limits of Functions at Infinity 1h23m
Exercise: What determines the existence of a horizontal asymptote in a function as x approaches infinity?
Video class: Calculus 1 Lecture 3.6: How to Sketch Graphs of Functions 1h32m
Exercise: What is one of the initial steps in curve sketching before using calculus?
Video class: Calculus 1 Lecture 3.7: Optimization; Max/Min Application Problems 1h34m
Exercise: How do you maximize the volume of a box with a fixed perimeter?

Video class: Calculus 1 Lecture 4.1: An Introduction to the Indefinite Integral 2h45m
Exercise: What is the primary purpose of the rectangular method explained in the video transcript?
Video class: Calculus 1 Lecture 4.2: Integration by Substitution 1h33m
Exercise: What is the main purpose of integration by substitution?
Video class: Calculus 1 Lecture 4.3: Area Under a Curve, Limit Approach, Riemann Sums 2h07m
Exercise: What is the sum of a series from k=1 to n of k^3 using the closed formula?
Video class: Calculus 1 Lecture 4.4: The Evaluation of Definite Integrals 30m
Exercise: What is the definite integral of a constant function from a to b?
Video class: Calculus 1 Lecture 4.5: The Fundamental Theorem of Calculus 2h46m
Exercise: How do you find the total area between f(x) = 1 - x^2 and the x-axis from 0 to 2?

Video class: Calculus 1 Lecture 5.1: Finding Area Between Two Curves 1h33m
Exercise: How do you find the area between two curves?
Video class: Calculus 1 Lecture 5.2: Volume of Solids By Disks and Washers Method 2h47m
Exercise: What is the volume of the solid formed by revolving a region around the x-axis?
Video class: Calculus 1 Lecture 5.3: Volume of Solids By Cylindrical Shells Method 54m
Exercise: _What is the advantage of using the cylindrical shells method over the disk or washer method when finding the volume of a solid of revolution around the y-axis?
Video class: Calculus 1 Lecture 5.4: Finding the Length of a Curve on a Plane 2h17m