Duration of the online course: 40 hours and 9 minutes
3.67
(3)
Build calculus skills fast with this free online course: functions, limits, derivatives, and integrals, plus practice exercises to boost grades and confidence.
In this free course, learn about
Number systems, integers, inequalities, and solving absolute value equations
Functions: domain/range, vertical line test, reflections, composition, and inverses
Inverse trig, exponential, and logarithmic functions; domains and key identities
Logic basics: conditionals and proof by mathematical induction
Sequences: limits, convergence, monotone sequences, and the Monotone Sequence Theorem
Applications of derivatives: related rates, extrema, MVT/Rolle, curve sketching, optimization
Advanced limit tools: L'Hopital's rule and Newton's method for root finding
Approximations: linearization, differentials, Taylor polynomials/series
Integrals: sigma notation, Riemann sums, definite integrals, properties, and area under curves
Antiderivatives, FTC (linking derivatives & integrals), and substitution (u-substitution)
About the free online course
Calculus becomes much easier when the foundations feel natural. This free online course is designed to help you move from core pre-calculus ideas into the essential toolkit of Calculus I, with clear explanations and plenty of guided practice. You will strengthen the skills that make everything else possible: working confidently with the number system, inequalities, absolute value, and the function mindset needed to model change.
From there, you will develop a solid understanding of limits, including one-sided behavior, infinite limits, and limits at infinity. You will learn how limit laws streamline algebraic evaluation, how the squeeze theorem justifies tricky conclusions, and how continuity connects graphs and formulas through important results like the intermediate value theorem. These concepts are presented to help you reason about what a function is doing, not just manipulate symbols.
Next, you will build fluency with derivatives as rates of change and slopes of tangent lines. Along the way, you will practice the full set of differentiation tools: rules for powers and combinations, product and quotient techniques, chain rule thinking, implicit differentiation, and derivatives involving trigonometric, inverse trig, exponential, logarithmic, and even hyperbolic functions. You will also see how derivatives support real problem-solving, including related rates, approximation ideas, and numerical methods.
As your skills grow, you will connect derivatives to behavior and decision-making: maxima and minima, critical points, concavity, and curve sketching. You will learn when key theorems apply and how they support reliable conclusions, then use these insights in applications such as optimization. When limits produce indeterminate forms, you will gain an additional strategy with L’Hopital’s rule and asymptote analysis to better interpret function behavior.
Finally, you will transition into integration, building intuition for area under curves and the meaning of definite integrals. You will work with integral properties, antiderivatives, and the fundamental theorem of calculus to connect accumulation and rate of change. The course supports steady progress through review points and exam-style questions, making it a strong choice for high school or early college learners who want a structured path to genuine Calculus I confidence.
Course content
Video class: Calculus I: Introduction10m
Exercise: What aspect of calculus deals primarily with finding the area under curves?
Video class: Calculus I: The Number System26m
Exercise: What is an integer in the number system?
Video class: Calculus I: Inequalities25m
Exercise: Consider the inequality x^2 - 3x - 10 < 0. What are the solutions for x?
Video class: Calculus 1: Absolute Values09m
Exercise: What is the absolute value of -5?
Video class: Calculus I: Absolute Value (Examples)22m
Exercise: Solve the absolute value equation: |3x - 6| = 9.
Video class: Calculus I: Properties of Absolute Values and an interesting example with Triple Absolute Values!13m
Exercise: Understanding Absolute Value Properties
Video class: Calculus I: Functions19m
Exercise: Considering the definition of a function, which of the following graphs represents a function, as per the vertical line test?
Video class: Calculus I: Understanding and Plotting Common Functions27m
Exercise: Which function represents a reflection about the x-axis?
Video class: Calculus I: Compositions of Functions08m
Exercise: Given the functions \( f(x) = x + 2 \) and \( g(x) = x^2 \), what is the composition \( f(g(x)) \) and what is its domain?
Video class: Calculus I: Inverse Functions23m
Exercise: What is required for a function to have an inverse?
Video class: Calculus I: Inverse Trigonometric Functions29m
Exercise: What is the correct expression for the inverse function of sine, commonly known as arcsine?
Video class: Calculus I: Exponential and Logarithmic Functions31m
Exercise: What is the domain of an exponential function?
Video class: Calculus I: Additional Examples in Pre Calculus Topics (Exam style questions)37m
Exercise: What is the domain of the function f(g(x)) defined by f(x) = e^(-x^2 + 1) and g(x) = √(x^2 - 4x + 3)?
Video class: Calculus I: Conditional Statements22m
Exercise: What is a true statement about material conditionals?
Video class: Calculus I: Principle of Mathematical Induction11m
Exercise: Which of the following steps is correctly described in the principle of mathematical induction?
Video class: Calculus I: Examples of the Principle of Mathematical Induction55m
Exercise: What is the first step in proving a statement by mathematical induction?
Video class: Calculus I: Convergence of a Sequence14m
Exercise: Which of the following describes the concept of a limit in a sequence?
Video class: Calculus I: Monotonic Sequences and the Monotone Sequence Theorem08m
Exercise: What is a monotonic sequence?
Video class: Calculus I: Examples of Sequence Questions (With Induction!)32m
Exercise: Consider the sequence given by the formula: \( a_n = \frac{3n^2 + 5n}{2n^2 + n + 1} \). What is the limit of the sequence \( a_n \) as \( n \) approaches infinity?
Video class: Calculus I: The Limit of a Function21m
Exercise: What is the concept of a limit in calculus?
Video class: Calculus I: One Sided Limits15m
Exercise: Consider the function f(x) defined as follows: f(x) = 3x + 1 for x < 2 and f(x) = 7 for x ≥ 2. What is the one-sided limit of f(x) as x approaches 2 from the left?
Video class: Calculus I: Example on Evaluating Infinite Limits and Vertical Asymptotes13m
Video class: Calculus I: Limit Laws11m
Exercise: Which of the following is a correct application of the limit laws when evaluating limits algebraically?
Video class: Calculus I: Limit Laws Examples27m
Video class: Calculus I: The Squeeze (Sandwich) Theorem14m
Exercise: According to the squeeze theorem, if we have two functions f(x) and h(x) such that f(x) ≤ g(x) ≤ h(x), and both the limit of f(x) and h(x) as x approaches a are L, what can we conclude about the limit of g(x) as x approaches a?
Video class: Calculus I: Squeeze Theorem Examples12m
Video class: Calculus I: Continuity and The Intermediate Value Theorem42m
Exercise: A function is said to be continuous at a point if which of the following conditions is true?
Video class: Calculus I: Limits at Infinity and Horizontal Asymptotes34m
Video class: Calculus I: Limits at Infinity and Horizontal Asymptotes (Examples)40m
Exercise: What is the value of the following limit as x approaches infinity: \( \lim_{{x \to \infty}} \frac{x^3 - 2x}{\sqrt{x^2 + 1} - x} \)?
Video class: Calculus I: Derivatives and Rates of Change31m
Video class: Calculus I: Examples on Derivatives and Rates of Change27m
Exercise: Using the definition of the derivative, differentiate the function f(x) = x^(1/3).
Video class: Calculus I: Higher Order Derivatives15m
Video class: Calculus I: Differentiation Rules29m
Exercise: Which rule of differentiation states that the derivative of a constant is always zero?
Video class: Calculus I: Examples on Differentiation Rules17m
Video class: Calculus I: Product/Quotient rules Normal lines24m
Exercise: Using the product rule, differentiate the function h(x) = (2x^3)(4x^2). Which of the following is the correct derivative?
Video class: Calculus I: Product and Quotient rules Tangent/Normal lines (Examples)35m
Video class: Calculus I: Differentiation of Trigonometric Functions34m
Exercise: What is the derivative of cosecant of x with respect to x?
Video class: Calculus I: Differentiation of Trigonometric Functions (Examples)48m
Video class: Calculus I: The Chain Rule12m
Exercise: What is the derivative of the function h(x) = (x^2 + 1)^(1/2) using the chain rule?
Video class: Calculus I: Chain Rule Examples21m
Video class: Calculus I: Implicit Differentiation27m
Exercise: Which of the following statements correctly describes implicit differentiation?
Video class: Calculus I: Derivatives of Inverse Trigonometric Functions13m
Video class: Calculus I: Derivatives of Inverse Trigonometric Functions (Examples)09m
Exercise: What is the derivative of f(x) = cos⁻¹(x) - 5 * tan⁻¹(x) with respect to x?
Video class: Calculus I: Derivative of inverse functions28m
Video class: Calculus I: Derivative of exponential and logarithmic functions21m
Exercise: What is the derivative of the function f(x) = 6^x?
Video class: Calculus I: Logarithmic differentiation23m
Video class: Calculus I: Related Rates1h03m
Exercise: In a related rates problem, a 10-foot ladder is leaning against a wall. The bottom of the ladder slides away from the wall at a rate of 3 feet per second. How fast is the top of the ladder sliding down the wall when the bottom of the ladder is 6 feet away from the wall?
Video class: Calculus I: Midterm Review!2h34m
Video class: Calculus I: Taylor Polynomials25m
Exercise: What is a Taylor series used for?
Video class: Calculus I: Linear approximations and differentials42m
Video class: Calculus I: Taylor Polynomials Examples35m
Exercise: What is the second degree Taylor polynomial T2(x) for the function f(x) = ln(x) about x = 1?
Video class: Calculus I: Partial derivatives31m
Video class: Calculus I: Hyperbolic Trigonometric Functions34m
Exercise: Which of the following is the standard definition of the hyperbolic cosine function, cosh(x)?
Video class: Calculus I: Derivatives of Hyperbolic Functions15m
Video class: Calculus I: Inverse Hyperbolic Trigonometric Functions21m
Exercise: What is the derivative of the inverse hyperbolic sine function (sinh⁻¹(x)) with respect to x?
Video class: Calculus I: Maxima and Minima19m
Video class: Calculus I: Extreme Value Theorem15m
Exercise: Which of the following statements is TRUE regarding the Extreme Value Theorem?
Video class: Calculus I: Critical points and Extrema27m
Video class: Calculus I: Rolle's theorem17m
Exercise: Which of the following is a condition that must be satisfied for Rolle's Theorem to be applicable to a function f(x) on the interval [a, b]?
Video class: Calculus I: The Mean Value Theorem26m
Video class: Calculus I: Curve Sketching (Part 1)33m
Exercise: In the context of curve sketching, if the first derivative of a function f(x) changes from positive to negative as x passes the critical number c, what can be inferred about the point at x = c?
Video class: Calculus I: Curve Sketching (Part 2)27m
Video class: Calculus I: Curve sketching (Examples)47m
Exercise: When using the second derivative test to determine the concavity of a function at a certain point, which of the following statements is true?
Video class: Calculus I: L'hopital's rule13m
Video class: Calculus I: L'Hopital's rule (Examples)48m
Exercise: Evaluate the limit as x approaches 0 of the expression 3x^2 - 2x + 1 / 2x^2 + x - 1.
Video class: Calculus I: Slant asymptotes44m
Video class: Calculus I: Optimization problems54m
Exercise: You have a piece of material 800 feet long to construct a rectangular enclosure along one side of a building. This side of the building does not need fencing. If L is the length of the fence parallel to the building and W is the width of the ends perpendicular to the building, what arrangement will maximize the area that can be enclosed?
Video class: Calculus I: Newton's Method13m
Video class: Calculus I: Sigma notation and summation40m
Exercise: Using sigma notation, express the sum of the squares of the first 10 natural numbers.
Video class: Calculus I: The area under a curve30m
Video class: Calculus I: The definition of a Definite Integral21m
Exercise: What is the value of a definite integral of the constant function f(x) = 5 from x = 2 to x = 6?
Video class: Calculus I: The definition of a Definite Integral (Examples)24m
Video class: Calculus I: Properties of definite integrals21m
Exercise: Which of the following statements about definite integrals represents a property that allows splitting the integral into multiple parts with respect to an intermediate point?
Video class: Calculus I: Antiderivatives and Indefinite Integrals55m
Video class: Calculus I: The Fundamental Theorem of Calculus33m
Exercise: What does the first part of the Fundamental Theorem of Calculus state about the relationship between integrals and derivatives?
Video class: Calculus I: Fundamental theorem of calculus (Examples)1h23m
Video class: Calculus I: The substitution rule50m
Exercise: When applying the substitution rule or 'u-substitution' to evaluate integrals, which of the following statements is generally true?
Video class: Calculus I: Final Exam Review2h28m
This free course includes:
40 hours and 9 minutes of online video course
Digital certificate of course completion (Free)
Exercises to train your knowledge
100% free, from content to certificate
Does this Introductory Calculus course cover both derivatives and integrals?
Yes. It covers limits, differentiation rules and applications, then definite and indefinite integrals, area under curves, and the Fundamental Theorem of Calculus.
What precalculus topics are included before limits and derivatives?
The course reviews number systems, inequalities, absolute values, functions, inverse functions, trigonometry, exponentials, logarithms, sequences, and induction.
Does the course teach calculus applications such as optimization and related rates?
Yes. It includes related rates, maxima and minima, curve sketching, optimization, linear approximations, Taylor polynomials, and Newton's Method.
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