Rounded trefoil knot The notebook Calculus I
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First-semester university calculus Self-paced course

Calculus I

A rigorous introduction to limits, derivatives, and integrals of functions of one variable.

6 chapters on the roadmap 2 available now · 6 lessons

About the course

How fast is a quantity changing? How much change has accumulated? Calculus makes these questions precise. A limit describes approaching behavior, a derivative measures instantaneous change, and a definite integral measures accumulation. The Fundamental Theorem of Calculus explains how differentiation and integration are connected.

These notes develop the ideas together with their uses. You will learn to calculate, interpret, and justify: to find an answer, explain what it means, and establish why the reasoning is valid. Definitions, theorem hypotheses, and notation matter throughout the course.

The six chapters below give the route through Calculus I. Begin with functions, domains, and representations. The lesson outlines describe the planned course. The existing derivatives lessons are available for readers who already know limits and continuity; they will be refined as the course is developed.

The course develops calculus of real-valued functions of one real variable, including algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions. Its core runs from limits and continuity through differentiation, the Fundamental Theorem of Calculus, substitution, and basic integral applications.

Start with functions

Learning objectives

By the end of Calculus I, you should be able to:

  • Read and describe functions. Identify domains, compositions, and inverses, and connect formulas with graphs, tables, and contextual meanings.

  • Reason about limits and continuity. Evaluate finite, one-sided, infinite, and at-infinity limits; identify discontinuities and asymptotes; and use continuity to justify the existence of a solution.

  • Explain and compute derivatives. Use the limit definition and differentiation rules, including the chain rule and implicit differentiation, and interpret first and higher derivatives.

  • Draw conclusions from derivatives. Analyze monotonicity, extrema, and concavity; apply central theorems with their hypotheses; and construct and assess a linear approximation.

  • Model changing quantities. Formulate and solve motion, related-rates, and optimization problems, respecting units, constraints, and the domain of the model.

  • Explain accumulation. Interpret Riemann sums and definite integrals, distinguish signed accumulation from geometric area, and explain both parts of the Fundamental Theorem of Calculus.

  • Integrate and apply. Find elementary antiderivatives, use substitution, and set up definite integrals for net change, distance, area, average value, and volume.

  • Justify a solution. State assumptions, use consistent notation, explain the mathematical steps, and check an answer against its context or an independent calculation.

Table of contents

Six chapters organize the core course. Begin with the functions lesson; the derivatives chapter also has existing lessons. The remaining lessons are being developed, and the outlines below show the planned course.

A connected path through Calculus I

Each chapter builds tools that the next chapter needs. Open a chapter's planned lesson outline to see how its central ideas develop.

  1. Available · 1 lesson

    Functions, limits, and continuity

    Elementary functions and their graphs; one-sided, infinite, and at-infinity limits; limit laws, the squeeze theorem, continuity, and the Intermediate Value Theorem.

    Goal: Evaluate limits, locate discontinuities and asymptotes, and justify an existence claim using continuity.

    Open module →
    Planned lesson outline · 6 lessons
    1. Functions, domains, and representations

      Establish the language needed to state every later definition and model.

    2. Approaching a value and one-sided limits

      Distinguish nearby behavior from the function's value at a point.

    3. The precise meaning of a limit

      Make informal claims precise and explain what numerical evidence can and cannot establish.

    4. Limit laws and useful techniques

      Justify algebraic simplification, squeeze arguments, and standard trigonometric limits before using them in derivative calculations.

    5. Infinite limits, limits at infinity, and asymptotes

      Separate unbounded behavior near a point from long-term behavior as the input grows.

    6. Continuity and the Intermediate Value Theorem

      Classify discontinuities and turn continuity into a justified existence argument.

  2. Available · 5 lessons

    Derivatives: from rates to local linearity

    Rates and tangent lines; the derivative definition; elementary and higher derivatives; product, quotient, and chain rules; implicit, inverse, and logarithmic differentiation.

    Goal: Explain and calculate a derivative, find a tangent approximation, and recognize a failure of differentiability.

    Open module →
    Planned lesson outline · 5 lessons
    1. Average rates, instantaneous rates, and tangent slopes

      Motivate the derivative through quantities whose meaning and units can be checked.

    2. The derivative definition and differentiability

      Compute from first principles and identify why corners and some joins fail to have derivatives.

    3. Differentiation rules and higher derivatives

      Develop efficient calculations from the definition and interpret repeated differentiation.

    4. Implicit, inverse, and logarithmic differentiation

      Handle relationships and function forms that are awkward to differentiate directly.

    5. Differentiability and local linear approximation

      Connect the derivative to a local model; revisit practical estimates in the next chapter.

  3. Forthcoming

    What derivatives tell us about functions

    Extreme Value, Rolle's, and Mean Value Theorems; monotonicity and concavity; curve sketching; linear approximation and differentials; L'Hôpital's rule. Possible extensions include Newton's method and introductory Taylor polynomials.

    Goal: Analyze a graph and construct a local approximation while checking the hypotheses of the method used.

    Planned lesson outline · 4 lessons
    1. Extreme values, Rolle's Theorem, and the Mean Value Theorem

      Explain what hypotheses allow us to infer from derivatives and where those conclusions can fail.

    2. Monotonicity, concavity, and curve sketching

      Combine first and second derivative information into a justified description of a graph.

    3. Linear approximation and differentials

      Use a tangent model for numerical estimates and explain the limits of its accuracy.

    4. Indeterminate forms and L'Hôpital's rule

      Use derivative information to evaluate eligible limits while checking the rule's conditions.

  4. Forthcoming

    Applications of derivatives

    Position, velocity, and acceleration; related rates; local and absolute extrema; constrained optimization with one independent variable.

    Goal: Translate a contextual problem into a derivative calculation and interpret its solution with units and domain restrictions.

    Planned lesson outline · 3 lessons
    1. Motion and the meaning of derivatives

      Interpret position, velocity, acceleration, and the distinction between velocity and speed.

    2. Related rates

      Differentiate a relationship between changing quantities before substituting a particular instant.

    3. Optimization

      Build an objective function, identify the feasible domain, and justify local or absolute extrema.

  5. Forthcoming

    Integrals and accumulation

    Antiderivatives and initial conditions; Riemann sums; definite integral properties; signed area and net change; accumulation functions and both parts of the Fundamental Theorem of Calculus.

    Goal: Move between a rate and its accumulated change, explaining why differentiation and integration are connected.

    Planned lesson outline · 4 lessons
    1. Antiderivatives and initial conditions

      Recover a quantity from its rate and account for the constant of integration.

    2. Riemann sums and the definite integral

      Construct accumulation from small contributions and understand its limiting definition.

    3. Integral properties, signed area, and net change

      Separate cancellation from total area and interpret accumulation consistently.

    4. The Fundamental Theorem of Calculus

      Explain and prove the connection between continuous rates, accumulation functions, and antiderivatives.

  6. Forthcoming

    Basic integration and its applications

    Elementary antiderivatives and substitution, including changed bounds; total change and total distance; area between curves, average value, and volumes by cross-sections, disks, washers, and shells.

    Goal: Evaluate basic integrals and set up a definite integral from a geometric or rate-based description.

    Planned lesson outline · 4 lessons
    1. Elementary integrals and substitution

      Reverse familiar derivative rules and handle changes of variable and integration bounds correctly.

    2. Net change, total distance, and average value

      Select the appropriate integral when a rate changes sign or a quantity is averaged over an interval.

    3. Area between curves

      Choose a variable of integration and split a region when its boundary order changes.

    4. Volumes by slices and revolution

      Derive volume integrals from cross-sections and choose disks, washers, or shells to match the geometry.

How the notes develop understanding

  • Understand the claim. Read a definition or theorem together with its assumptions. Explain what it says in your own words and how it connects to an earlier idea.

  • Work with purpose. Examples illustrate a specific method or distinction. Activities ask a mathematical question, and their explanations remain usable without the interaction.

  • Exercises review learning. Use them to retrieve definitions, interpret representations, and practise techniques introduced in the lesson.

  • Problems challenge understanding. Expect unfamiliar combinations, proof, counterexamples, or modeling choices. The most challenging problems are identified and their statements remain visible.

  • Explain before checking. Attempt a complete solution before opening a hint or solution. Use chapter checkpoints and the final cumulative review to identify what needs another pass.

Prerequisites

You do not need prior calculus for the full course. Be comfortable with:

  • Algebra: equations, inequalities, factoring, fractions, exponents, and logarithms.

  • Functions: notation, domains and ranges, graphs, transformations, composition, and inverses.

  • Trigonometry: radians, the unit circle, sine, cosine, tangent, and basic identities.

To use the existing derivatives lessons now, first be comfortable with limits and continuity, including the standard limits those lessons cite.

University objectives behind the course

The core follows the first-semester single-variable coverage represented by the sources below. Some first-semester courses also introduce Newton's method or Taylor polynomials alongside approximation. Integration by parts, advanced integration methods, improper integrals, infinite series, and parametric or polar calculus belong to the later course in this sequence.