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University Calculus I–II Self-paced course

Introductory Calculus

Understand change, accumulation, and approximation in a first-year university Calculus I–II sequence.

13 modules on the roadmap 1 available now · 5 lessons

About the course

Calculus starts with two questions: how fast is something changing, and how much has accumulated? Derivatives describe instantaneous change; definite integrals describe accumulation. The Fundamental Theorem of Calculus connects the two. A third question carries us further: how accurately can we approximate a function or a quantity?

These course notes are organized around the learning objectives of introductory university calculus in the United States and Canada. The roadmap covers the full first-year single-variable sequence, with a clear boundary between Calculus I and Calculus II. The aim is to understand the ideas, calculate confidently, build useful models, and explain why a solution is valid.

The derivatives module is available now, with five lessons. Start there if you already know limits and continuity; otherwise follow the roadmap from the beginning as modules become available. The current lessons include challenging problems and optional proof extensions. The other modules are forthcoming, and their outcomes below describe the intended complete course.

These notes follow a standard first-year university sequence with trigonometry, suitable for mathematics, science, and engineering. Calculus I develops change and accumulation; Calculus II develops integration, models, and approximation. The division below is a study guide, since universities place some topics in different terms.

Start the derivatives module

Learning objectives

By the end of the complete sequence, you should be able to:

  • Interpret functions, limits, and continuity using formulas, graphs, tables, and words, and calculate limits with an appropriate method.
  • Explain a derivative as an instantaneous rate and tangent slope; compute derivatives from the definition and with differentiation rules.
  • Use derivatives and the hypotheses of calculus theorems to analyze graphs, estimate values, and solve optimization and related-rates problems.
  • Explain a definite integral as a limit of sums and as net accumulation, and use both parts of the Fundamental Theorem of Calculus.
  • Choose and carry out integration methods, and assess numerical approximations and convergence of improper integrals.
  • Build and interpret calculus models for motion, area, volume, and other accumulated quantities, with correct units and assumptions.
  • Interpret direction fields and solve elementary separable differential equations and initial-value problems.
  • Determine convergence of sequences and series, work with power and Taylor series, and justify approximation errors.
  • Communicate complete mathematical reasoning, check theorem conditions, and assess the plausibility of an answer using estimates or technology.

Table of contents

Follow the modules in order. Each module states what you should be able to do; only modules with available lessons can be opened.

Calculus I: change and accumulation

Build the foundations, learn differentiation and its applications, and connect derivatives to definite integrals.

  1. Forthcoming

    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.

  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 →
  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; an introduction to Taylor polynomials, Newton's method, and L'Hôpital's rule.

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

  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.

  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.

  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.

Calculus II: integration, models, and series

Extend integration, study elementary differential equations, and use convergence and series to understand approximation.

  1. Forthcoming

    Techniques of integration

    Integration by parts; trigonometric integrals and substitutions; partial fractions; choosing and combining integration methods.

    Goal: Select a suitable integration method and check an antiderivative by differentiation.

  2. Forthcoming

    Further applications of integrals

    Arc length and surface area; work, variable forces, and hydrostatic force; mass, moments, and centers of mass. Probability densities provide a syllabus-dependent application.

    Goal: Derive an integral from small contributions and interpret the result in the original setting.

  3. Forthcoming

    Numerical integration and improper integrals

    Midpoint, trapezoidal, and Simpson's rules with error estimates; unbounded intervals and singular integrands; convergence and comparison of improper integrals.

    Goal: Approximate an integral to a stated accuracy and determine whether an improper integral converges.

  4. Forthcoming

    Elementary differential equations

    Solution checking, direction fields, and Euler's method; separable equations and initial values; exponential growth and decay, equilibrium, and logistic models. First-order linear equations are an optional addition.

    Goal: Solve a separable initial-value problem and compare its solution with the qualitative behavior of the model.

  5. Forthcoming

    Sequences and infinite series

    Sequence limits and partial sums; geometric, telescoping, harmonic, and p-series; divergence, integral, comparison, limit comparison, ratio, root, and alternating-series tests; absolute and conditional convergence.

    Goal: Choose and justify a convergence test and bound the error of an alternating-series approximation.

  6. Forthcoming

    Power series and Taylor approximation

    Radius and interval of convergence, including endpoints; differentiation and integration of power series; Taylor and Maclaurin polynomials and series; remainder estimates and applications.

    Goal: Represent a function by a power series where valid and justify the accuracy of a polynomial approximation.

  7. Forthcoming

    Parametric and polar curves

    Parametric motion, tangents, and arc length; polar graphs and area. This module supplies a common addition whose placement varies by university.

    Goal: Apply differentiation and integration to a curve given parametrically or in polar coordinates.

How the notes develop understanding

  • Each lesson connects a concept to graphical, numerical, verbal, and symbolic representations, then develops the relevant calculation.
  • Practice progresses from short conceptual checks and routine calculations to applications and mixed problems that require choosing a method.
  • Module checkpoints combine explanation, computation, and modeling. Cumulative reviews revisit earlier ideas and require complete solutions.
  • Proofs of central results support the main course. Advanced counterexamples and analysis problems are marked as optional enrichment.

Prerequisites

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

  • Algebra, including equations, inequalities, and manipulating fractions.
  • Function notation and graphs, including polynomial, exponential, and logarithmic functions.
  • Basic trigonometry, including sine, cosine, and radians.

To start the available derivatives module, you will also need limits and continuity, including the standard trigonometric and exponential limits.

University objectives behind the course

This roadmap is a synthesis of the university sources below. Taylor polynomials, L'Hôpital's rule, integral applications, and differential equations may move between terms. Parametric and polar curves and probability applications are syllabus-dependent. A university requiring introductory partial derivatives needs an additional multivariable bridge; that material is outside this single-variable sequence.

References

Companion reading and sources for further exploration.

  • Calculus, Volume 1

    OpenStax

    A free companion text for limits, differentiation, and integration, with examples and exercises.

  • Active Calculus — Single Variable

    Matt Boelkins

    A free text built around questions and activities for developing a conceptual understanding of calculus.

  • Problems in Real Analysis: Advanced Calculus on the Real Axis

    Teodora-Liliana Rădulescu, Vicențiu D. Rădulescu, and Titu Andreescu · Springer, 2009

    Chapter 5 supplies optional proof extensions for the derivatives module. The core course follows the introductory university objectives above.