Calculus I
Courseware

Learnvia’s Calculus I courseware provides a research-informed, mastery-based approach to help students develop both conceptual understanding and procedural fluency.
Request a Live Demo

Content Scope and Sequence

1.1 Functions and their representations
1.2 Linear functions
1.3 Polynomial functions
1.4 Rational and piecewise-defined functions
1.5 Combining and composing functions
1.6 Inverse functions
1.7 Exponential functions
1.8 Logarithmic functions
1.9 Trigonometric functions

2.1 Preview to calculus
2.2 Average and instantaneous velocity
2.3 Limits of a function
2.4 Evaluating limits algebraically
2.5 Limits at infinity, infinite limits, and asymptotes
2.6 Continuity

3.1 Derivatives and the tangent problem
3.2 Derivative as a function
3.3 Basic rules of finding derivatives
3.4 Product and quotient rules
3.5 Chain rule
3.6 Derivatives of trigonometric functions
3.7 Implicit differentiation
3.8 Derivatives of exponential functions
3.9 Derivatives of logarithmic functions

4.1 Related Rates
4.2 Linear approximation and differentials
4.3 Analyzing functions (Global and local extrema)
4.4 Analyzing functions (Mean Value Theorem)
4.5 Analyzing functions (First derivative test, increasing and decreasing functions)
4.6 Analyzing functions (Second derivative test and concavity)
4.7 Analyzing functions (Sketching curves)
4.8 Applied optimization
4.9 Indeterminate forms and L’Hopital’s rule

5.1 Antiderivatives and indefinite integrals
5.2 Antiderivatives of trigonometric, exponential, and reciprocal functions
5.3 Integrals and the area problem
5.4 The definite integral
5.5 The Fundamental Theorem of Calculus
5.6 Net Change Theorem
5.7 Integration by substitution

6.1 Additional questions: Functions and their representations
6.2 Additional questions: Linear functions
6.3 Additional questions: Polynomial functions
6.4 Additional questions: Rational and piecewise-defined functions
6.5 Additional questions: Combining and composing functions
6.6 Additional questions: Inverse functions
6.7 Additional questions: Exponential functions
6.8 Additional questions: Logarithmic functions
6.9 Additional questions: Trigonometric functions
6.9 Additional questions: Derivatives and the tangent problem
6.10 Additional questions: Derivative as a function
6.11 Additional questions: Basic rules of finding derivatives
6.12 Additional questions: Product and quotient rules
6.13 Additional questions: Chain rule
6.14 Additional questions: Derivatives of trigonometric functions
6.15 Additional questions: Implicit differentiation
6.16 Additional questions: Derivatives of exponential functions
6.17 Additional questions: Derivatives of logarithmic functions
6.18 Additional questions: Related Rates
6.19 Additional questions: Linear approximation and differentials
6.20 Additional questions: Analyzing functions (Global and local extrema)
6.21 Additional questions: Analyzing functions (Mean Value Theorem)
6.22 Additional questions: Analyzing functions (First derivative test, increasing and decreasing functions)
6.23 Additional questions: Analyzing functions (Second derivative test and concavity)
6.24 Additional questions: Analyzing functions (Sketching curves)
6.25 Additional questions: Applied optimization

Download the table of contents

Pedagogical Approach

Courseware Description

Calculus 1 is a configurable, 14-week courseware designed to strengthen student success in gateway math and support retention in STEM pathways. Interactive lessons, scaffolded activities, and AI-powered feedback help students develop conceptual understanding and problem-solving skills across core topics including functions, limits, continuity, differentiation, applications of the derivative, and an introduction to integration.  Developed by educators and continuously updated to reflect best practices, the courseware blends conceptual and procedural focus, provides real-time guidance and targeted support, and promotes engagement and achievement in foundational calculus learning.

Learnvia Calculus 1 Courseware Components

Structured for Learning

The courseware is organized into chapters, modules, and short interactive activities that follow the rhythm of a college course. The default 14-week design can be easily adapted for alternative term lengths. Each chapter represents a week of instruction and includes several modules focused on two to four clearly defined learning outcomes.

Modules feature concise, ten-minute activities such as lessons, homework, and quizzes that encourage consistent progress and effective time management. This “bite-sized” format supports attention, pacing, and student motivation, allowing learners to engage meaningfully in shorter study sessions while retaining core ideas.

Designed for Clarity and Consistency

Each module follows a consistent pattern that builds conceptual understanding step by step:

  • Lessons introduce new concepts through concise explanations, animated figures, and embedded questions that check understanding in real time.
  • Homework provides randomized, auto-scored practice with hints, explanations, and unlimited attempts. Students can seek support but must solve a new version to earn credit, reinforcing mastery.
  • Quizzes offer brief, auto-graded assessments aligned with lessons and homework, allowing students to demonstrate comprehension with confidence.

Research-Informed Learning Cycle

Each week, students move through a scaffolded learning cycle:
  1. Engage and explore through guided lesson activities.
  2. Practice and apply with structured, feedback-rich homework.
  3. Assess and reflect through quizzes and reviews that consolidate learning.
This cycle reflects backward design principles where quizzes define learning goals, homework builds toward those goals, and lessons establish the conceptual foundation.

Pedagogical Foundations

Learnvia Calculus 1 is intentionally:
  1. Chunked: Short, focused activities sustain attention and promote incremental learning.
  2. Scaffolded: Support gradually decreases from lessons to homework to quizzes, developing student independence.
  3. Aligned: Every component connects through shared learning outcomes and knowledge–skill–ability (KSA) targets.

Integrated Support and Feedback

Built-in AI tutoring and discussion forums provide just-in-time support, while formative analytics give instructors real-time insight into student progress. Instant, actionable feedback helps close learning gaps before high-stakes assessments. This integrated support system encourages persistence and empowers both students and instructors to make data-informed decisions.

Mobile and Accessible

Learnvia Calculus 1 is designed with a mobile-first approach that ensures seamless functionality on smartphones, tablets, and iPads. The responsive interface adjusts to any screen size, allowing students to review lessons, complete assignments, and check progress from any location. Optimized visuals, accessible navigation, and touch-friendly interactions create a consistent learning experience across all devices. The design supports students who balance coursework with work or personal responsibilities, offering flexibility without compromising quality.

Topics Covered

  • Introduction to functions
  • Limits, continuity, and differentiation
  • Applications of the derivative, including optimization, related rates, and curve sketching
  • Introduction to definite integrals and integration

Key Features

  • Balanced emphasis on conceptual understanding and procedural fluency
  • AI-powered feedback and real-time guidance
  • Continuously updated content aligned with best practices in calculus instruction
  • Structured, mobile-accessible design supporting flexible and inclusive learning

Developed by educators and grounded in learning science, Learnvia Calculus 1 enhances instruction, promotes engagement, and helps more students succeed in foundational mathematics. The courseware unites pedagogy, technology, and accessibility to create an integrated learning environment that supports both teaching and mastery in Calculus 1.

Contributors

Curriculum Committee 
Matt Boelkins, Grand Valley State University 
David Bressoud, Macalester College
Dennis Davenport, Howard University
Roberto Pelayo, University of California, Irvine
Alicia Prieto-Langarica, Youngstown State University
April Strom, Chandler-Gilbert Community College

Authors
Jessica Bernards, Portland Community College and University of Oregon
Chris Chan, Learnvia
Ashlee Kalauli, Hawai’i Community College
Ishwari Kunwar, Fort Valley State University
Vijay Kunwar, Albany State University
Brittni Lorton, Community College of Denver
Molly Lynch, Hollins University
Edward Mosteig, Loyola Marymount University
Roberto Pelayo, University of California, Irvine
Kathryn Stewart, Learnvia
Tam Tran, San Diego City College

Reviewers
Mark Atkins
Dr. April Crenshaw, Chattanooga State Community College
Sharona Krinsky, Cal State, Los Angeles and the Center for Grading Reform
Kyle Kundomal, Collin College
Sara Lapan, University of California, Riverside
Liam O’Brien, M.S. in Mathematics, University of New Mexico
Joseph Petrillo, Alfred University
Laura Watkins, Glendale Community College

Math Advisory Board
John Mackey, Carnegie Mellon University
Michael Starbird, The University of Texas at Austin
Michael Young, Carnegie Mellon University

Download the Learnvia VPAT

Request a Live Demo