Calculus Foundations
Overview
Study the mathematics of continuous change, focusing on rates of change (derivatives) and accumulation of quantities (integrals).
Core Concepts Covered
- ✔ Limits and Continuity
- ✔ Differentiation Rules and Applications
- ✔ Integration Techniques (U-Substitution, Parts)
- ✔ The Fundamental Theorem of Calculus
- ✔ Sequences, Series, and Convergence
Interactive Application
Each concept includes clear explanations, step-by-step examples, and guided practice with linked calculators for hands-on reinforcement.
Select a concept or jump directly into a calculator below to begin.
Lessons
Functions and Graphs: The Language of Calculus
Understand what a function is, how to read a graph, and why this visual language is the foundation for every calculus concept that follows.
Slopes and Rates of Change
Master the concept of slope as a rate — the mathematical measure of how fast something changes — and lay the groundwork for derivatives.
Linear vs. Nonlinear Relationships
Compare straight-line functions with curves — and understand why nonlinearity is what makes calculus necessary in the first place.
Introduction to Limits
Understand what a limit means intuitively — approaching a value without necessarily reaching it — and learn why limits are the rigorous foundation beneath all of calculus.
Instantaneous Rate of Change: The Bridge to Derivatives
See how average rates of change over shrinking intervals converge to an exact instantaneous rate — the key insight that makes derivatives possible.
Introduction to Derivatives: Rules That Make Calculus Practical
Learn the Power Rule and basic derivative rules that let you find instantaneous rates of change without computing limits by hand every time.
Understanding Derivatives: What the Slope Function Tells You
Learn to read a derivative to find where functions increase, decrease, reach peaks and valleys — the practical skill of interpreting derivatives without graphing by hand.
Introduction to Integrals: Measuring Accumulated Change
Discover how integration measures total accumulated quantities — area under a curve, total distance traveled, net value built up over time.
Understanding Integrals: Area, Accumulation, and the Riemann Sum
Build geometric intuition for integrals by connecting Riemann sums (rectangles under a curve) to the exact definite integral — and learn when the area interpretation applies.
The Fundamental Theorem: Connecting Derivatives and Integrals
See how differentiation and integration are inverse operations — two sides of the same coin — and why this connection is the most important theorem in calculus.