From foundations to the Fundamental Theorem — every calculus concept explained visually. Limits, derivatives, integrals, and applications with interactive graphs.
Build the intuition behind calculus before the formulas.
Calculus is the mathematics of change — velocities, accelerations, tangent lines, slopes, areas, volumes, and more. It differs fundamentally from precalculus: precalculus is static, calculus is dynamic. The key bridge between them is a single powerful idea: the limit.
Calculus is the mathematics of continuous change. Before derivatives or integrals, we explore the core question: how do quantities change relative to each other?
f(x) is the universal language of calculus. Understanding how to read, evaluate, and interpret function notation is the essential first step.
A refresher on slope, linear functions, and how graphs encode information — the language we will use throughout calculus.
f(g(x)) means: apply g first, then feed the result into f. Composition is everywhere in calculus — the Chain Rule is built entirely on it.
The average rate of change measures how much a function changes over an interval. It is the slope of the secant line connecting two points on the curve.
What happens as the interval shrinks to zero? The average rate of change approaches the instantaneous rate — the derivative.
Two kinds of lines on a curve. A secant cuts through at two points — measuring average change. A tangent touches at one point — measuring instantaneous change.
Infinity is not a number — it is a direction. Understanding how functions behave as x grows without bound prepares you for limits at infinity.
What does it mean to approach a value without reaching it? This question defines limits — and limits define derivatives and integrals.
Understand the foundation of calculus — what functions approach, not just what they equal.
A limit describes the value a function approaches as the input gets close to a point — not necessarily the value at that point.
Sometimes a function approaches different values from the left and right. One-sided limits capture each direction separately.
Limit laws allow us to compute limits algebraically by breaking complex expressions into simpler parts.
What happens to a function as x grows without bound? Limits at infinity describe end behavior and horizontal asymptotes.
A function is continuous at a point if its limit equals its value there. Discontinuities break this condition in different ways.
Master the instantaneous rate of change and its applications.
The derivative is defined as the limit of the difference quotient. This formal definition connects limits to the slope of the tangent line.
The tangent line at a point has slope equal to f′(a). It locally approximates the curve and is foundational to linearization.
The simplest and most used differentiation rule. If f(x) = x^n, then f′(x) = n·x^(n−1).
When differentiating a product of two functions, you cannot just multiply the derivatives.
Differentiating a ratio of two functions requires careful application of the quotient rule.
The chain rule handles composite functions: when one function is nested inside another.
The sign of f′ tells us whether the function is rising or falling.
Find local maxima and minima using the first and second derivative tests.
Set up and solve real-world problems by finding the maximum or minimum of a function.
Position, velocity, and acceleration are linked by derivatives. Calculus gives us the tools to analyze any motion.
When y cannot be isolated, differentiate both sides with respect to x and solve for dy/dx.
Two quantities change over time. Use implicit differentiation to find how their rates are related.
Accumulate quantities and compute areas under curves.
Slice the area under a curve into rectangles and add them up. As the rectangles get thinner, the approximation becomes exact.
The definite integral gives the exact accumulated area (signed) between a curve and the x-axis over [a, b].
The indefinite integral (antiderivative) finds a family of functions whose derivative is the given function.
U-substitution reverses the chain rule. Substitute u = g(x) to simplify the integral.
Compute the area between curves using definite integrals.
Rotate a region around an axis to generate a solid. Use the disk or washer method.
Integrate velocity to find total displacement or total distance traveled.
The average value of a function on [a, b] is computed using a definite integral.
The bridge connecting differentiation and integration.
The Fundamental Theorem of Calculus unifies derivatives and integrals into a single elegant result — the deepest result of introductory calculus.
An accumulation function A(x) = ∫ₐˣ f(t) dt tracks the running total of f from a to x.
Understand how the graphs of f, f′, and ∫f are related. This synthesizes everything in calculus.
Use the calculus visualizer to see derivatives, integrals, and limits come to life.
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