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Orientation

The idea map

Calculus isn't a pile of formulas — it's a small tower of ideas, each standing on the one below. Here's the whole tower at a glance. Every node is a door.

The ground floor

FunctionsRules that turn inputs into outputs.GraphsFunctions, made visible.Slope & rateHow fast one thing changes with another.

The key that unlocks everything

LimitsGetting infinitely close — rigorously. Powers both ideas below.

The two big ideas

DerivativesInstant rate of change. Slope at a point. “How fast, right now?”IntegralsAccumulated total. Area under a curve. “How much, overall?”

Fluency

Derivative rulesThe power rule and friends — patterns over memorization.AntiderivativesDerivatives, run backwards. The +C family.Antiderivative rulesRaise the power, divide by the new power.

The grand connection

The fundamental theoremDerivative and integral are inverse operations. One subject, not two.

Out into the world

Optimization & related ratesPeaks, valleys, and linked rates.Real-world applicationsMotion, money, medicine, machine learning.The playgroundTouch every idea on this map.

Derivatives zoom in on an instant; integrals zoom out to a total; limits make both moves legal; the fundamental theorem proves they're one move, mirrored.