Calculus
Functions, slope, and sine and cosine; then limits, the derivative, the integral and the fundamental theorem — ending at the gradient, the idea that trains neural networks.
12 stops27 min
The lessons, in order
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Functions
the machine metaphor
A function is a machine: numbers in, numbers out — and its graph is a portrait of every answer at once.
2min2:07
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Slope
how steep, how fast
Rise over run: one number that says steep or gentle, uphill or down. Straight lines have one slope; curves are a mystery…
2min2:24
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Sine and cosine
the numbers behind rotation
Cosine and sine are a turning point's coordinates. They rotate any arrow, and a dot product feels only the gap between two turns.
2min2:18
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Rates of change
slopes in the wild
Speed is the slope of a distance graph. Every 'how fast is it changing?' is secretly a slope question.
2min2:09
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The limit
sneaking up on a point
Two points squeezed to one gives 0/0 nonsense — unless you sneak up and read where the journey is heading.
2min1:33
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The derivative
slope at an instant
The settled slope at a single point gets a name — and becomes a whole new function. Your speedometer computes one every second.
2min2:17
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The power rule
a beautiful shortcut
The exponent hops down front and drops by one: xⁿ → n·xⁿ⁻¹. Whole families of curves, differentiated instantly.
2min2:13
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Where slopes vanish
maxima and minima
At the very top of a hill the tangent goes flat. Set the derivative to zero and you can find the best of anything.
2min2:07
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Area under a curve
why accumulate?
Distance hides under the speed graph; totals hide under rates. But how do you measure a region with a curvy top?
2min2:10
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The integral
infinitely thin slices
Slice the region into rectangles, let them get infinitely thin, and the total settles on one exact number: the integral.
2min2:11
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The bridge
the fundamental theorem
Derivatives and integrals are opposites — one undoes the other. The bridge that makes calculus practical.
2min2:28
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The gradient
slope grows up
Two slopes stack into one arrow that points straight uphill. Flip it, and you have the engine of machine learning.
3min2:48
