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Unit circle diagram showing a shaded sector of area x/2 and arc length x to a point P, illustrating the arc-length and area based definition of sine and cosine rather than a right triangle

How Do You Actually Define Sine and Cosine? (No...

The triangle-ratio definition of sine and cosine breaks down as a foundation for calculus. Here's how Spivak rebuilds them rigorously from circular area and arc length — and the addition...

How Do You Actually Define Sine and Cosine? (No...

The triangle-ratio definition of sine and cosine breaks down as a foundation for calculus. Here's how Spivak rebuilds them rigorously from circular area and arc length — and the addition...

A rising curve with the area beneath it shaded from a point a to a point x, and a dotted arrow linking that accumulated area back up to the curve's height at x, illustrating the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus: Why Integr...

A close look at Spivak's Chapter 14: why the area-accumulation process of integration and the slope-finding process of differentiation turn out to be exact inverses, with the proof idea and...

The Fundamental Theorem of Calculus: Why Integr...

A close look at Spivak's Chapter 14: why the area-accumulation process of integration and the slope-finding process of differentiation turn out to be exact inverses, with the proof idea and...

Line-art diagram showing a curve with upper and lower sum rectangles from a partition, illustrating the rigorous definition of the integral

What Is an Integral, Really? (It's Not Just "Ar...

A rigorous look at how Spivak's Calculus actually defines the integral through upper and lower sums over partitions — and why continuity is what makes the definition work.

What Is an Integral, Really? (It's Not Just "Ar...

A rigorous look at how Spivak's Calculus actually defines the integral through upper and lower sums over partitions — and why continuity is what makes the definition work.

Line art showing an increasing function f and its inverse f-inverse reflected across the diagonal line y=x, illustrating invertibility in calculus

How Do You Know a Function Has an Inverse? The ...

The horizontal line test is just a picture. Here's the rigorous calculus theorem — tying a derivative's sign to strict monotonicity — that actually proves when a function has an...

How Do You Know a Function Has an Inverse? The ...

The horizontal line test is just a picture. Here's the rigorous calculus theorem — tying a derivative's sign to strict monotonicity — that actually proves when a function has an...

Diagram of a convex curve with a chord connecting two points on it lying above the graph between them, illustrating the definition of convexity from Spivak's Calculus

Convexity & Concavity, Precisely: What Makes a ...

The rigorous definition of a convex function—chords above the graph, tangent lines below it, and the second-derivative test—worked out through Spivak's example f(x) = 1/(1+x^2).

Convexity & Concavity, Precisely: What Makes a ...

The rigorous definition of a convex function—chords above the graph, tangent lines below it, and the second-derivative test—worked out through Spivak's example f(x) = 1/(1+x^2).

Line-art diagram illustrating the Mean Value Theorem: a curve from point a to point b with a straight secant line connecting the endpoints and a dashed tangent line parallel to it touching the curve at an interior point x

The Mean Value Theorem: Why Average Speed Guara...

How Rolle's Theorem and the Mean Value Theorem let you prove, rigorously, that an average rate of change is always achieved exactly at some instant — with the classic driving...

The Mean Value Theorem: Why Average Speed Guara...

How Rolle's Theorem and the Mean Value Theorem let you prove, rigorously, that an average rate of change is always achieved exactly at some instant — with the classic driving...