Jugami Calculus

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Calculus

The mathematics of change, one tiny step at a time.

This is a free calculus course for high school students in grades 11–12, and for anyone getting ready for a first college calculus class. Each lesson explains one big idea with graphs you can play with, fully worked examples, and practice problems with step-by-step answers you can reveal.

Start with LimitsSee the roadmap
A downward parabola with a point P at x = 1. Four secant lines from P to points at x = 5, 4, 3 and 2 have slopes 0, one half, 1 and one and a half, and close in on the dashed tangent line at P, whose slope is 2.Psecant slopes 0, ½, 1, 1½ …… close in on the tangent slope 2
Secant lines through P tilt toward the tangent line. Calculus is built on that “closing in.”

IntroductionWhat is calculus?

Glance at a car’s speedometer and it says 50 miles per hour, right now. But speed is distance divided by time, and in a single instant the car covers no distance in no time. That’s 0 ÷ 0, which tells you nothing. So what does the speedometer mean?

Calculus answers questions like that. Its trick is to zoom in: measure the speed over a tenth of a second, then a hundredth, then a thousandth, and watch which number the answers close in on. That “closing in” number is called a limit, and every other idea in calculus is built on it.

From there calculus has two big tools. Derivatives measure how fast something is changing at one instant, like the slope of a curve at a single point. Integrals add up infinitely many tiny pieces to find a total, like the area under a curve or the distance traveled from a changing speed. The Fundamental Theorem of Calculus shows that these two tools undo each other.

Isaac Newton in England and Gottfried Wilhelm Leibniz in Germany each worked out these ideas, independently, in the second half of the 1600s. Today engineers, doctors, economists, game designers and climate scientists use them every day.

The planRoadmap

Like Newton’s Principia, this course is arranged in books, and each one depends on the one before. Limits is live now. The other books are being written and will appear here when they’re ready.

  1. Live Book I

    Limits

    What a function is heading toward, even when it never gets there. Holes, jumps, one-sided limits and limits at infinity, with an interactive explorer, worked examples, practice and a quiz.

  2. Coming soon Book II

    Derivatives

    The slope of a curve at a single point, built from limits of secant slopes. Then the rules that make derivatives quick to find.

  3. Coming soon Book III

    Applications of Derivatives

    Using derivatives to find the highest and lowest points, describe motion, and solve related-rates problems.

  4. Coming soon Book IV

    Integrals

    Adding up thin rectangles to find areas and totals, and what happens as the rectangles get thinner.

  5. Coming soon Book V

    The Fundamental Theorem

    Why the slope problem and the area problem turn out to be two sides of the same coin.

  6. Coming soon Book VI

    Newton and Leibniz

    The two inventors of calculus, their rival notations, and the famous priority dispute.

Before you startWhat you need to know first

Calculus leans hard on algebra. You’ll be in good shape if you can:

  • simplify expressions, factor quadratics like x² − x − 6 = (x − 3)(x + 2), and cancel common factors in fractions;
  • read and sketch graphs of functions, and use function notation such as f(3);
  • find the slope of a line from two points;
  • work with sine and cosine in radians (needed for just one example in Limits).

Rusty? The Jugami Algebra lessons on equations and inequalities are a good warm-up.

ColophonAbout these lessons

Every worked example, practice answer and quiz answer is checked twice: once by algebra and once numerically, by plugging in values of x closer and closer to the target. The interactive graphs are drawn live from the real functions, so what you see is what the math says. Quiz best scores are saved only in your own browser.

Credits: all graphs, icons and the share image are original SVG drawn in code for Jugami. The layout borrows the look of an old printed math book, a nod to Newton’s Principia (1687). Written by Isaac, Jugami’s calculus teacher.

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