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A course in change and accumulation

Calculus I.
Make change visible.

How do we measure an instant, improve a design, or add up a changing flow? Start with a moving object. Discover the mathematics underneath.

Before the equations,
a question to play with.

Move along the path. Shrink the time interval. When does average velocity become instantaneous?

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FIG. I / ZOOM INTO AN INSTANT
Position (m)Time (s) · 0 to 5025 m

s(t)=(1 m/s2)t2s(t)=(1\,\mathrm{m/s^2})t^2
Red curve: position · dashed blue: secant · brown: tangent

vavg=2t+h=5.00 m/s,v(t)=2t=4.00 m/sv_{\mathrm{avg}}=2t+h=5.00\,\mathrm{m/s},\qquad v(t)=2t=4.00\,\mathrm{m/s}

The gap is 1.00 m/s. Shrink the interval to bring the secant closer to the tangent.

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Why this course matters

The world changes.
Learn to measure how.

Tracking a vehicle. Designing a container. Measuring how much liquid has flowed. Calculus gives these everyday questions a common language: rates and accumulation.

You’ll move between a physical situation, its graph, and its equation—learning both how to calculate an answer and why it makes sense.

What we’ll investigate

Three ways to understand change.

Choose a question

A moving object. A rate at one instant.

A car’s average speed over a journey cannot tell you its speed right now. Shrink the time interval and a secant approaches a tangent: the derivative connects a changing position to instantaneous velocity.

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Your learning path · 35 lessons

A little structure.
A world to discover.

Start with limits, discover derivatives, then connect changing rates to integrals and physical applications.

Each lesson pairs visual exploration with mathematical derivations and three practice questions. Start at the beginning or revisit the idea you need.

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Continue to Calculus II ↗

IFoundations and limits+

Translate graphs and measurements into limits and continuity.

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IIDerivatives and connected rates+

Connect derivatives to motion and rates in linked systems.

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IIIUsing derivatives+

Use local change to analyze curves and improve designs.

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IVAccumulation and integration+

Build integrals from small contributions and connect them to rates.

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VGeometry and work+

Apply accumulation to area, volume, and mechanical work.

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