Calculus: Motion, Change, and Proof
Discover how mathematics models a dynamic world through the twin concepts of rate and accumulation. Move from the intuitive mechanics of curves and optimization to rigorous theorems and foundational proofs.
Like Duolingo, but for Calculus: Motion, Change, and Proof. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
24 levels across 4 sections, about 48 minutes end to end, roughly 10 days at five minutes a day. It moves through Capturing Instantaneous Change, Analyzing Function Behavior and Extremes, Accumulation and the Integral Bridge, and The Rigorous Analytic Foundations. It starts from scratch.
Free forever · No credit card · iPhone & Android
Key ideas in Calculus: Motion, Change, and Proof
- Average speed only compares starting and ending points over a span of time
- A driver can speed up or slow down drastically inside an interval while maintaining the same overall average
- Instantaneous speed describes motion at a single exact moment, which average speed averages out
- A secant line passes through two distinct points on a curve
- Shrinking the gap between the two points tilts the secant line toward a single limiting line
- A tangent line touches a smooth curve locally at exactly one point, matching the curve's direction right there
- The difference quotient calculates the slope between x and x + h
- Plugging in h = 0 immediately creates 0/0, which is undefined
- Algebraic simplification (like expanding or factoring) cancels out the h in the denominator before evaluating
- The numerator f(x + h) - f(x) measures vertical change (rise)
- Dividing by h turns the total change into an average rate of change (slope)
- Taking the limit as h approaches 0 turns the average rate into an instantaneous rate (the derivative)
- The original function f(x) gives the vertical position or value at x
- The derivative function f'(x) takes the same x and outputs the steepness and direction (slope) at that exact point
- A negative derivative value means the graph is sloping downwards from left to right
- The power rule states that the derivative of x^n is n * x^(n-1)
You've tried the other tabs
Thirty open tabs. Four facts you actually kept.
You watched. You nodded. By Sunday it was gone.
One answer, then back to scrolling.
Eight weeks. You meant to finish. You didn't.
Tomo gives Calculus: Motion, Change, and Proof the Duolingo treatment: levels, streaks, and quick quizzes that test what you just learned. That game loop is what the tabs above never had, so it's the one you actually finish.
Here's what playing it feels like
A real question from this course. Take your best guess.
You finish a 60-mile drive in one hour, but zoomed at 75 miles per hour for a few minutes. How is this possible?
Get it right to open this lesson and 23 more in the app.
How to learn Calculus: Motion, Change, and Proof, step by step
Go in order. 24 short levels across 4 sections, about 48 minutes in all, five minutes a day.
- 1
Capturing Instantaneous Change
- Finding Instantaneous Speed and Slope
- Differentiating Functions Systematically
- 2
Analyzing Function Behavior and Extremes
- Optimizing Outcomes with Critical Points
- The Mean Value Theorem and Monotonicity
- 3
Accumulation and the Integral Bridge
- Measuring Area via Riemann Sums
- Unifying Rates and Areas with the Fundamental Theorem
- 4
The Rigorous Analytic Foundations
- The Epsilon-Delta Limit and Formal Continuity
- Topological Foundations and Existence Theorems
4 sections · 8 units · 24 levels. Built to play, not to enroll.
Questions about Calculus: Motion, Change, and Proof
How do I learn Calculus: Motion, Change, and Proof?
Go in order: start with Capturing Instantaneous Change, then Analyzing Function Behavior and Extremes, then Accumulation and the Integral Bridge, then The Rigorous Analytic Foundations. On Tomo that's 24 short levels, about 48 minutes in all, played five minutes a day. Start free and play the first level in under a minute.
Is there a Duolingo-style app for Calculus: Motion, Change, and Proof?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Calculus: Motion, Change, and Proof is one of its free courses, and you can type any other topic to get a course of your own.
Is the Calculus: Motion, Change, and Proof course free?
Yes. Calculus: Motion, Change, and Proof is playable on Tomo's free plan, which doesn't expire. No credit card, no waitlist. Download the app and start your first lesson.
How long does it take to learn Calculus: Motion, Change, and Proof?
24 levels across 4 sections, about 48 minutes of play in total. It's built for five minutes a day, so you finish it a little at a time instead of all at once.
Do I need any background to start Calculus: Motion, Change, and Proof?
No. It's pitched at a beginner level, so it starts where you are and builds up. If you can tap a phone, you can start.
Tomo is in ChatGPT
Tomo is also a ChatGPT plugin: the same playable courses, built and played inside your chat.
Add Tomo to ChatGPT- What to ask
- “Make me a Tomo course on black holes.” “Turn this chat into a Tomo course.”
- What you get
- Quick questions with instant feedback, hearts and stars, ready to play in about a minute.
- Where it goes
- No sign-up to start. Connect your Tomo account and your streak carries on in the app.
You pick the voice
Calculus: Motion, Change, and Proof is taught in the Explain Like I'm 5 style: no big words. promise.. Want a different feel? In the app you can spin up the same topic in any of Tomo's teaching styles. Same facts, totally different vibe.
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Start Calculus: Motion, Change, and Proof today.
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