Calculus and Trigonometry: Motion and Change
Understand how the mathematics of angles, waves, and slopes work together to describe a continuously changing world. Learn to calculate instantaneous speeds, trace periodic motion, and accumulate curves to solve practical problems without getting lost in abstract notation.
Like Duolingo, but for Calculus and Trigonometry: Motion and Change. 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 Seeing Curves and Circles in Motion, Calculating Exact Rates, Adding Up Slices That Never Stay Still, and Connecting Foundations to the Real World. It starts from scratch.
Free forever · No credit card · iPhone & Android
Key ideas in Calculus and Trigonometry: Motion and Change
- Magnifying a tiny section of any smooth curve flattens its visible bend
- Calculus treats curves as locally straight lines at microscopic scales to measure instantaneous behavior
- Average speed connects two separate points on a distance-time graph across an interval
- Speedometer readings show instantaneous speed, which matches the slope of the curve at that single exact point
- Calculus uses single-point slopes to describe what is happening in the exact present
- The sign of a tangent line's slope tells you direction: positive slopes rise, negative slopes fall
- The steepness of the tangent line tells you pace: steeper slopes mean faster change
- A zero slope means the quantity is level and not changing at that split second
- A peak or crest on a curve flattens out at the very top, giving a slope of zero
- A valley or turnaround point also levels out, yielding a slope of zero before reversing
- Points of ascent or descent always have non-zero slopes
- An original function outputs the position or height for any given point
- The derivative function takes the same input point and outputs the slope of the original graph at that location
- A derivative provides an automatic formula for slopes without needing to calculate tangents by hand each time
- Trigonometric angles measure continuous turning or spinning, not just the fixed corners inside a triangle
- A full rotation is 360°, and multiple spins accumulate past 360° (e.g., 3 turns plus 90° is 1,080° + 90° = 1,170°)
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 and Trigonometry: Motion and Change 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.
Why does calculus treat curves as straight lines at microscopic scales?
Get it right to open this lesson and 23 more in the app.
How to learn Calculus and Trigonometry: Motion and Change, step by step
Go in order. 24 short levels across 4 sections, about 48 minutes in all, five minutes a day.
- 1
Seeing Curves and Circles in Motion
- Zooming In on Instant Change
- Angles as Continuous Rotation
- 2
Calculating Exact Rates
- Rules for Differentiating Curves
- The Slopes of Oscillating Waves
- 3
Adding Up Slices That Never Stay Still
- Accumulation and the Area Under a Curve
- Integrating Waveforms and Shifting Shapes
- 4
Connecting Foundations to the Real World
- The Master Link and Approximation
- Optimization and Real-World Modeling
4 sections · 8 units · 24 levels. Built to play, not to enroll.
Questions about Calculus and Trigonometry: Motion and Change
How do I learn Calculus and Trigonometry: Motion and Change?
Go in order: start with Seeing Curves and Circles in Motion, then Calculating Exact Rates, then Adding Up Slices That Never Stay Still, then Connecting Foundations to the Real World. 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 and Trigonometry: Motion and Change?
Yes. Tomo works like Duolingo: short daily lessons, streaks and quick quizzes. Calculus and Trigonometry: Motion and Change is one of its free courses, and you can type any other topic to get a course of your own.
Is the Calculus and Trigonometry: Motion and Change course free?
Yes. Calculus and Trigonometry: Motion and Change 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 and Trigonometry: Motion and Change?
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 and Trigonometry: Motion and Change?
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 and Trigonometry: Motion and Change is taught in the Professor style: clear, structured, thorough. 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 and Trigonometry: Motion and Change today.
Download Tomo, search Calculus and Trigonometry: Motion and Change, and play your first lesson in under a minute.