Calculus: Rates and Totals
Discover how the twin engines of calculus—derivatives and integrals—make sense of a moving world. Learn how to track instantaneous change to find optimal decisions, then reverse the process to reconstruct complete totals from tiny slices.
Like Duolingo, but for Calculus: Rates and Totals. Tomo turns the whole topic into a game you play five minutes a day, until it actually sticks.
A short one: 11 levels across 2 sections, about 22 minutes end to end, roughly 4 days at five minutes a day. It moves through Tracking the Pace of Change and Rebuilding Totals from Slices. It starts from scratch.
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Key ideas in Calculus: Rates and Totals
- Average speed takes total distance over total time, hiding bursts of speeding and stops along the way
- The derivative measures the rate of change at one specific, frozen instant in time
- A car's speedometer reading at any second represents an instantaneous rate of change, which is the derivative
- Smooth curves exhibit local linearity: magnification flattens out curvature
- The slope of this tiny, straight-looking segment equals the derivative at that exact point
- This straight line approximation matches the tangent line at that point
- The derivative function itself is a formula that outputs the slope for any x-coordinate you feed into it
- Evaluating the derivative at one number yields a single slope value
- It acts as a complete map of steepness across the entire original curve
- When an amount is growing or rising, its rate of change (derivative) is positive
- When an amount is shrinking or falling, its rate of change (derivative) is negative
- As a value climbs toward a smooth peak, its rate of change is positive; after the peak, its rate of change turns negative.
- To switch from a positive slope to a negative slope, a smooth curve's derivative must cross through zero at the very top.
- At the exact peak or valley of a smooth curve, the tangent line is completely flat (slope = 0).
- Peaks and valleys on a smooth curve occur only where the derivative equals zero or where the derivative does not exist.
- Setting the derivative equal to zero turns the task of testing every possible option into solving a single algebraic equation.
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: Rates and Totals 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.
What does a derivative measure when you look at an object in motion?
Get it right to open this lesson and 10 more in the app.
Where Calculus: Rates and Totals takes you
- 1
Tracking the Pace of Change
- Zooming In to Catch Instant Speed
- Finding Peaks, Valleys, and Optimal Choices
- 2
Rebuilding Totals from Slices
- Accumulation as Area Under the Curve
- Running the Process in Reverse
2 sections · 4 units · 11 levels. Built to play, not to enroll.
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