Calculus
Limits, rates, and shapes of change — earned, not delivered.
the shape of this subject
Calculus builds intuition for change. That intuition can only form by wrestling with limits, rates, and accumulation yourself. Brainback lets you struggle at the right seam, then hands you the concept name — never the final derivative.
Because everything from physics to econ to ML depends on you knowing what a derivative geometrically IS — not just how to compute one.
- discipline
- STEM
- parent subject
- Mathematics
- typical weekly load
- 3-5 problem sets · 4-6 hours of practice · 2 recall sessions on tricky bits
- covered on plan
- all plans (free forever)
Every calculus concept Brainback covers.
Add Calculus as a Learn subject and every problem in these areas routes to the Socratic tutor — attempt-first, hint-laddered, output-guarded.
What one turn in calculus actually looks like.
Attempt → correction → guiding question. No answer. Ever. Verbatim from a beta session.
Related rates: a ladder 10 ft long leans against a wall, bottom slides at 2 ft/s. dx/dt = 2, x = 6. — Attempt: 'By Pythagoras x² + y² = 100. Differentiate w.r.t. t: 2x·dx/dt + 2y·dy/dt = 0. Need y at x=6…'
You picked the right identity — good. Now: what's y when x = 6? Compute that BEFORE plugging into dy/dt.
no answer · one guiding question · your next move
How real attempts start. How Brainback nudges.
- I let u = sin(x) and got du = cos(x) dx…
- By l'Hôpital, take derivatives of top and bottom…
- Which quantities change with t? Which are constant?
- You differentiated implicitly — good. Which variable is the input?
- You have du/dt on both sides — solve for what?
the anti-pattern
What calculus looks like when the AI does the thinking.
Answer machines will happily compute the derivative for you. The professor's exam will ask you to explain what it means. Guess which one you didn't practice.
- M1Applying the chain rule to only one composition layer.
- M2Confusing dy/dx with dy/dt in related-rates problems.
- M3Forgetting +C on indefinite integrals and losing points on principle.
- M4Reading a limit form and jumping to L'Hôpital before checking it's actually indeterminate.
the recall card
What a spaced-repetition card looks like in calculus.
Every wrong attempt and low-scored explain-back becomes a card. SM-2 schedules them — the tricky ones come back tomorrow, the solid ones drift out to next week, then next month.
Given [scenario], set up the related-rates equation. Then differentiate. Show every step.
covers
Every mainstream calculus course you're likely to take.
- AP Calculus AB
- AP Calculus BC
- MATH 141 / Calc I
- MATH 142 / Calc II
- MATH 243 / Multivariable
Related subjects.
The subject where the shortcut has always been most expensive.
Where symbolic manipulation becomes second nature — or doesn't.
See the diagram. Not just look at it.
Unit circle, but also unit intuition.
Vectors are directions, not numbers.
The subject where wrong feels right — until Brainback slows you down.
calculus — ready when you are
Start your first calculus session.
Attempt-first. Never the answer. Add it as a Learn subject in 30 seconds.