Starting from nothing, single-variable calculus on the Consiliae map stands on 116 concepts, from counting to polar curves: about 43 hours, roughly 2 months at 5 hours a week. Every chapter you already hold comes off that total. With solid algebra the route is 52 concepts, about 24 hours. With solid precalculus it is 10 concepts, about 7 hours, and that is the calculus chapter on its own. Choose the last maths you were comfortable with to see your own figure.
Your distance
What’s the last maths you were comfortable with?
116 concepts between zero and calculus: about 43 hours, roughly 2 months at 5 hours a week.
That is the worst case: every concept on the route. The scan takes off the ones you already know.
Check my real starting pointWhich calculus do you need?
The calculus chapter on the map has 11 sections: limits, continuity, the foundations of differentiation, differentiation rules, applications of differentiation, the foundations of integration, the definite integral, integration techniques, applications of integration, sequences and series and parametric and polar calculus. Not everyone needs all 11. If you want to follow the derivatives in an economics module or a machine learning text, the first 5 are the target. If you need to integrate as well, it is the first 9. Sequences and series and parametric and polar calculus are worth checking against your syllabus before you budget for them.
| Finish line | From scratch | Solid algebra | Solid precalculus |
|---|---|---|---|
| Differentiation (sections 1 to 5) | 276 concepts, about 135 hours | 148 concepts, about 91 hours | 110 concepts, about 74 hours |
| Integration as well (sections 1 to 9) | 381 concepts, about 195 hours | 223 concepts, about 135 hours | 171 concepts, about 115 hours |
| The whole chapter (all 11 sections) | 452 concepts, about 240 hours | 292 concepts, about 180 hours | 222 concepts, about 150 hours |
Reading across a row shows how much your starting point is worth. Reading down a column shows what each extra finish line costs. Both gaps are large, which is why a single figure for “learning calculus” tells you very little until you know which row and which column you are in.
From hours to a calendar
Divide by the hours you will keep up in an ordinary week, not the hours you plan in a good one. The table below does it for the whole chapter at three common paces.
| Starting point | Hours | 3 h a week | 5 h a week | 8 h a week |
|---|---|---|---|---|
| From scratch | 43 | 3 months | 9 weeks | 5 weeks |
| Solid algebra | 24 | 8 weeks | 5 weeks | 3 weeks |
| Solid trigonometry | 8 | 3 weeks | 2 weeks | 1 week |
| Solid precalculus | 7 | 2 weeks | 1 week | 1 week |
These figures assume the weeks run on without a break. A month off in the middle does not simply pause the clock: some of what came before has to be relearned when you return, and those hours appear nowhere in the table. The research on spacing points the same way. Cepeda and colleagues (2008) found that the best gap between study sessions grows with how long you need to keep the material, which on a route lasting many months suggests planning for steady weekly sessions with scheduled review, and pricing any planned break into the estimate before you start.
Where the hours go
From scratch, 106 of the concepts on the route sit before the calculus chapter and 10 inside it. Measured in hours the split is about 37 hours before and about 7 hours inside, because calculus concepts are rated harder and given longer each. Anyone starting from solid precalculus still has that whole calculus block ahead of them, and it is the larger share of the hours from scratch as well.
What the hour totals hide is how deep each calculus idea reaches back. The chain rule, for example, stands on 101 earlier concepts on the map, and 87 of them come before the calculus chapter begins: 38 from arithmetic, 10 from pre-algebra, 11 from algebra, 13 from geometry, 7 from trigonometry and 8 from precalculus. A weak link anywhere in that chain shows up as a mistake in a derivative, even though the faulty step is algebra or trigonometry.
Following the direct links on the map shows where earlier topics enter. Sum formula for sine sits directly under derivative of sin x. Sum formula for cosine sits directly under derivative of cos x. Reciprocal trig functions: csc, sec, cot sits directly under derivative of cot x, derivative of csc x, derivative of inverse trig: arccsc and 2 more. Pythagorean identity (sin^2 + cos^2 = 1) sits directly under derivative of inverse trig: arccos, derivative of inverse trig: arcsin, derivatives of inverse hyperbolic functions and 2 more. Pythagorean identity (1 + tan^2 = sec^2) sits directly under trig substitution: √a^2 + x^2, trig substitution: √x^2 - a^2 and trigonometric integrals: powers of secant and tangent. Composition of functions (algebraic) sits directly under chain rule, continuity of composite functions and limit of a composite function. Inverse function: definition sits directly under derivative of an inverse function (inverse function theorem), derivative of inverse trig: arccos, derivative of inverse trig: arccot and 5 more.
In all, 10 concepts from the earlier chapters are cited directly by a calculus concept. The most cited are “Series: definition” (cited by 2), “Area of a rectangle” (cited by 2), “Evaluating a function at a point” (cited by 2) and “Sector area formula” (cited by 1). Those are the topics to be sure of before you start.
How much of each earlier chapter calculus uses
You do not need every topic from school maths before calculus. Some chapters feed it heavily and others barely at all. These are the concepts from each earlier chapter that sit somewhere on the route, against the chapter’s full size on the map:
- Arithmetic: 44 of 155
- Pre-algebra: 7 of 92
- Algebra: 13 of 202
- Geometry: 27 of 269
- Trigonometry: 13 of 102
- Precalculus: 2 of 79
Geometry is the clearest case. Calculus draws on its foundations, lines, triangles, pythagorean theorem, quadrilaterals and polygons, circles, mensuration and coordinate geometry sections, and on nothing from congruence, similarity or geometric proof. The precalculus that matters is sequences, series, combinatorics and logic and proof; complex numbers is not on the route at all. If you are rebuilding from a long way back, starting maths again as an adult covers the earlier chapters in more depth.
Doing calculus or using it
The table above measures topics covered, and two people can cover the same topics with different aims. The first aim is procedural: given a function, differentiate or integrate it correctly, recognise which technique a problem calls for, and pass a standard exam. The second is modelling: read a situation described in words, see that the question is really about a rate of change or an accumulated total, and set up the derivative or integral yourself. Exams mostly test the first, while physics, engineering, economics and machine learning lean on the second. If you need a qualification, a placement test or a prerequisite ticked off, the procedural aim is enough. If calculus is a tool for something else, budget extra time after the topics are covered for working problems set in words, since the hour estimates here stop at the topics.
Working to a fixed date
Work backwards from the date and be prepared to cut. Some of the route can wait for a first pass without harming your ability to take a derivative or evaluate an integral: the formal epsilon-delta definition of a limit, the polar coordinates material in trigonometry (on the map it feeds only the final calculus section, parametric and polar calculus), and most of a geometry course. Some of it cannot be cut: fractions and exponents, solving equations, function notation, composition and inverses, the unit circle and the trigonometric identities. When the date forces a choice, cut from the first group and leave the second alone. On the map, nothing in calculus depends on the epsilon-delta definition, and the polar material feeds one section.
How these figures are worked out
The route is every concept the calculus chapter depends on, followed back through the prerequisite links until nothing is left, so a topic appears only if some calculus concept actually needs it. Each concept carries a difficulty level, and each level is given a fixed time budget covering the lesson, graded practice and a first review: L1 15 minutes, L2 20 minutes, L3 30 minutes, L4 40 minutes and L5 50 minutes. Those budgets are our own judgement, not an average measured from learners, and they do not include relearning anything forgotten after a long gap. The starting points in the picker assume every chapter before yours is fully solid. If you learnt the material years ago, a chapter can survive mostly intact with one link inside it gone.
