First-year calculus: what changes after high school, the 26 ideas in order, and how to study it week by week

First-year calculus covers what Grade 12 calculus covered and then about twice as much, at twice the pace: limits with their formal definition, derivatives and their rules, the Mean Value Theorem and curve sketching, L'Hôpital's rule, optimization and related rates, then antiderivatives, Riemann sums, the Fundamental Theorem, substitution, parts, areas, volumes and improper integrals. What changes is not the ideas but the speed, the proofs and the expectation that you practise outside class. Here is the map and the plan.

What changes after high school (and why calculus 1 feels hard)

Four things change at once, and the shock is that they change together.

The pace. A Grade 12 calculus course spends two or three weeks on the derivative rules. A first-year course does them in one, then moves to implicit differentiation and the derivatives of exponentials, logarithms and trigonometric functions the week after. The ideas you met over a semester in high school come back in six weeks, and the second half of the course is ideas you have never seen.

The proofs. In Grade 12 a limit was a number you found. In first year a limit has a formal definition, with ε and δ, and you can be asked to use it. The Mean Value Theorem is stated as a theorem with hypotheses, and an exam question can ask you to check that the hypotheses hold before you apply it. Proofs are examinable. Nobody tells you this on day one.

The textbook is the course. Lectures assume you have read the section before you arrive. A lecture on substitution that you walk into cold is an hour of watching someone else do algebra. The same lecture after twenty minutes with the section the night before is an hour of seeing what you half understood confirmed and corrected.

Nobody collects the homework. The problem sets are suggested, not marked, and the exam assumes you did them. The students who struggle in first-year calculus are, in our experience, rarely short of ability. They are short of a practice habit that matches the pace, because high school never asked them to build one. Office hours exist from week one and almost nobody uses them in week one.

If the first six weeks feel like review, good: the twelve ideas calculus assumes are in the precalculus guide, and if any of them are shaky, that is where to spend the first weekend.

The course map: 26 ideas in the order they are taught

These are our own titles for the ideas, in the order a one-semester first-year course teaches them. Five groups; the numbers are the order.

Limits and continuity

1. Limits of functions and limit laws
2. The formal (ε-δ) definition of the limit
3. Continuity and the Intermediate Value Theorem

The derivative

4. The derivative as a limit
5. Power, product and quotient rules
6. The chain rule
7. Implicit differentiation
8. Derivatives of exponential and logarithmic functions
9. Derivatives of trigonometric functions
10. Hyperbolic functions and their derivatives

Applications of the derivative

11. Rolle's theorem and the Mean Value Theorem
12. Critical points and extrema
13. Concavity and points of inflection
14. Limits at infinity, asymptotes and curve sketching
15. Indeterminate forms and L'Hôpital's rule
16. Optimization
17. Related rates
18. Linear approximation and Newton's method

The integral

19. Antiderivatives and the indefinite integral
20. Riemann sums and the definite integral
21. The Fundamental Theorem of Calculus
22. Integration by substitution
23. Integration by parts

Applications of the integral

24. Area between curves
25. Volumes of solids of revolution
26. Improper integrals

Write the 26 lines on one page and keep it at the front of the binder. Mark each idea with one of three words as the term goes on: can do, shaky, cannot. That page is the course on one screen, and chapter 3 of the free playbook is the method for keeping it honest.

Where Grade 12 and AP leave you

Your own course order will differ a little from the list above, but the overlap with high school is close to the same everywhere. Ontario's MCV4U covers limits informally, the derivative from first principles, every derivative rule, the derivatives of exponential, logarithmic and trigonometric functions, extrema, concavity, curve sketching, optimization and related rates; the rest of that course is vectors, which first-year calculus does not use. What MCV4U covers has its own map. AP Calculus AB, by our reading of the College Board's published unit list, reaches further: it includes the Mean Value Theorem, L'Hôpital's rule, linear approximation, Riemann sums, the Fundamental Theorem, substitution, areas and volumes. AP Calculus AB and BC has the detail.

Idea group	MCV4U	AP Calculus AB	New to everyone
Limits and continuity (1–3)	Limits informally, continuity	Limits, continuity, the Intermediate Value Theorem	The ε-δ definition
The derivative (4–10)	4 to 9	4 to 9	Hyperbolic functions
Applications of the derivative (11–18)	12, 13, 14, 16, 17	11 to 17 and linear approximation	The MVT as a theorem (MCV4U), Newton's method
The integral (19–23)	None	19 to 22	Integration by parts; the whole half, for MCV4U
Applications of the integral (24–26)	None	24, 25	Improper integrals

Two readings of the table. If you come from MCV4U, the first six weeks are review at speed and the second half of the term is new, so your practice habit has to be in place by week six, not built then. If you come from AP AB, almost everything is review except ε-δ, hyperbolics, Newton's method, parts and improper integrals, and the danger is the opposite one: a term of coasting, then a final that asks for proofs you never practised. Whether AP credit lets you skip the course is your program's rule, and programs differ.

The proofs: what "examinable" means

Two proofs come up in almost every first-year course, and "examinable" means something specific for each.

The ε-δ definition. You will be asked to state the definition, and you will be asked to prove a limit of a simple function, usually a linear one, from the definition. Here is the whole of one such question, worked.

Prove that the limit of 3x − 1 as x approaches 2 is 5.

The definition says: for every ε > 0 there is a δ > 0 such that whenever 0 < |x − 2| < δ, we have |(3x − 1) − 5| < ε. So the job is to find δ in terms of ε.

Start from the thing you want small: |(3x − 1) − 5| = |3x − 6| = 3|x − 2|. That is 3 times the thing you control. So if |x − 2| < δ, then |(3x − 1) − 5| < 3δ. To make 3δ equal to ε, choose δ = ε/3.

The proof, written out: let ε > 0 and choose δ = ε/3. If 0 < |x − 2| < δ, then |(3x − 1) − 5| = 3|x − 2| < 3δ = ε. Done. Check it: with ε = 0.3, δ = 0.1; take x = 2.05, so |x − 2| = 0.05 < 0.1, and 3(2.05) − 1 = 5.15, which is within 0.15 of 5, inside the 0.3 asked for.

The pattern to learn is the pattern, not the numbers: rewrite the output gap as a multiple of the input gap, then choose δ to cancel the multiple. For a linear function the multiple is the slope.

The Mean Value Theorem. You will be asked to state it exactly: if f is continuous on [a, b] and differentiable on (a, b), there is a c in (a, b) with f′(c) = (f(b) − f(a))/(b − a). Then you will be asked to use it, and the marks are for checking the hypotheses before you do. A function that is not differentiable somewhere inside the interval, |x| on [−1, 1] for instance, is the standard trap: the conclusion fails, and the question wants you to say which hypothesis failed.

How to study a proof. Not by reading it. Read it once, close the book, and rebuild it from the statement alone on a blank page. Where you get stuck is the step you did not understand; open the book to that step only, then close it and start again from the top. Chi and colleagues (1994) had students explain each line of a text to themselves as they read, and the students who did so understood it better than students who read it twice; a proof rebuilt from its statement is that self-explanation with the book closed, and teaching it out loud is the same method spoken. Put the statement of each theorem on a recall card, because stating it exactly is marks on its own.

Worked examples: three ideas that are new

Three of the ideas that were not in Grade 12, each done fully so you can check your own version against it.

L'Hôpital's rule (idea 15). Find the limit of (eˣ − 1)/x as x approaches 0.

Substituting x = 0 gives (1 − 1)/0 = 0/0, an indeterminate form, so the rule applies. Differentiate the top and the bottom separately (not as a quotient): the top becomes eˣ, the bottom becomes 1. The limit of eˣ/1 as x approaches 0 is e⁰ = 1. So the limit is 1. Check with a number: at x = 0.01, (e^0.01 − 1)/0.01 is about 1.005, close to 1 and getting closer as x shrinks. The common wrong line is applying the rule to a limit that is not 0/0 or ∞/∞; always write the indeterminate form down first.

The Fundamental Theorem of Calculus (idea 21). Evaluate the integral of 3x² from 0 to 2.

An antiderivative of 3x² is x³ (differentiate x³ and you get 3x² back). The theorem says the definite integral is the antiderivative at the top limit minus the antiderivative at the bottom: 2³ − 0³ = 8 − 0 = 8. Check it against the Riemann picture: the area under 3x² from 0 to 2 sits under a curve that rises from 0 to 12, so a value of 8 over a width of 2 (an average height of 4) is plausible.

Integration by parts (idea 23). Find the integral of x eˣ dx.

Parts says the integral of u dv equals uv minus the integral of v du. Choose u = x (it gets simpler when differentiated) and dv = eˣ dx. Then du = dx and v = eˣ. So the integral is x eˣ − ∫ eˣ dx = x eˣ − eˣ + C.

Check by differentiating the answer: the derivative of x eˣ is eˣ + x eˣ by the product rule, the derivative of −eˣ is −eˣ, and the sum is x eˣ, which is what we started with. Every integration answer can be checked this way, and the check is part of the answer on an exam. The common wrong line is choosing u = eˣ, which makes the new integral harder, not easier.

How to study first-year calculus

Every one of the 26 ideas is either a rule, a technique or a theorem, and each kind has a method.

For every rule and technique: the procedure cycle. Read one worked example from the section slowly, line by line, saying why each line follows. Close it and redo the same problem from the question alone. Then do a sibling problem with different numbers from the exercise set. Sweller and Cooper (1985) found that students who studied worked examples before attempting problems learned the procedure with less effort and fewer errors than students who only solved problems, and the cold redo turns the example into something you can produce. How to study math has the full cycle, including how to fade the example away.

The error log. Every wrong line, with the right line beside it and a name for the mistake: "chain rule, forgot the inside derivative"; "parts, chose u badly". Read the log once before each problem set. By week six it is a list of your own habits, which no textbook can give you.

Mixed sets from week three. The exercise set gives you twenty chain-rule problems under a heading that says "chain rule". The exam gives you a derivative with no heading. Rohrer and Taylor (2007) had students practise the same problems either blocked by type or shuffled; the blocked group did better in practice and worse on a test a week later. From week three, build your own set of eight problems drawn from every idea so far, shuffled, and do it with the notes closed. Why mixing problem types works has the how.

Proofs rebuilt from memory, as above: statement on a card, proof rebuilt on a blank page, the stuck step found and fixed.

The textbook before lecture. Twenty minutes the night before: survey the section's headings and boxed theorems, turn each heading into a question, read for the answers. You arrive with the questions, and the lecture answers them. You are not trying to learn the section the night before; you are trying to make the lecture land.

The office-hours habit. Go in week two, before anything is wrong, with the error log open. "Here are the four wrong lines from this week's set, and I think three are the same mistake" is a conversation an instructor can have in ten minutes. "I don't get integration" is not. The students who use office hours from week two are the ones who are not in them in a panic in week eleven.

Spacing. Cepeda and colleagues (2006) pulled together hundreds of comparisons and found that the same study time spread over separate days produced better recall than the same time in one block, at every delay they measured. In a 13-week course this is not a technique, it is the structure: every idea returns in the mixed set the week after it is taught, and again in the midterm review.

One sentence on solver apps: they will give you the answer to a problem, and an answer you did not produce is not an attempt, so it teaches nothing you can reproduce on the exam. The guide to what a solver can and cannot do for you, in the AI section of this blog, is the honest version.

The weekly plan that survives midterms

Thirteen weeks. Each row is three things: the reading before lecture, the problem set, and the recall task that closes the week. The mixed set starts in week four, once there are enough ideas to shuffle, and it includes every idea taught so far, not only this week's.

Week	Ideas	Reading before lecture	Problem set	Recall task
1	1	Limit laws; the factoring and rationalising tricks	Ten limits, including two that need factoring	Blank page: the limit laws from memory
2	2, 3	The ε-δ definition; continuity and the IVT	Two ε-δ proofs for linear functions; four continuity questions	Rebuild the ε-δ proof of a linear limit
3	4, 5, 6	The derivative as a limit; the rules	One first-principles derivative; twelve rule problems	State each rule with one example of your own
4	7, 8, 9	Implicit; exponential, log and trig derivatives	Eight problems, then the first mixed set of eight from ideas 4 to 9	Error log read once; the mixed set's misses redone cold
5	10, 11	Hyperbolics; Rolle's theorem and the MVT	Four hyperbolic derivatives; three MVT applications with hypotheses checked	Rebuild the MVT's statement and one application from memory
6	12, 13, 14	Extrema, concavity, curve sketching	Three full curve sketches; a mixed set of eight from ideas 4 to 14	Sketch one curve from its equation alone, then compare
7	Midterm	None	A timed past set under exam conditions, two days out	The one-week plan, compressed: diagnostic, spaced sessions, light review
8	15, 16, 17	L'Hôpital; optimization; related rates	Six limits with the form written first; three optimization and three related-rates problems	Redo the midterm's missed questions cold
9	18, 19	Linear approximation and Newton's method; antiderivatives	Two Newton iterations by hand; twelve antiderivatives, each checked by differentiating	Blank page: the antiderivative table from memory
10	20, 21	Riemann sums; the FTC	One Riemann sum computed by hand; eight definite integrals by the FTC	State the FTC exactly; explain in a sentence what it connects
11	22, 23	Substitution; parts	Eight substitutions, six by parts, every answer checked by differentiating; a mixed set of eight from ideas 19 to 23	Error log read once; the misses redone
12	24, 25, 26	Areas; volumes; improper integrals	Three areas, three volumes, four improper integrals	Sketch the region for each area and volume problem before integrating
13	Review	None	Two mixed sets of twelve, across all 26 ideas, timed	The one-page map marked can do, shaky, cannot; the shaky list is the final week's plan

Two rules keep the plan alive. First, a missed week moves forward and never resets: if week five went wrong, week six's reading is week six's, and week five's problem set is added to the next mixed set rather than done in full. Second, the grid lives on paper next to the course map. A study plan you will keep is the weekly grid that holds this table and the rest of your week together.

The course on one screen

• Five groups: limits and continuity (1 to 3), the derivative (4 to 10), its applications (11 to 18), the integral (19 to 23), its applications (24 to 26).
• New to everyone: the ε-δ definition, hyperbolic functions, the MVT as a theorem, Newton's method, integration by parts, improper integrals. New to MCV4U students: the whole integral half.
• The proofs are examinable: state the definition or theorem exactly, check the hypotheses, rebuild the proof with the book closed.
• Every technique: worked example, redo cold, sibling, error log. Mixed sets from week four.
• The textbook before lecture, twenty minutes.
• Office hours in week two, with the error log, before anything is wrong.