"Learn Calculus 1" is searched by students starting the course at college, by students taking AP Calculus AB, by self-learners who want the subject for data science, engineering or interest, and by parents of a 17-year-old who has just hit the first topic that did not yield to memory. The course is more learnable than its reputation: it has two ideas, the derivative and the integral, a short list of rules, and a set of applications that repeat. What makes it hard is the algebra inside it and the habit of learning the rules without the ideas. This page gives the topics in order, the prerequisites with a check, the resources that are genuinely good and free, a ten-week plan, how to study each topic, the five places students fail, and how the course maps to the school exams. The printable is the plan as a checklist.
Download the Calculus 1 checklist and ten-week plan (PDF) (the ten topics with their skills as tick boxes, the prerequisite check with answers, the plan week by week, the resources, and the five failure points; two pages, free).
What Calculus 1 covers, in order
| Topic | What it is | What it assumes |
|---|---|---|
| 1. Limits and continuity | What a function approaches; one-sided limits; limits at infinity; the limit laws; continuity and the intermediate value theorem | Algebra: factoring, rational expressions; function graphs |
| 2. The derivative | The derivative as the limit of the difference quotient; as the slope of the tangent; as an instantaneous rate; differentiability | Topic 1; slope |
| 3. The rules | Power, constant multiple, sum, product and quotient rules; derivatives of eˣ, ln x, sin x, cos x, tan x, and the inverse trigonometric functions | Topic 2; the trigonometric values |
| 4. The chain rule | Derivatives of composite functions; combined with every rule above | Function composition |
| 5. Implicit differentiation and related rates | Differentiating equations that are not solved for y; rates of change of connected quantities (the ladder, the cone, the shadow) | Topic 4; setting up equations from a picture |
| 6. Curve sketching | Increasing and decreasing; critical points; the first and second derivative tests; concavity and inflection; the mean value theorem; L'Hôpital's rule in many courses | Topics 3 and 4; solving f′(x) = 0 |
| 7. Optimisation | Maximum and minimum problems in context (the largest box, the cheapest fence, the shortest path) | Topic 6; writing a quantity as a function of one variable |
| 8. Antiderivatives | Reversing differentiation; the + C; initial-value problems; basic rules | Topic 3 read backwards |
| 9. The definite integral | Riemann sums; the integral as area and as accumulated change; properties | Topic 8; summation |
| 10. The fundamental theorem | F(b) − F(a); accumulation functions; substitution; the first areas | Topics 8 and 9 |
A university Calculus 1 usually ends at topic 10; AP Calculus AB continues into areas between curves, volumes and separable differential equations; integrals in full are in integrals in calculus explained.
The prerequisite check
Ten questions, no calculator, fifteen minutes. These are the skills topics 1 to 7 use on every page.
- Factor x² − 5x + 6 and x³ − 8.
- Simplify (x² − 9)/(x − 3) for x ≠ 3.
- Write 1/√x and x²·x⁻⁵ as single powers of x.
- Expand (x + h)² − x², then divide by h.
- Solve x² − 3x − 10 = 0 and 2x + 1 > 7.
- If f(x) = 3x + 2 and g(x) = x², find f(g(x)) and g(f(x)).
- Give sin(π/6), cos(π/3), tan(π/4) and sin(π/2).
- Simplify ln(e³) and e^(ln 5); write log₂ 8 as a number.
- Find the equation of the line through (1, 4) with slope −2.
- Sketch y = x² − 4 and mark where it crosses the axes.
Answers: 1) (x − 2)(x − 3); (x − 2)(x² + 2x + 4). 2) x + 3. 3) x^(−1/2); x⁻³. 4) 2xh + h², then 2x + h. 5) x = 5 or −2; x > 3. 6) 3x² + 2; (3x + 2)². 7) ½, ½, 1, 1. 8) 3; 5; 3. 9) y = −2x + 6. 10) Crosses the x-axis at ±2, the y-axis at −4.
Nine or ten: start the plan. Six to eight: a fortnight on the missed ones first. Five or fewer: a month of algebra and trigonometry before calculus will save a semester; the plan in how to learn calculus on your own includes that month.
The resources that are actually good, and free
| Resource | What it is best for |
|---|---|
| OpenStax Calculus Volume 1 | A complete free textbook with worked examples, exercises and answers to the odd-numbered ones; the backbone of the plan |
| Paul's Online Math Notes (Lamar University) | The clearest written explanations of every topic, with practice problems and full solutions; the first place to go when a textbook paragraph does not land |
| Khan Academy, AP Calculus AB and Calculus 1 | Short videos and graded practice per skill; best for drilling the rules and the chain rule |
| MIT OpenCourseWare 18.01 Single Variable Calculus | Full lectures, problem sets and exams with solutions; best for the ideas, and for exam practice |
| 3Blue1Brown, Essence of Calculus | Twelve short videos on why calculus works; watch one before each new topic |
| Desmos | Graph the function and its derivative side by side; the fastest way to see what a derivative is |
| Past AP Calculus AB free-response questions (College Board) | The applications (related rates, optimisation, accumulation) in exam form, with scoring guidelines |
The ten-week plan
An hour a day, five days a week. Each week: read the topic (OpenStax or Paul's Notes), study three worked examples covering the steps, do twenty problems with the answers hidden until done, and finish with a closed-book set of five.
| Week | Topics | The check at the end of the week |
|---|---|---|
| 1 | Limits and continuity | Evaluate a limit by factoring; one at infinity; decide where a piecewise function is continuous |
| 2 | The derivative definition; the power rule | Differentiate x² from the definition; differentiate a polynomial; find a tangent line |
| 3 | Product, quotient and chain rules; eˣ and ln x | Differentiate x² eˣ, (x + 1)/(x − 1), (3x² + 1)⁵, ln(x² + 1) |
| 4 | Trigonometric derivatives; implicit differentiation | Differentiate sin 3x, x cos x; find dy/dx for x² + y² = 25 |
| 5 | Related rates | The ladder, the cone, the two cars: set up, differentiate, substitute |
| 6 | Curve sketching: critical points, the tests, concavity | Sketch x³ − 3x with every feature labelled |
| 7 | Optimisation; the mean value theorem | The open box, the fence, the closest point; verify the MVT on an interval |
| 8 | Antiderivatives; initial-value problems | Integrate a polynomial, eˣ, 1/x, cos x; find the function given its derivative and one point |
| 9 | The definite integral; the fundamental theorem | Area under x² from 0 to 2; a total from a rate |
| 10 | Substitution; areas; a full past exam | ∫ 2x(x² + 1)⁵ dx; the area between y = x and y = x²; an AP AB free-response set under time |
How to study each topic
- The idea before the rule. Watch the 3Blue1Brown episode or read the first page of Paul's Notes until you can say in one sentence what the topic is for. A derivative is a rate; an integral is a total.
- Three worked examples, covered. Cover each step, predict it, uncover, compare. Only then problems.
- Twenty problems with answers hidden. Mark them all at the end, not one by one; rework every wrong one from scratch the next day.
- A closed-book set of five. No notes, no formula sheet. This is the only step that tells you whether you have learned it.
- The error notebook. Every wrong answer with the correct working beside it and one line on why. The same five errors recur; the notebook makes them visible.
- Algebra before numbers. Solve in symbols, substitute at the end. A calculus problem is one line of calculus and five of algebra; the algebra is where it goes wrong.
The five places students fail
| Where | Why | The fix |
|---|---|---|
| Limits | Treating a limit as "plug in the number" and stopping when it gives 0/0 | Factor, rationalise or use the squeeze; 0/0 means "work harder", not "undefined" |
| The chain rule | Not seeing the inside function, or forgetting to multiply by its derivative | Say "outside, inside" aloud for a week; write u for the inside |
| Related rates | Substituting the numbers before differentiating | Differentiate the general equation first; substitute only at the end |
| Optimisation | Not reducing to one variable; not checking the endpoints | Draw it, name the variables, use the constraint to eliminate one, test every critical point and endpoint |
| Integrals | Dropping + C; forgetting the inside's derivative; sign errors with sin and cos | The six mistakes in the integrals guide, read before every integration set |
How Calculus 1 maps to the school exams
| Exam | Covers | Beyond Calculus 1 |
|---|---|---|
| AP Calculus AB | All ten topics | Areas, volumes, separable differential equations |
| AP Calculus BC | All of AB | Series, parametric and polar, integration techniques |
| A-level Mathematics | Year 12: topics 1 to 10 at a lighter level; Year 13: the rest | Techniques of integration, differential equations |
| IB Analysis and Approaches SL / HL | SL: the core; HL: all ten in depth | HL: techniques, more applications |
| CBSE Class 12 | Chapters 5 to 8: continuity and differentiability, applications of derivatives, integrals, applications of integrals | The same core, with a quarter of the paper on integrals |
Why the subject is worth the effort, and who actually needs it, is in why do we learn calculus; how hard each level really is, in is calculus hard.
Download the Calculus 1 checklist and ten-week plan (PDF)
Where to get help
A 1-on-1 online calculus class with a tutor who holds a master's in mathematics: the prerequisite check in the first class, the algebra gaps closed first, the ten topics in order with every line of working read on the shared whiteboard, and the AP free-response or exam-board questions from the first month. Before school, after school or weekends, US time or UK time; the recording and notes after every class; the same tutor through the exam. US$10 to $13 an hour, billed monthly. The free 30-minute first class is a real lesson on the topic the student is stuck on.
Questions parents ask
1What does Calculus 1 cover?
2How long does it take to learn Calculus 1?
3What do I need to know before Calculus 1?
4Can I learn Calculus 1 on my own?
5Is Calculus 1 the same as AP Calculus AB?
See how we teach this, 1-on-1 online →
Shobha
Founder and lead tutor, Science with Shobha
Teaching maths, science and English to children online since 2018. 500+ students, 20+ tutors, families in nine countries.