How to use this guide
Where this guide comes from. It follows the course's published description (a first single-variable calculus course that moves from limits to derivatives to integrals, with applications of each and a short introduction to functions of several variables) and the standard topics for a university calculus course. Instructors' emphasis and notation vary: check your own course outline. If your outline leaves a topic out, such as partial derivatives or improper integrals, skip that tab.
The order. 01 to 03 build the language: functions, limits and continuity. 04 to 09 are differential calculus: the rules, then everything derivatives are used for. 10 to 12 are integral calculus: sums, the Fundamental Theorem, substitution and applications. 13 is a first look at two variables. The course is more rigorous than a business or life-science calculus, so you will also see the formal idea of a limit and short proofs of "why this works".
Inside a topic: read the idea, look at the rules, then open the worked example and follow every step. Close it and redo it on paper. Then change the numbers in the lab until the answer stops surprising you, and finish with the traps.
How the math is written here. Powers and subscripts are written the way you type them: x², e2x, xn. A fraction is a ÷ b, a/b or stacked, and √(…) covers everything under the root. f′ and dy/dx both mean the derivative. All angles are in radians, as calculus requires. ln is the natural log, and arcsin, arccos and arctan are the inverse trig functions (also written sin⁻¹, cos⁻¹, tan⁻¹).
Then practise. The Drill, the multiple-choice bank and the flashcards are all levelled:
| Level | What it asks | Example |
|---|---|---|
| Beginner | A definition, or one step: apply one rule | Differentiate 4x³ at x = 2 |
| Intermediate | A standard problem in a few steps | Find f′(1) for a product with an exponential |
| Exam level | A full exam-style question that strings several steps together | Find the slope of a curve defined implicitly, or a limit that needs L'Hôpital twice |
| Hard | Twists: topics combined, setting the problem up yourself, and the traps that cost the most marks | Volume of a region rotated about the x-axis, or an optimization with a cost that changes by region |
The question generator starts on Auto: it begins at Beginner and moves up a level after three correct answers in a row. You can pick a level yourself at any time. The multiple-choice bank and flashcards are ordered from easy to hard, so working through them in order builds each idea from the ground up.
In the labs, a number you type is accepted when it is within the stated rounding. Text answers ignore capitals, spaces and hyphens, so "None", "none" and "NONE" are all fine.
Exam format, weight and allowed aids differ between sections and terms. Check your own course outline.
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This is the first section. The full guide has every topic, the labs and the practice questions.
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