Short answer: a typical Calculus I exam tests five areas: limits, derivative rules, applications of derivatives, basic integrals and the Fundamental Theorem of Calculus. If you can answer the 20-question mini-exam below in about 40 minutes, scoring 16 or more, you are in good shape. Below 12 means you should focus on your weakest area before anything else.
How to use this page
- Read the worked problems to refresh each area.
- Take the 20-question mini-exam at the bottom without notes.
- Note which section your mistakes come from. That is your study priority.
Worked problems by area
1. Limits
limx→1 (x² + 2x − 3)/(x − 1) = lim (x + 3)(x − 1)/(x − 1) = 4. Review: 12 common limit mistakes.
2. Derivative rules
d/dx [x³ sin x] = 3x² sin x + x³ cos x (product rule). d/dx (5x² − 1)4 = 40x(5x² − 1)3 (chain rule). Review: chain rule practice problems.
3. Applications of derivatives
Tangent line to y = x² at x = 3: slope 2·3 = 6, point (3, 9), so y = 6x − 9.
Critical points of f(x) = x³ − 3x: f′(x) = 3x² − 3 = 0 → x = ±1. f″(x) = 6x, so x = −1 is a local max and x = 1 a local min. For word problems, see the 5-step optimisation method.
4. Basic integrals
∫(4x³ − 2x + 5) dx = x⁴ − x² + 5x + C. ∫cos(3x) dx = ⅓ sin(3x) + C (reverse chain rule).
5. Fundamental Theorem of Calculus
∫13 2x dx = [x²]13 = 9 − 1 = 8. And d/dx ∫0x cos t dt = cos x.
Common exam traps
- Leaving 0/0 as an answer on limit questions.
- Forgetting the inner derivative in the chain rule.
- Finding critical points but not checking endpoints on a closed interval.
- Forgetting + C, or adding + C to a definite integral.
Practice section: the 20-question mini-exam
Questions 1–4 are limits, 5–9 derivatives, 10–13 applications, 14–17 integrals, 18–20 the Fundamental Theorem. Time yourself: aim for 40 minutes. Your score appears at the end, with an explanation after every answer.
Want a personal plan instead of a single score? The free diagnostic adapts to your answers and tells you what to study first. See also the calculus readiness page.