MCV4U practice exam: Grade 12 Calculus and Vectors questions and exam rules

The daily Calculus and Vectors (MCV4U) paper has 8 original questions worth 25 to 35 marks. It is practice for the rate of change and derivatives strands: first principles, the product, quotient and chain rules, tangents, rates of change and curve sketching. It has no vectors, and MCV4U has no integration, so skip its integration, area, differential equation and parametric questions.

How the MCV4U exam counts in 2026-27

In Grade 12 mathematics in Ontario, your final mark is now 65% classroom work, a 25% written exam, and 10% attendance and participation. The exam is written on official exam days in your board's exam period. You must take Advanced Functions (MHF4U) before MCV4U or at the same time. These figures come from the Ministry of Education's September 2026 update to Growing Success.

What's on the MCV4U exam: the three strands

MCV4U practice exam questions with answers

We wrote these for this page. Try each before reading the working.

1. The derivative from a limit

Question: Use the limit definition to find the slope of the tangent to f(x) = x² at x = 3.

The slope is 6. Cancel the h before you let it go to zero; substituting h = 0 straight away gives 0/0. The Limit Calculator checks limits like this.

2. Product and chain rules

Question: Differentiate f(x) = e^2x cos x.

Use the product rule, and the chain rule on e^2x, which gives the factor 2. Check with the Derivative Calculator. Try a rules question from our Calculus course, then press Mark.

3. Optimization

Question: A closed cylindrical can must hold 500 cm³. What radius uses the least metal?

From πr²h = 500, h = 500/(πr²), so the surface area is A = 2πr² + 2πrh = 2πr² + 1000/r. Then A′ = 4πr − 1000/r², which is zero when r³ = 250/π, so r ≈ 4.30 cm and h ≈ 8.60 cm. A″ = 4π + 2000/r³ is positive, so this is a minimum. The height equals the diameter.

Here x is the radius in centimetres and y is the surface area in square centimetres.

4. Dot and cross products

Question: For u = (2, −1, 3) and v = (1, 4, −2), find u · v, the angle between them, and u × v.

u · v = 2 − 4 − 6 = −8. |u| = √14 and |v| = √21, so cos θ = −8 ÷ √294 and θ ≈ 117.8°; a negative dot product means an obtuse angle. u × v = ((−1)(−2) − (3)(4), (3)(1) − (2)(−2), (2)(4) − (−1)(1)) = (−10, 7, 9). Check: its dot product with u and with v is 0, so it is perpendicular to both. The Vectors Calculator works these out.

5. A line meeting a plane

Question: Where does the line r = (1, 2, 0) + t(1, −1, 2) meet the plane x + y + z = 7?

Write the line's coordinates in terms of t: x = 1 + t, y = 2 − t, z = 2t. Substitute into the plane: (1 + t) + (2 − t) + 2t = 7, so 3 + 2t = 7 and t = 2. The point is (3, 0, 4). Check: 3 + 0 + 4 = 7.

Use the daily paper as a practice exam

1. Open today's MCV4U paper and work to the suggested time, 35 to 45 minutes.
2. Skip the integration, area, differential equation and parametric questions: they aren't in MCV4U.
3. Mark it and note which strand each lost mark belongs to.
4. Practise vectors separately with the Vectors Calculator, because the paper has none.

Common MCV4U mistakes

• Missing the inside derivative in the chain rule.
• Not justifying a maximum or minimum with the second derivative or a sign chart.
• Mixing up the dot and cross products: the dot product is a number, the cross product a vector.
• Getting the cross product's middle component sign wrong. Check by dotting with both vectors.
• Leaving out units and context in an optimization answer.

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