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Calculus Practice Tests

These Grade 12 Math practice tests connect limits with derivatives, rates of change, optimization, motion, antiderivatives, definite integrals, accumulation, area, and the Fundamental Theorem of Calculus.

Skills covered

  • Using power, product, chain, trigonometric, exponential, and logarithmic derivative rules
  • Interpreting derivatives as slopes, velocity, acceleration, marginal change, and monotonicity
  • Finding antiderivatives and evaluating definite integrals with the constant of integration
  • Applying the Fundamental Theorem of Calculus to accumulation functions and net signed area
  • Solving optimization, motion, concavity, average-value, and tangent-line problems

How to use these practice tests

Decide whether the question asks for local change or accumulated change. Then apply a derivative or integral rule and interpret the result with its sign, units, interval, and context.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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Grade 12 Calculus Practice Tests

Practice Grade 12 Math questions about derivatives, derivative rules, rates of change, extrema, antiderivatives, definite integrals, the Fundamental Theorem of Calculus, area, motion, and applications.

Choose the best answer. Identify the derivative or integral structure, apply the matching rule, and interpret the result as a slope, rate, accumulation, area, or motion quantity.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

A rectangle has perimeter 20. Which dimensions maximize its area?

Please select an answer first.

Correct answer: A

The correct answer is 5 by 5.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: For a fixed perimeter, a square has the greatest area.
  3. Check the interpretation: The correct answer is 5 by 5.

For a fixed perimeter, a square has the greatest area.

2. QuestionMultiple choice

If C(x)=x2+10x+4C(x)=x^2+10x+4, what is marginal cost at x=3?

Please select an answer first.

Correct answer: B

The correct answer is 16.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: Marginal cost is C'(x)=2x+10, so C'(3)=16.
  3. Check the interpretation: The correct answer is 16.

Marginal cost is C'(x)=2x+10, so C'(3)=16.

3. QuestionMultiple choice

A particle has velocity v(t)=2t−4v(t)=2t-4. When is it at rest?

Please select an answer first.

Correct answer: C

The correct answer is t=2.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: Rest occurs when velocity is zero: 2t-4=0.
  3. Check the interpretation: The correct answer is t=2.

Rest occurs when velocity is zero: 2t-4=0.

4. QuestionMultiple choice

If acceleration is a(t)=6ta(t)=6t, what is the change in velocity from t=0 to t=2?

Please select an answer first.

Correct answer: D

The correct answer is 12.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: Change in velocity is ∫0^2 6t dt=3t^2 evaluated at 2, which is 12.
  3. Check the interpretation: The correct answer is 12.

Change in velocity is ∫0^2 6t dt=3t^2 evaluated at 2, which is 12.

5. QuestionMultiple choice

What is the derivative of the accumulated area A(x)=∫1x(t2+1) dtA(x)=\int_1^x (t^2+1)\,dt?

Please select an answer first.

Correct answer: A

The correct answer is x2+1x^2+1.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: The Fundamental Theorem says A'(x) equals the integrand evaluated at x.
  3. Check the interpretation: The correct answer is x2+1x^2+1.

The Fundamental Theorem says A'(x) equals the integrand evaluated at x.

6. QuestionMultiple choice

What is the critical point of f(x)=x3−3xf(x)=x^3-3x?

Please select an answer first.

Correct answer: B

The correct answer is x=-1 and x=1.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: f'(x)=3x^2-3=3(x-1)(x+1), so critical points are ±1.
  3. Check the interpretation: The correct answer is x=-1 and x=1.

f'(x)=3x^2-3=3(x-1)(x+1), so critical points are ±1.

7. QuestionMultiple choice

What is the tangent-line approximation to f near x=a used for?

Please select an answer first.

Correct answer: C

The correct answer is Estimating nearby function values.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: The tangent line locally approximates a differentiable function.
  3. Check the interpretation: The correct answer is Estimating nearby function values.

The tangent line locally approximates a differentiable function.

8. QuestionMultiple choice

If f′(x)=0f'(x)=0 for every x in an interval, what can be concluded?

Please select an answer first.

Correct answer: D

The correct answer is f is constant on that interval.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: A zero derivative means no change in output across the interval.
  3. Check the interpretation: The correct answer is f is constant on that interval.

A zero derivative means no change in output across the interval.

9. QuestionMultiple choice

What does an inflection point indicate?

Please select an answer first.

Correct answer: A

The correct answer is A change in concavity.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: Concavity changes from up to down or down to up at an inflection point.
  3. Check the interpretation: The correct answer is A change in concavity.

Concavity changes from up to down or down to up at an inflection point.

10. QuestionMultiple choice

If f′′(x)>0f''(x)>0 on an interval, the graph is concave how?

Please select an answer first.

Correct answer: B

The correct answer is Up.

  1. Identify the calculus structure: Determine whether the problem asks for a derivative, rate, extremum, antiderivative, area, accumulation, or motion quantity.
  2. Apply the calculus rule: A positive second derivative means slopes are increasing and the graph is concave up.
  3. Check the interpretation: The correct answer is Up.

A positive second derivative means slopes are increasing and the graph is concave up.

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