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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

What does f′(x)>0f'(x)>0 on an interval indicate?

Please select an answer first.

Correct answer: A

The correct answer is f is increasing on the 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 positive derivative means output increases as the input increases.
  3. Check the interpretation: The correct answer is f is increasing on the interval.

A positive derivative means output increases as the input increases.

2. QuestionMultiple choice

What does f′(x)<0f'(x)<0 on an interval indicate?

Please select an answer first.

Correct answer: B

The correct answer is f is decreasing on the 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 negative derivative means output decreases as the input increases.
  3. Check the interpretation: The correct answer is f is decreasing on the interval.

A negative derivative means output decreases as the input increases.

3. QuestionMultiple choice

At an interior differentiable local maximum, what is usually true?

Please select an answer first.

Correct answer: C

The correct answer is f′(c)=0f'(c)=0.

  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 differentiable local extremum has a horizontal tangent.
  3. Check the interpretation: The correct answer is f′(c)=0f'(c)=0.

A differentiable local extremum has a horizontal tangent.

4. QuestionMultiple choice

What is the critical number of f(x)=x2−6x+4f(x)=x^2-6x+4?

Please select an answer first.

Correct answer: D

The correct answer is 3.

  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)=2x-6, so f'(x)=0 at x=3.
  3. Check the interpretation: The correct answer is 3.

f'(x)=2x-6, so f'(x)=0 at x=3.

5. QuestionMultiple choice

What is the slope of the tangent to y=x2y=x^2 at x=3?

Please select an answer first.

Correct answer: A

The correct answer is 6.

  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 derivative is 2x, and 2(3)=6.
  3. Check the interpretation: The correct answer is 6.

The derivative is 2x, and 2(3)=6.

6. QuestionMultiple choice

If position is s(t)=t3−2ts(t)=t^3-2t, what is velocity at t=2?

Please select an answer first.

Correct answer: B

The correct answer is 10.

  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: Velocity is s'(t)=3t^2-2, so s'(2)=10.
  3. Check the interpretation: The correct answer is 10.

Velocity is s'(t)=3t^2-2, so s'(2)=10.

7. QuestionMultiple choice

If velocity is positive and acceleration is positive, what happens to speed?

Please select an answer first.

Correct answer: C

The correct answer is The object is moving in the positive direction and speeding 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: Positive velocity and acceleration point in the same direction.
  3. Check the interpretation: The correct answer is The object is moving in the positive direction and speeding up.

Positive velocity and acceleration point in the same direction.

8. QuestionMultiple choice

What does the Mean Value Theorem connect?

Please select an answer first.

Correct answer: D

The correct answer is An instantaneous derivative with an average rate of change.

  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: Under its hypotheses, some tangent slope equals the secant slope over the interval.
  3. Check the interpretation: The correct answer is An instantaneous derivative with an average rate of change.

Under its hypotheses, some tangent slope equals the secant slope over the interval.

9. QuestionMultiple choice

What is the absolute maximum of f(x)=x2f(x)=x^2 on [-1,2]?

Please select an answer first.

Correct answer: A

The correct answer is 4.

  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: Check endpoints and critical points: f(-1)=1, f(0)=0, f(2)=4.
  3. Check the interpretation: The correct answer is 4.

Check endpoints and critical points: f(-1)=1, f(0)=0, f(2)=4.

10. QuestionMultiple choice

What is the minimum of f(x)=x2−4x+7f(x)=x^2-4x+7?

Please select an answer first.

Correct answer: B

The correct answer is 3.

  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 vertex occurs at x=2, and f(2)=4-8+7=3.
  3. Check the interpretation: The correct answer is 3.

The vertex occurs at x=2, and f(2)=4-8+7=3.

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