StudyBrixFree worksheets, study guides and courses

Mathematics · Grades 11–12

Calculus: meaning, conditions and checks

A derivative describes local rate of change where it exists.

Make a worksheet

Understand the idea

Derivative

A derivative describes local rate of change where it exists.

Integral

A definite integral accumulates signed contribution; geometric area may require separating signs.

Examples and reasoning

The derivative of a constant function is

0

0

For f(x) = x³, find f′(2).

f′(x) = 3x²; 3 × 2² = 12.

12

Check your understanding

  1. Which expression is an antiderivative of 2x?

    1. x³ + C
    2. 2x + C
    3. 2 + C
    4. x² + C

  2. Why include + C in an indefinite integral?

  3. If velocity is negative on an interval, its signed integral there contributes

    1. zero in every case
    2. negative displacement
    3. necessarily negative distance
    4. positive mass

  4. Does f′(a) = 0 alone prove a local maximum?

    1. Yes, always.
    2. No.
    3. Only when the page has a graph.
    4. Only when a is positive.

Answers and marking notes

Answers and marking notes
  1. x² + C

    x² + C

  2. Functions differing by an additive constant have the same derivative, so the family of antiderivatives includes an arbitrary constant.

    Functions differing by an additive constant have the same derivative, so the family of antiderivatives includes an arbitrary constant.

  3. negative displacement

    negative displacement

  4. No.

    No.

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.

Learning preview · teacher review required. Grade bands are approximate, not official curriculum alignment.