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Mathematics · Grades 11–12

Derivatives, accumulated change and units

Connect calculations to their meanings and domains.

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Derivative

A derivative describes a local rate of change with respect to the independent variable.

Units

If position is measured in metres and time in seconds, its time derivative has units m/s.

Signed integral

A definite integral of velocity gives signed displacement, not automatically total distance.

Critical points

A zero derivative is a candidate for an extremum, not a guarantee. Check the surrounding behavior.

Worked examples

The tangent to f(x) = x³ at (2, 8) has slope 12. Find the y-intercept of this tangent line.

Use y = 12x + b and the point: 8 = 24 + b, so b = -16.

-16

For s(t) = 2t² + 3t metres, find velocity at t = 4 s.

ds/dt = 4t + 3, so the velocity is 19 m/s.

19 m/s

Check your understanding

  1. What does the integral of signed velocity over a time interval give?

    1. Always total distance
    2. Displacement
    3. Acceleration at the last instant

  2. Evaluate the integral of 3x² from x = 0 to x = 2.

  3. For f(x) = x³, what happens at x = 0 despite f′(0) = 0?

    1. It must be a maximum.
    2. The function continues increasing through the point.
    3. It must be a minimum.

Answers and marking notes

Answers and marking notes
  1. Displacement

    Positive and negative contributions combine as signed changes in position.

  2. 8

    An antiderivative is x³; evaluate 8 − 0.

  3. The function continues increasing through the point.

    A horizontal tangent does not by itself establish an extremum.

Further reading and review references

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

Independent educational review required. Grade bands are approximate; these are not officially aligned curricula.