Mathematics · Grades 11–12
Introductory calculus: interpret the result
A derivative describes local rate; a definite integral describes signed accumulation.
Understand the idea
Rate and accumulation
A derivative describes local rate; a definite integral describes signed accumulation.
Use conditions
Differentiability implies continuity, but continuity alone does not imply differentiability.
Keep units
Derivative units are output per input; integral units multiply output and input units.
Worked examples
For s(t) = t² + 3t, find s′(2).
s′(t) = 2t + 3, so s′(2) = 7.
7
If s is distance in metres and t is time in seconds, the units of ds/dt are
The derivative compares change in distance with change in time.
metres per second
Try it yourself
Evaluate the definite integral of 3 from x = 1 to x = 5.
Which is an antiderivative of 4x³?
- x⁴ + C
- 4x⁴ + C
- 12x² + C
- x³ + C
Explain why f′(a) = 0 does not by itself prove a local maximum or minimum.
For f(x) = x³, find the slope of the tangent at x = −2.
Answers and marking notes
Answers and marking notes
- 12
A rectangle of height 3 and width 4 has area 12.
- x⁴ + C
Differentiating x⁴ gives 4x³.
- A stationary point can be neither, as for f(x) = x³ at x = 0.
Credit a valid counterexample or need to examine sign/shape around the point.
- 12
f′(x) = 3x²; 3 × 4 = 12.
Further reading and review references
These are reference links, not copied exercises, institutional endorsement or proof of independent review.
Preview material · independent educational review required. Grade bands are approximate, not official curriculum alignment.