WebLab STEM

Mathematics · 7. Calculus

Derivatives, tangents and signed area

Explore secant-gradient limits and Riemann sums.

45 minutesContent version: 2.1
Start the interactive pathwayOpen the virtual experiment

Driving question

When can a sequence of approximations support an exact limiting result?

Curriculum coverage

Vietnam 2018 and Cambridge AS/A Level: derivatives, tangents and definite integrals.

Model scope and limitations

Numerical visuals suggest limits; exact conclusions require differentiability/integrability conditions and algebraic reasoning.

Learning objectives

  • Interpret derivative as instantaneous rate of change.
  • Connect derivative sign with increasing/decreasing behaviour.
  • Distinguish a definite integral from non-negative geometric area.

Prerequisite knowledge

  • Functions, gradients and basic limits

Core knowledge

Derivative at a point

f′(x) is the limit of secant gradients as the increment approaches zero.

Stationary point

f′(x) = 0 identifies a stationary point, but sign or second-derivative tests classify it.

Signed area

Regions below the x-axis contribute negatively to a definite integral.

Worked example

For f(x)=x² at x=3, the secant gradient [f(3+h)−f(3)]/h = 6+h tends to 6 as h→0.

Misconceptions to avoid

  • f′(a)=0 identifies a stationary point but does not by itself classify it.
  • A definite integral is signed area; total geometric area requires splitting at roots and taking magnitudes.

Virtual experiment procedure: Derivative and integral explorer

  1. Choose f(x), a point x and progressively reduce the secant increment h.
  2. Record gradients and compare them with algebraic f′(x).
  3. Increase the number of Riemann rectangles and observe convergence.

Safety and cautions

  • Extremely small h can cause floating-point rounding error.
  • A graph does not replace checking the function domain.

Evidence to collect

  • An h–secant-gradient table approaching zero from both signs.
  • A comparison of numerical approximation with algebraic derivative/integral, including rounding limits.
Open the virtual experiment

Diagnostic check

1. What is the geometric meaning of f′(a)?

  1. Tangent gradient at x = a
  2. Area under the graph
  3. y-intercept
Show answer and explanation

Correct answer: Tangent gradient at x = a

At a differentiable point, the derivative is the tangent gradient.

2. If f′(x) > 0 on an interval, how does f behave?

  1. Increasing
  2. Decreasing
  3. Always zero
Show answer and explanation

Correct answer: Increasing

A positive derivative means a positive rate of change.

3. How does a region below the x-axis contribute to an integral?

  1. Negatively
  2. As an absolute positive value
  3. It contributes nothing
Show answer and explanation

Correct answer: Negatively

A definite integral represents signed area.

Application practice

1. For f(x) = x², what is f′(3)?

  1. 6
  2. 9
  3. 3
Show answer and explanation

Correct answer: 6

f′(x) = 2x, so f′(3) = 6.

2. Is f′(a) = 0 sufficient to conclude a is a maximum?

  1. No, use a sign or another classification test
  2. Yes, always
  3. Yes, if a is positive
Show answer and explanation

Correct answer: No, use a sign or another classification test

A stationary point can be a maximum, minimum or stationary inflection.

3. How do you find total geometric area when a graph crosses the x-axis?

  1. Split at roots and add absolute integral values
  2. Integrate once and ignore the sign
  3. Count only the part above the axis
Show answer and explanation

Correct answer: Split at roots and add absolute integral values

Below-axis regions integrate negatively but geometric area is positive.

Academic references

  1. OpenStax Precalculus 2e — functions, geometry and probability

This lesson is maintained against the cited sources below. Simulations are learning models and do not replace supervised physical-laboratory safety procedures.

Content publisher
WebLab STEM
Technical maintainer
Đức Tiến — Control & Automation Engineer