Mathematics 11–12 · Year 11
Gradient of a tangent as the limit of secant gradients
Introduction to differentiation: The derivative (Mathematics Advanced, Year 11)
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The idea
The derivative at a point is the limit of the gradient of a chord as the second point slides towards the first, which the learner sees both on a drawn curve and in a table.
What you need
- Three printed graphs of y = x^2 on 1 cm grid paper for -2 <= x <= 3, drawn with 1 unit = 1 cm on both axes (the protractor step needs equal scales)
- A ruler, a sharp pencil and a protractor
- A scientific calculator
How to do it
- On copy 1 draw the chord from (1, 1) to (2, 4) and compute its gradient from the coordinates.
- On copy 2 draw the chords from (1, 1) to (1.5, 2.25) and to (1.1, 1.21) and compute both gradients.
- Compute (f(1 + h) - f(1)) / h for h = 0.1, 0.01 and 0.001 on the calculator and tabulate.
- On copy 3 draw by eye the tangent at (1, 1), measure its angle to the x-axis with the protractor and compute the tangent of that angle.
- Expand ((1 + h)^2 - 1) / h algebraically and compare its limit with the table and the drawn tangent.
What you should see
The chord gradients are 3 (h = 1), 2.5 (h = 0.5), 2.1 (h = 0.1), 2.01 (h = 0.01) and 2.001 (h = 0.001); the algebra gives (2h + h^2) / h = 2 + h, so the limit is exactly 2. The exact tangent makes 63.4 degrees with the x-axis; a tangent drawn by eye between 62 and 65 degrees gives a gradient between 1.88 and 2.14. These angles hold only when both axes use the same scale; with the y-axis at 0.5 cm per unit the same tangent is drawn at 45 degrees. The learner knows it worked when the numerical table converges to the algebraic answer and the drawn tangent falls inside that range.
What changes
- What you change
- the step h between the two chord points
- What you measure
- gradient of the chord
- What you keep the same
- the function y = x^2
- the fixed point x = 1
Common misconceptions
Each of these ideas is wrong, and the activity is a chance to test it.
- Setting h = 0 in the difference quotient gives 0/0, so the derivative does not exist; the limit exists although the quotient at h = 0 does not.
- A tangent touches a curve at only one point; the tangent to y = x^3 at x = 1 crosses the curve again at x = -2.
Safety card
Hazards
No hazard is listed.
Controls
No control is listed.
Note
No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; ordinary classroom supervision.
Curriculum references
The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.
- Mathematics Advanced 11–12 Syllabus (2024), Year 11 focus area Introduction to differentiation; Year 11 taught from Term 1 2026, Year 12 from Term 4 2026, first HSC examination 2027 (the 2017 syllabus is still taught to Year 12 until then); page read 2026-09-22MAV-11-06
- Australian Curriculum v9No Australian Curriculum v9 code is listed.
Sources
The pages the author read to write this activity.