IB Maths AA: Introductory Calculus — Tutor Lesson Plan (derivatives, tangents, stationary points)

2026-10-28

The first question in an IB calculus problem is not how to differentiate but whether the question wants a slope or an extreme value. One cubic separates the two.

Objectives — the student can apply the power rule and explain it as an instantaneous rate, write the tangent at a point, and locate stationary points and classify them.

01 Prerequisites (3-minute check)

If any of these fails, fix it before the lesson starts.

Skill	Quick check	Pass mark
Power rule	For f of x = x cubed – 3x + 2, what is f dash of x?	3x squared – 3
Substitution	What is f of 2?	4
Line equation	Slope 9 through (2, 4)	y = 9x – 14

02 Lesson flow (60 minutes)

Time	Stage	Tutor does	Student does
0-10 min	Slope to derivative	From secant gradient to tangent gradient	State the difference
10-24 min	Power rule	x cubed – 3x + 2 becomes 3x squared – 3	Differentiate term by term
24-36 min	Tangent line	m equals f dash of 2, point (2, f of 2)	Write y = 9x – 14
36-48 min	Stationary points	f dash equals zero gives x = plus or minus 1	Classify each
48-56 min	IB answer format	Method and reason, not answer alone	Rewrite one answer
56-60 min	Exit check	Three questions marked	Finish alone

03 Teaching script (read it out)

Scene 1: From secant to tangent: what a derivative is

The secant gradient is f of x plus h minus f of x, all over h. As h shrinks the secant approaches the tangent, and the limit is f dash of x.
For x cubed – 3x plus 2 the power rule gives 3x squared – 3, a different slope at every point.

Student explains why f dash still contains x.

Scene 2: The tangent line: slope first, point second

At x = 2 the gradient is f dash of 2, that is 3 times 4 minus 3, so 9, and the point is f of 2, that is 8 minus 6 plus 2, so 4.
Point-slope form gives y minus 4 equals 9 times x minus 2, so y = 9x – 14, and the -14 is computed, not recalled.

Student calculates the y-intercept of the tangent (-14).

Scene 3: Stationary points: zero derivative is only a candidate

3x squared – 3 = 0 gives x = 1 or -1. At x = 1 the value is 0, a minimum; at x = -1 it is 4, a maximum.
Test the sign: at x = 0 the derivative is -3, so the curve falls there, which makes -1 a maximum and 1 a minimum.

Note: IB says justify, so the sign test must be written down.

04 Question types and worked steps

Type	Example	Steps	Answer
Differentiate	f of x = x cubed – 3x + 2	Multiply by the index, drop it by one	f dash = 3x squared – 3
Tangent	at x = 2	m equals f dash of 2, point (2, f of 2)	y = 9x – 14
Stationary points	f dash equals zero	Solve then test the sign	x = 1 minimum 0, x = -1 maximum 4
Rate of change	distance function to speed	Differentiate position	State the units

Sanity check (teach this): The tangent meets the curve at the point of contact, so substituting x = 2 into y = 9x – 14 must return f of 2, which is 4.

05 Independent practice (3 questions)

Question	Answer	What to watch
Differentiate x cubed – 3x + 2	3x squared – 3	Constant term becomes zero
Gradient of the tangent at x = 2	9	Differentiate before substituting
The two stationary points	(1, 0) and (-1, 4)	Say which is the maximum

06 Common mistakes and what to say

Mistake	Student writes	Say this
Keeping the constant	f dash written 3x squared – 3 + 2	A constant has no slope, so it differentiates to zero.
Using f of x as the gradient	m written 4	The gradient always comes from f dash, not f.
Stopping at the stationary point	only x = plus or minus 1	IB wants the sign test, which is the justify mark.
Losing the units	speed written as distance	Differentiating changes the unit; speed is metres per second.

07 Homework (paper, section and question numbers)

IB maths chapter index — Check the three questions in the index

08 Exit check (last 5 minutes)

Question	Answer	Decision
Value of f of 2	4	All three right: move to integration
Tangent at x = 2	y = 9x – 14	One wrong: re-teach point-slope form
Both stationary points and their nature	(1, 0) minimum, (-1, 4) maximum	Two wrong: repeat scene 2

09 Tutor FAQ

Does IB examine the limit definition?

Paper 1 rarely asks for the derivation, but Paper 3 and exploration work can ask what the limit means, so scene 1 stays.

Can a calculator differentiate?

It can check an answer, but the marks are for the working, not the output.

How does this chapter relate to functions?

The functions chapter reads shape, calculus reads slope and extremes, and one cubic links both.

Find a tutor — free

A TutorZone lesson-prep guide. Written by the TutorZone team; questions and lesson flow are original and are not taken from any school or publisher material.