Revise Optimization and kinematics for IB Mathematics: Analysis and Approaches (5.4)

Optimisation is the process of finding the best possible value of a quantity — the largest area, the greatest volume, the least cost or material. The strategy is always the same. Write the quantity you care about (the objective) as a function; if it depends on more than one variable, use a constraint to eliminate all but one; then differentiate, set the derivative to zero to find the stationary points, and justify whether each is the maximum or minimum you want. The justification is not optional: solving $f'(x)=0$ only finds candidates, and IB awards a reasoning mark for showing which candidate is genuinely the extremum.

Kinematics describes the motion of an object without worrying about the forces behind it. In this course the motion is along a straight line, and calculus links its three descriptions. The displacement $s(t)$ says where the particle is relative to the origin; differentiating gives the velocity $v(t)$, which says how fast and in which direction it moves; differentiating again gives the acceleration $a(t)$, which says how the velocity is changing. Reading motion is then a matter of interpreting these functions: the particle is at rest where $v=0$, its extreme displacements occur where $v=0$, and the sign of $v$ tells you the direction of travel.

The core relationships are: Velocity: $v(t) = \dfrac{ds}{dt} = s'(t)$ Acceleration: $a(t) = \dfrac{dv}{dt} = v'(t) = \dfrac{d^2s}{dt^2} = s''(t)$ And, going the other way, by integration: Velocity: $v(t) = \int a(t)\,dt$ Displacement: $s(t) = \int v(t)\,dt$

Optimisation and kinematics feel like different topics, but they are the same technique in different clothing. Finding the maximum displacement of a particle means finding a stationary point of $s(t)$ — solving $s'(t)=v(t)=0$ — which is exactly what you do to maximise an area or volume in optimisation. In both cases you differentiate, set the derivative to zero, and then justify or interpret what that stationary point means. Recognising this shared structure means you are really only learning one method, applied to two families of question.

On Paper 1 the marks are analytic: each is tied to a specific line of working. A method mark (M) rewards a correct approach even if the arithmetic later slips; an accuracy mark (A) rewards a correct result and is usually dependent on the method mark being earned. Follow-through (FT) means a correct final step performed on your own earlier (wrong) value still scores, ISW ('ignore subsequent working') means a correct answer is not un-marked by a later fumble, and equivalent correct forms are accepted. All of that protection exists only if your method is on the page. Study how each mark below is earned by a specific line.

Everything here is one skill — differentiate, set the derivative to zero, interpret the result — applied to two contexts. That skill returns immediately in integration, where you reverse it to recover velocity and displacement from acceleration (remembering $+C$), and in the area-under-a-curve problems that follow. Master the habit — model the quantity, reduce to one variable or differentiate for $v$ and $a$, solve, then justify or interpret — and the rest of calculus becomes variations on a method you already own.