For greatest displacement on \([0,T]\), solve \(v(t)=0\) in the interval, then compare \(s(t)\) at those times and at \(0,T\). Negative acceleration means velocity is decreasing; it does not always mean speed is decreasing.

Use radians for trigonometric models unless the question states otherwise. On a GDC, check the whole time interval, retain extra digits during working, and report the requested accuracy.

The following complete IB questions combine differentiation with integration, rational functions and differential equations. Return to the differential-equation question after studying Euler’s method and first-order equations.

Applications of Differentiation

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

Quantity	Relationship
Velocity	\(v(t)=s'(t)\)
Acceleration	\(a(t)=v'(t)=s''(t)\)
Speed	\(\|v(t)\|\)
Displacement from the initial position	\(s(t)-s(0)=\displaystyle\int_0^t v(u)\,\mathrm{d}u\)

Displacement, Velocity and Acceleration

Displacement, Velocity and Acceleration