O-Level Differentiation and integration: worked solution

Question

A particle's acceleration is a = −2 cos(t/2) m/s², t seconds after leaving O. Sketch the acceleration-time graph over 0 ≤ t ≤ 2π.

a = -2 cos(t/2) for 0 <= t <= 2pi. As t goes 0 to 2pi, t/2 goes 0 to pi. Key values: t = 0, a = -2 cos 0 = -2 (minimum); t = pi, a = -2 cos(pi/2) = 0; t = 2pi, a = -2 cos(pi) = 2 (maximum). The graph is a smoothly increasing curve (an inverted, stretched cosine) starting at (0, -2), passing through (pi, 0), and ending at (2pi, 2), with horizontal tangents at the two endpoints.