B3: a) Differentiate from first principles the function f(x) = x² + 2x – 1. b) A particle is moving in a straight line such that at time t (sec) its acceleration is given by a(t) = 2t – 1 m/s². Derive a general expression for v(t), the velocity at time t, and the specific solution for v(0) = -1 m/s. c) [5] Evaluate (3x2 — х + 2) dx -1

Find the position function of the particle if the particle is initially located at the origin and has an initial velocity of 2 m/s.

A model rocket is fired vertically upward from rest. Its acceleration for the first three seconds is a(t) = 90t, at which time the fuel is exhausted and it becomes a freely "falling" body. Eighteen seconds later, the rocket's parachute opens, and the (downward) velocity slows linearly to -11 ft/s in 5 seconds. The rocket then "floats" to the ground at that rate. (a) Determine the position function s and the velocity function v (for all times t). 4512 if 0 sts 3 405 if 3 26 15r3 if 0 sts 3 405 if 3 26