A rocket is departing Earth towards Mars. The following function represents the rocket's vertical velocity for the first 3 hours: v(t) = t - t²e²² in km/h. (a) Approximate the distance travelled by the rocket in the first 3 hours using a right-hand Riemann sum with 4 intervals: Distance (b) Approximate the velocity function with a Taylor polynomial of degree 2 centred at a = 0: v(t)~+t+t². (c) Use the polynomial approximation to obtain a new approximation of the distance travelled by the rocket in the first 3 hours: Distance km. (d) Calculate the exact distance travelled by the rocket in the first 3 hours: Distance = km. km.

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