Mathematics

Mathematics

Problem

For 0≤t≤60 \le t \le 6 seconds a particle moves with velocity v(t)=(t−1)(t−4)v(t) = (t - 1)(t - 4) meters per second. Identify the subinterval(s) of [0,6][0,6] on which the particle is speeding up. Express your answer using interval notation.
Intuitive explanation

The key is recognizing that speeding up means velocity and acceleration point in the same direction—both positive or both negative. Find where v(t)v(t) changes sign by solving (t−1)(t−4)=0(t-1)(t-4)=0, then compute a(t)=v′(t)a(t) = v'(t) to determine the sign of acceleration on each subinterval. A particle speeds up only when vv and aa have matching signs.

Intuitive explanation

The key is recognizing that speeding up means velocity and acceleration point in the same direction—both positive or both negative. Find where v(t)v(t) changes sign by solving (t−1)(t−4)=0(t-1)(t-4)=0, then compute a(t)=v′(t)a(t) = v'(t) to determine the sign of acceleration on each subinterval. A particle speeds up only when vv and aa have matching signs.

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