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I actually came across the question below while studying a book:

A particle moves along a line so that its velocity at time $t$ is $v(t) = t^2 - t - 6$ (measured in meters per second). (a) Find the displacement of the particle during the time period $1 \leq t \leq 4$. (b) Find the distance traveled during this time period.

In the book's solution to the (b) part of the question it explained that: Note that $v(t) = t^2 - t - 6 = (t - 3)(t + 2)$ and so $v(t) \leq 0$ on the interval $[1, 3]$ and $v(t) \geq 0$ on $[3, 4]$.

I have realized that sometimes the factorized quadratics have some geometric (sometimes as areas etc.) explanations attached to them especially when inequalities are involved.

  • What's the meaning of the explanation?
  • When it comes to interpreting these factors in area problems say, what considerations are made?
  • What other information can be glean from quadratic equations that can aid in better solving calculus problems?
  • Really how much information can be gleaned from a quadratic equation?
octopus
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    Since $t$ is at least $1$, the factor $t+2$ is always positive. Hence $v(t)$ has the same sign as $t-3$, i.e., positive when $t>3$ and negative when $t<3$. Thus the velocity is negative for the first two seconds, after which it is positive. – Semiclassical Nov 03 '21 at 00:43
  • @Semiclassical, wow that's a nice observation. Never thought of it that way. It makes sense. Thanks. – octopus Nov 03 '21 at 21:59

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