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?