A particles moves on a horizontal line so that its coordinate at time $t$ is $x = \ln (1 + 2t) − t^2 + 2, t ≥ 0.$
a. Find the velocity and acceleration functions.
b. When is the particle moving forward and when is it moving backward?
c. When is the particle speeding up and when is it slowing down?
I found the velocity and acceleration. For question b, do I do sign analysis $x(t)$ or $x'(t)$. explanation for c would also be appreciated
For part B, if the particle is moving backward it has negative velocity and positive if moving forward, so you want to do sign analysis on $x'(t)$. The derivative $x'(t)$ can be rewritten as a rational function, i.e. $x'(t) = f(t)/g(t)$ where $f$ and $g$ are some polynomials. Those polynomials are of small degree, so it's easy to figure out for which $t$ either one is positive and negative. Use that information to to figure out when $x'(t)$ is positive and negative.
Part C is a similar kind of problem. The particle is speeding up if its acceleration $x''(t)$ is positive and slowing down if its negative. The approach is identical to the above: express it as a rational function, look at the two polynomials individually, and deduce from there.