Question

for time 0<= t <= 10, a particle moves along the x-axis with position given by x(t) = t^3 - t^2 + 8t +5. A) During what time intervals is the speed of the particle increasing? B) What is the position of of the particle when it is farthest to the left?

Answer 1

For the increasing portion you only need to find the second derivative, and find out where it's positive. That will tell you a POSITIVE rate of change of the velocity, i.e,. increasing. Also, the position when it is furthest to the left is where v=0. So find where the derivative vanishes and then plug in to the position to see which one is furthest "left".