Mathematics

Problem

A particle moves along a straight line with position function

s(t)=\begin{cases} t^{2}, & 0 \le t < 2,\\ 4t-4, & 2 \le t \le 5, \end{cases}

where

s

is measured in meters and

t

in seconds. a. Part 1: (a) Determine the velocity

v(t)

for each sub-interval of

t

. b. Part 2: (b) Compute the left-hand and right-hand limits of

v(t)

t

approaches

2

. State whether the velocity is continuous at

t=2

. c. Part 3: (c) Decide whether

v(t)

is differentiable at

t=2

. Provide a clear justification based on your work from part (b). d. Part 4: (d) Find the particle’s average velocity on the interval

0 \le t \le 5

and explain how the discontinuity (or lack thereof) in velocity at

t=2

affects this average value.