A particle moves along a straight line

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

A particle moves along a straight line. Its velocity

v

(in meters per second) is a differentiable function of time

t

(in seconds) with the following properties: •

v(0)=0

, and

v

increases on

(0,2)

, attaining a local maximum at

t=2

. •

v

decreases on

(2,5)

, attaining a local minimum at

t=5

. •

v

increases on

(5,8)

, attaining another local maximum at

t=8

. • For

t>8

v

is negative and strictly decreasing. Assume that

v

is continuous and differentiable on

[0,\infty)

and that no other local extrema occur. (a) For each critical time

t=2,5,8

, classify the extremum of

v

and explain your classification. (b) Determine every time in

(0,\infty)

at which the particle’s position function

s(t)

must have an inflection point. Provide a complete justification that includes a discussion of concavity sign changes for

s

. (c) On which intervals is the graph of

s

concave up? On which intervals is it concave down? State reasons tied directly to the behavior of

v

. (d) Suppose the acceleration

a(t)

is known to be continuous. Explain how continuity of

a

influences your concavity and inflection conclusions in parts (b) and (c).