A particle moves along the –axis so that its velocity at time seconds is modeled by…

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A particle moves along the

x

–axis so that its velocity at time

t

seconds is modeled by the differentiable function

v(t)=t^{3}-6t^{2}+8t, \qquad 0\le t\le 5.

Velocity is measured in meters per second. (a) Determine the particle’s acceleration

a(t)

for

0\le t\le 5

. (b) Construct a complete sign chart for

v(t)

and

a(t)

on the interval

0\le t\le5

. Clearly indicate every value of

t

at which either function is zero. (c) Using your sign chart, determine each open interval on which the particle’s speed is increasing and each open interval on which it is decreasing. State your conclusions using interval notation and justify each one by referring explicitly to the signs of

v

and

a

. (d) Identify every critical time in

(0,5)

at which the speed attains a local extremum, classify each as a local minimum or local maximum of speed, and compute the corresponding speed value.

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