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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

A particle moves on a vertical line so that its coordinate at time t is y = t3 – 12t + 3, t ≥ 0.

(a) Find the velocity and acceleration functions.

(b) When is the particle moving upward and when is it moving downward?

(c) Find the distance that the particle travels in the time interval 0 ≤ 1 ≤ 3.

(d) Graph the position, velocity, and acceleration functions for 0 ≤ t.≤ 3.

(e) When is the particle speeding up? When is it slowing down?

(a)

To determine

To find: The velocity and the acceleration functions.

Explanation

Given:

The equation of motion of a form is, y=t312t+3.

Recall:

(1) If x(t) is a displacement of a particle and the time t is in seconds, then the velocity of the particle is v(t)=dxdt.

(2) If v(t) is the velocity of a particle and the time t is in seconds, then the acceleration of the particle is a(t)=dvdt.

Calculation:

Obtain the velocity of the particle at time t.

v(t)=ddt(y)=ddt(t312t+3)=3t3112(

(b)

To determine

To find: When is the particle moving upward and when is it moving downward.

(c)

To determine

To find: The distance that the particle travels in the time interval 0t3.

(d)

To determine

To sketch: Graph the position, velocity, and acceleration function for 0t3.

(e)

To determine

To find: When is the particle speeding up and when it is slowing down.

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