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What are the answers (part a & b) to this question and why? Deals with Particle Position fxn and its Acceleration fxn

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Scholar
Karma Points: 206
(Wright State University - Main Campus)
Respect (100%):
Date Posted: 7/20/2008 6:18:13 PM  Status: Closed
What are the answers (part a & b) to this question and why? Deals with Particle Position fxn and its Acceleration fxn
Course Textbook Chapter Problem
Calculus Calculus: Concepts and Contexts (3rd) by Stewart 2.9 8E
Question Details:
I don't understand exactly how to do this two-part problem... Can anyone help me out? Please answer both parts a & b.

The question is:

A particle is moving along a horizontal straight line. The graph of its position function is shown:



***I apologize for the bad graph. Ignore the arrow, don't know where that came from. Just so you know, the graph is a smoothly curve graph, with maximum at 2, inflection at 3, minimum at 4, and the a somewhat exponential curve from (4, infinity).  Ignore the inflection after 4, the graph is smooth after 4... Hope this helps.***

(a) When is the particle moving toward the right and when is it moving toward the left?

(b) When does the particle have positive acceleration and when does it have negative acceleration?
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Answers:

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Oracle
Karma Points: 15,022
Date Posted: 7/26/2008 2:04:19 AM  Status: Live
Asker's Rating: Lifesaver   
Response:
a) the particle moving toward the right if ds/dt > 0,
from the graph, we know 0< t < 2, and t > 4
the particle moving toward the left if ds/dt < 0,
from the graph, we know 2 < t < 4
b) the particle has positive acceleration if the curve is concave upward,
 from the graph, we know t > 3
the particle has negative acceleration if the curve is concave downward,
 from the graph, we know 0 < t < 3
ATShank's Comment:
Thx

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Sage
Karma Points: 5,455
(Melbourne University)
Date Posted: 7/26/2008 4:12:06 AM  Status: Live
Asker's Rating: Helpful   
Response:
a) the particle moving toward the right if gradient is increasing and larger than zero,
we see from the graph, from 0 to 2 and from 2 to ∞
the particle moving toward the left when gradient is decreasing and smaller than zero,
from the graph, we know 2 to 4


b) the particle has positive acceleration when the gradient of gradient(d2/dx2 is increasing
from the graph inflection point is at 3∴for t to ∞
 the particle has negative acceleration when the gradient of gradient(d2/dx2 is decreasing
 from the graph ∴for 0 to 3
ATShank's Comment:
Thank you. I am not sure what you mean by gradient, however.



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