MHB 2.1.2 AP Calculus Exam particle at rest

karush
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ok I chose e for the zeros
 
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Hi karush,

$x(t)$ represents the particle's location at time $t$, not its velocity. Hence, $t=a,b$ give the time values when the particle is located at $x=0$.

Do you know which tool from calculus will tell us about the particle's velocity, given that we know its position function?
 
I'll probably be studying that this spring semester!
 
If you are not now taking a Calculus class, where did you get this problem?
 
karush said:
I'll probably be studying that this spring semester!

wait ... after posting quite a few AP calculus type motion problems, are you saying you don’t know the derivative of a position function is a velocity function?
 
expanding we have
$x(t) = (t - a)(t - b) = t^2 - (a + b)t + ab$
the velocity will be dx(x(t)) so
$x'(t) = 2t - (a + b)$
particle will be at rest when x'(t) = 0 so if rewriting we have
$t-\dfrac{(a + b)}{2}=0$
thus

$\dfrac{(a + b)}{2}$ is when the particle is at rest which is b.
 
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visualize the graph of the position function $x=(t-a)(t-b)$, a open upward parabola with zeros at $t=a$ and $t=b$.

given the symmetry of the parabola, we know the vertex is midway between the zeros and the slope at the vertex equals zero $\implies v(t)=0$.

the value of $t$ midway between $a$ and $b$ would be $\dfrac{a+b}{2}$, correct?
 
strangely, I don't remember doing particle on x-axis problems when I took calculus...

but it never was a strong spot...:rolleyes:43/365
 
https://mathhelpboards.com/calculus-10/293-ap-calculus-exam-t-v-t-t-8-a-26976.html

post #5, provided a link to a pdf I strongly recommend you have a look at ...
 

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