33 terms

Particle Motion

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Given x(t), find v(t)
Take the derivative; v(t) = x'(t)
Given x(t), find a(t)
Take the second derivative; a(t) = x"(t)
Given x(t), find total distance traveled on [a,b]
--Method--
(1) evaluate x(t) at critical numbers & endpoints (a & b)
(2) find the absolute value of the difference between consecutive x(t)s from step 1
(3) find the sum of all the changes in position from step 2
Given v(t), find total distance traveled on [a,b]
Integral from a to b of |v(t)|
*don't forget absolute value*
Given x(t), find the displacement on [a,b]
x(b) - x(a)
When is the particle speeding up?
When v(t) and a(t) have the same signs.
When is the particle slowing down?
When v(t) and a(t) have opposite signs.
When is the particle moving right?
When v(t) > 0
When is the particle moving left?
When v(t) < 0
When is the particle at rest?
When v(t) = 0
Given x(t), find the average velocity on [a,b].
[x(b) - x(a)] / [b - a]
Given v(t), find the average velocity on [a,b].
[integral from a to b of v(t)] / [b - a]
Given v(t), find the average acceleration on [a,b].
[v(b) - v(a)] / [b - a]
Given a(t), find the average acceleration on [a,b].
[integral from a to b of a(t)] / [b - a]
When does the particle change direction?
When v(t) changes signs.
--Method--
(1) Look for where v(t) = 0 or is undefined.
(2) Create a sign chart to find where v(t) changes signs
Given v(t) and initial position x(a), find position when t=b
x(b) = x(a) + [integral from a to b of v(t)]
Given x(t), find the instantaneous velocity.
Take the derivative; instantaneous velocity = v(t) = x'(t)
Given f(x), find all relative maximums on [a,b]
These occur where f'(x) changes from positive to negative on (a,b).
(First Derivative Test)
Given f(x), find all relative minimums on [a,b]
These occur where f'(x) changes from negative to positive on (a,b).
(First Derivative Test)
Given f(x), find all absolute maximums on [a,b]
These occur where f(a), f(b), or f(c) is the greatest on [a,b].
c = any x-value of a critical point
Given f(x), find all absolute minimums on [a,b]
These occur where f(a), f(b), or f(c) is the least on [a,b].
c = any x-value of a critical point
Given f(x), find where the function is increasing on [a, b].
Where f'(x) > 0
Given f(x), find where the function is decreasing on [a,b]
Where f'(x) < 0
Given f(x), find where the function has a point of inflection on [a,b].
Find where f"(x) changes signs on (a,b)
Given f(x), find where the function is concave up on [a,b]
Find where f"(x)>0
Given f(x), find where the function is concave down on [a,b]
Find where f"(x)<0
If a linear approximation is an overestimate, then
A function is concave down.
If a linear approximation is an underestimate, then
A function is concave up.
Find horizontal tangent lines of f(x)
set numerator of f'(x) = 0
Find vertical tangent lines of f(x)
set denominator of f'(x) = 0
Given v(t), find when the instantaneous velocity is the same as the average velocity.
v(t) = [integral from a to b of v(t)] / [b-a]
Given v(t), find displacement on [a,b]
integral from a to b of v(t)
Given x(t), find when the instantaneous velocity is the same as average velocity
x'(t) = [x(b) - x(a)] / (b-a)