WEBVTT
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here we have the graph of a position function and
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we want to know when the particle is speeding up
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and when it's slowing down, it'll be speeding up
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when the velocity and acceleration have the same sign,
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either both positive or both negative. And it will
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be slowing down when they have different signs one positive
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and one negative. So what we're going to do
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is break our time, intervals down and analyze what's
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happening with the velocity and acceleration in each interval.
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So we have 0 to 1 1 to 2,
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2 to 3 and 3 to 4. Something is
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changing for each interval. So now we're going to
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find the sign of the velocity and the sign of
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the acceleration. When the position graph is increasing,
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the velocity is positive. So we see it increasing
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from 0 to 1, and we see it increasing
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from 3 to 4. When the position equation is
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decreasing, the velocity is negative, so we see
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a decreasing from 1 to 3. Now, the
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acceleration is the second derivative of position, and the
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second derivative relates to the con cavity of the curve
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con que but versus conch it down. If the
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curve is conquered down. The second derivative will be
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negative. And if the curve is Khan gave up
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, the second derivative will be positive. So we
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see that this curve is conclave, down from 0
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to 2. So even negative acceleration. And then
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it's Khan gave up from 2 to 4, so
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we have a positive acceleration. Now let's look and
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see when both of these signs are positive. Velocity
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and acceleration are both positive from 0 to 4,
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and they're both negative from 1 to 2. So
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that's an indication of speeding up. So we have
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speeding up from times 1 to 2 and from times
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3 to 4. Now, look for when they
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have opposite signs from 0 to 1 and from 2
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to 3. So that means it's slowing down 0
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to 1, 2 to 3. Okay, let's
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do something similar for the second graph. So again
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, we're going to be analyzing the velocity and the
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acceleration to figure out whether they're positive or negative.
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So we have the interval from 0 to 1 from
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1 to 2 from 2 to 3, and from
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3 to 4, we have the velocity and we
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have the acceleration. Okay, So this graph is
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decreasing from 0 to 3, and so it has
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a negative velocity from 0 to 3. This graph
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is increasing from 3 to 4, so it has
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a positive velocity from 3 to 4. This graph
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is conclave, up from 0 to 1, so
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it has a positive acceleration. Then it's Con Cape
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, down from 1 to 2, so it has
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a negative acceleration. Then it's called Cave up from
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2 to 4, so it has a positive acceleration
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again. Now let's look for where the signs are
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the same. The signs are the same from 1
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to 2 and 3 to 4, so that would
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be when it's speeding up. The signs are different
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from 0 to 1 and 2 to 3. That
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would be when it's slowing down.