WEBVTT
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So we are looking at the function f of x
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equals the absolute value of X-6. Uh And
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here is a graph of that function. So just
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blue graph Uh is the image of the function absolute
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value of X-6. For any uh X greater
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than six. Any X value greater than six.
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You are on this portion of the graph which is
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linear. It is a line and the slope of
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this line uh is one. So for any X
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value greater than six, uh the slope of the
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line is one. And for a line the slope
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of the line is going to be the derivative of
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the function if it's a linear function. So F
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prime of X. The derivative of this function equals
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one. Uh when X is greater than six.
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Okay, so for this portion of the X axis
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, the slope of the function is one. Since
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this linear. So the derivative of the function is
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one For X is greater than six. If we
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look at X values that are less than six,
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that's this portion of the X axis, Everything to
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the left of six. then this is the portion
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of the graph corresponding to X values that are less
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than six. You can clearly see that this portion
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of the graph is also linear and the slope of
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this portion of the graph is negative one. So
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f prime of X equals negative one. When X
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is less than six, this is supposed to be
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a comma f prime of X equals negative one.
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When X is less than six. Well, f
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is not differentiable at six because if f prime of
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x existed and was defined when x equals six uh
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than the derivative f prime of X to the right
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side of six. Uh And to the left side
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of six, uh would would have they would have
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to be approaching the derivative that prime of X as
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X approaches six from the positive side at prime of
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X Would have to exist. And it does it
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equals one. But as X approaches six from the
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negative side, uh F prime of X would also
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have to exist and it does equals negative one.
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But for F prime of X to exist at six
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, uh prime of X as X approaches from the
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positive side would have to be approaching the same limit
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as F prime of X. Uh Coming in from
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the negative side and they don't okay, on the
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positive side of six, at prime of X is
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one on the left side of six, F prime
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of X is negative one. So F prime of
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X is not going to be defined at six because
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when we approach from uh the positive And from the
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negative sides of six, the f prime of X
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approaches to different limits. The limit of F prime
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of X as X comes in from the positive side
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is one, the limit of F prime of X
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as X approaches from the negative side or the left
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side of six is negative one. So f prime
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of X at six is not defined. So next
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we actually want to define F prime of X.
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Uh and or find a formula for F prime of
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X and we actually already have it. Uh This
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fray here is the formula for F prime of X
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. The derivative of this function equals one when X
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is greater than six and equals negative one. When
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X is less than six. Last but not least
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. Uh We need to sketch the graph of F
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prime of X. Okay, so now we're going
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to graph this uh F prime of X function that
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prime of X equals one when X is greater than
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six. So when X is greater than six,
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f prime of X would equal one. So we're
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gonna put an open circle when X is six but
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for X greater than six, F prime of X
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equals one. When X is less than six to
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the left of six, that prime of X is
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going to equal negative one. So when X is
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less than six at prime of X equals negative one
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. Let's go ahead and get a line to do
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that for us. And once again ah we want
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we want an open circle here. Okay, so
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here is the graph of f prime of X.
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When X is greater in six. Okay, everything
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to the right of six, not including the six
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, that's why we have the open circle, F
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prime of X is one. Okay, so one
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on the vertical axis, S C F prime of
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X axis when X is less than six. So
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when we go to the left of six, that
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prime of X equals negative one. So for all
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X values to the left of six have prime of
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X is negative one. So here we are there
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once again at six. Um it's not included.
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Now notice once again we said that F prime of
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X was not able to be defined at six.
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That's why at six you have an open circle here
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. You have an open circle here and you do
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not have any other points for when X is exactly
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six because F prime of X is not defined when
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X equals six. Okay, as X approaches six
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from the right side, that prime of X uh
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is approaching one as X approaches six from the negative
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side from left side, that prime of X approaches
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the limit of negative one. Uh So you can't
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approach to different limit numbers. Uh when we come
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in from the two different sides uh and have a
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limit. So at prom index did not exist at
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six, that's why you do not have a point
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on the graph. Now, when X is six