BAM! Mr tarrou In this video we're going to be looking at the equation y is equal
to the absolute value of x plus 7 and work with one of the situations where you cannot find a
derivative now that we're being introduced to a definition of derivative i'm looking around
some of the board yet but now we're being introduced to the definition of a derivative
in in finding those derivative functions that give us slopes of another function
we need to understand that you cannot find a derivative at every single point if a
function goes vertical if a function has a removable discontinuity hole
if a function has a jump discontinuity where the left and the right hand limits are not equal
or where the function like at least as it's defined in the textbook i'm teaching from
you cannot find the derivative at the end point of a closed interval all of those situations are
discussed you'll find a link in the description below in my full length very detailed introduction
to the definition of a derivative where i cover all of the situations where you cannot find
a derivative at a point we're going to focus at the sharp end because it is a
pretty common problem on the ap test they seem to focus on that the most so let's remind ourselves
of what is the definition of a derivative so we have f let's just get a bigger piece of chalk too
much better we have f prime of x is equal to the limit
as delta x the horizontal distance between two points the change in the x-axis approaches zero
of f of x plus delta x minus f of x all over delta x and we discussed this in the previous video but
the derivative the point of using one reason we use and introduce derivatives with the definition
of a derivative is so that you see it's just a limit process applied to y sub 2 minus y sub 1
over a simplified denominator of x sub 2 minus x sub 1. so a derivative function f prime of x
is going to tell you the slope of function f or the antiderivative of f prime now i want to
find the derivative of or attempt to find the numerical value y prime or f prime i'll rename
this function f prime of well i want to find the derivative of this function f of x is equal to
the absolute value of x plus 7 specifically at the x coordinate of negative 7 and we're
going to be finding that derivative at x equals negative 7 because that's where the fun stuff is
that's where this derivative has a sharp bend so i don't want to find f prime of x for this problem
we're going to find f prime of 7. so i'm just going to simply take out these x's
and replace it with a 7 so that i don't get a derivative function i get a that'd be negative 7.
i'm going to try and find the slope of this absolute value function
right at the sharp bend okay so we're going to let f prime of 7 f prime of c
equal the limit as x delta x approaches 0 of f of c plus delta x or f of negative 7 plus delta x
minus f of c or in this case minus f of negative seven over delta x
great well this is saying let's go find function f
and function f of x is equal to the absolute value of x minus 7
go take that function and in place of x let's plug in the expression of negative 7 plus delta x
absolute value of x plus 7.
now after the minus sign we're going to be doing f of
negative 7 so we're going to take that x out and replace it with negative seven
okay now the parentheses here after the minus sign are a little bit redundant because basically every
math operation also acts as a grouping symbol so kind of double grouping symbols there but we
have the absolute value of negative seven plus delta x plus seven well yes of course negative
seven plus seven is going to cancel and over here after the minus sign we have the absolute value of
negative seven plus seven or the absolute value of zero so simplifying this up we have the limit
as delta x approaches 0 of well the absolute value of delta x
over delta x so as we try and find that f prime of c that f prime of negative 7 the
work that we need to find the slope of this absolute value function at the sharp bend it all
falls away very quickly but now here's where the interesting stuff comes up the reason why ap will
college board will put this kind of question on an ap exam every once in a while is because they want
you to realize they're seeing whether you realize that you remember that a derivative by definition
is the result that you arrive at whether it's a numerical value or derivative function it is the
result of a two-sided limit this is the limit as delta x approaches zero there's no plus or minus
up here so this is notation for a two-sided limit and does it exist what is the answer
well you need to realize that for this two-sided limit to exist you must have
the statement or the fact that the limit as delta x approaches 0 from the left
is the same and we're going to check it as the limit
as delta x approaches zero from the right
okay now this picture is all nice and all but it's now it's getting my way
remember what that looked like in a minute we'll need it so we're going to analyze the
left hand limit this one over here now how we're going to analyze this left-hand limit we're only
we've just learned how to analyze limits both graphically numerically and maybe maybe not maybe
we're now in the section about analyzing graphic analyzing limits maybe even analytically but this
is in here so we don't have that many tools in our toolbox to analyze the delta x approaching
0 from the left i kind of need to think about doing that like that t table and i'm only going
to pick one value just slightly to the left of zero but we're going to use an x value of
negative one over one negative one over 100. so this is going to be we're going to analyze
this by letting that delta x be a negative 1 over 100 just a little bit to the left of 0.
so when i let delta x
equal negative 1 over 100 this is going to be the absolute value of negative over a negative 1 over
100 is going to be equal to well positive 1 over 100 that's going to be divided by the negative
1 over 100 giving us an answer of negative 1. so this left hand limit
is i wish i had more room to write down here but this left hand limit is equal to negative one
the right hand limit which i guess i could just pick in another color and do this in orange the
right hand limit is going to be again we only have tables and graphs at our disposal here let's pick
a value of delta x just a teeny tiny bit to the right of zero like positive one over one hundred
that's going to give us the absolute value of 1 over 100
over 1 over 100 the absolute value of positive 100 is equal to 100 or 1 over 100
so does this two-sided lemon exist does f prime
of negative seven exists does the two-sided limit exist well the left-hand limit gave us an answer
of negative one the right-hand limit gave us an answer of positive one those two one-sided limits
are not equal therefore this limit does not exist and that's what i need you and my students to show
is if i ask you to justify why a derivative does not exist at a sharp bend i need to see some kind
of work from you that is showing that the left hand limit and the right hand limit are not equal
and you showing that understanding that the definition of a derivative and all derivatives
are based off of a two-sided limit and that only exists when the one-sided limits are equal
you