alright in this video I will show you guys how a calc 1 final exam might look like in college
and believe it or not this right here could be the hardest question so check
this out first we will have part a and don't get too excited because this is just gonna be
one point and what we're going to do is find f prime of x namely find the derivative
of f of course we can just use the power rule bring the power to the front and then minus 1
and we see that f prime of x this is just going to be one third x to the third power and of course
don't forget your plus c right just kidding put the two in front and minus one for the power so
we get two x and then plus c right again i know i should stop just two 2x just 2x just 2x all right
anyway and uh if you get this question wrong seriously then i don't know what to say anyway par
b right four points right what's four points um let's go ahead and prove the result from part a
man where to prove this result okay so we have to think about how to prove the
derivative right and the best way is of course to get this right here we can just use the definition
of derivative not so bad all right so i can do that totally f prime of x by definition this is
the limit as h approaching zero and then here we will have let me just write down the definition
for you guys f of x plus h and then minus f of x over h all right so let's continue don't forget to
write down the lym otherwise it will take you the the teacher is going to take away point 2
or whatever all right put x plus h inside so we get x plus h squared minus the original function
and divided by h and again write down the l i m h approaching zero open this we get x squared plus
2xh plus h squared and then minus x squared or over h well we see a nice cancellation this and
that cancel and we can also factor the h so go ahead and do that and we get 2x right here and
then plus h this h and h cancel which is very nice and then we get the limit as h approaching zero
and now we just have two x plus h but because h is approaching zero you put it here which is just 2x
plus zero which is just 2x all right good i feel so much better about my answer from part a now
right you think you are done no you think you have the five points yeah but here is part c ten point
and you know what we are going to do we are going to prove the result from part b so i'm just going
to write down let's go ahead and prove part b and now you might be thinking man are you serious
it's another proof already and now we have to prove the proof huh
we'll have a look this right here is technically a limit question right it's a limit of this and
then we end up with 2x so to prove the limit we actually have to use the epsilon delta
definition of a limit and yes that's actually on my shirt already did you guys notice that
all right now let me tell you sometimes if you don't know how to do a proof on the test
just go ahead and write down some definitions because that way you might be able to get some
partial credits but of course we have to know the definition first right so let me put that
down right here for you guys right here when we have a limit let's say we have x is approaching a
and then let's say we have some function f of x and let's say this gives us l right this right
here means for all epsilon greater than zero and that's exactly why i found my shirt and the rest
is actually on my back all right let me continue there exists a delta greater than zero such that
and you see this is the epsilon-delta definition the infamous definition anyway continue such that
the distance between x and a and we don't need to have x to be the same as a so you can write
it down like this absolute value of x minus a we want the distance to be less than delta
but they don't have to be equal so we want this to be greater than zero if this happens
well we will end up with the absolute value meaning the distance between
this and that so i'll put it down as f of x minus l this right here will be less than epsilon
yes that's the definition of a limit and by this you can see why i wasn't
joking when you have excellent delta definition questions on the k1 test
that's almost like a gpa killer right so no clipping in this video i meant to say wow
yeah you know what i mean by that but anyway let me show you guys how to work this cell all right
so go ahead prove we have to prove that this limit is equal to
2x so we use this and to do so we will start with let epsilon to be greater than zero
and the epsilon greater than zero now next he says there is a delta greater than zero and our job is
to find out what that delta is and what we can do is we can just say i like to use the word choose
right choose delta but the problem is that we don't know what delta is at the moment
how can we find it though that's bizarre let me tell you just put down equal and leave it blank
for now yes just leave it blank for now that's how you can do it so just trust me on that i
don't know what it will be but just trust me on that yeah okay so we have a delta already
and you have to keep in mind this delta will actually be based on epsilon because you have
the epsilon greater than zero already this is perhaps the only thing you can work with right
all right so choose delta to be blank and then this has to be based on epsilon but i don't know
what it is yet let's just continue so it says such that if this happens we shall get this
so we are going to suppose this inequality is true so i'm going to just write down then we are
going to suppose the inequality is true and you see the x is actually our h right here all right
so i will just put down h right here and then a minus and then we have the a which is actually our
zero here so you can see we want to have h minus zero in the absolute value we want this to be
greater than zero and we want this to be less than delta so suppose this is true then we hope to show
that this is also true right so i like to use the word check so we're going to check the inequality
function minus l right f of x minus l in our case we're looking at this as our function part
right this whole thing and then the l is our output right at the outcome which is the 2x so
let me just copy this down inside so we get x plus h squared minus x squared over h and then minus
our answer 2x we have to check if this inequality will be less than delta so it will be less than
epsilon will not have a look of course we can do the algebra so let's go ahead and do the algebra
again and you see here again this and that cancel we can factor an h and then we get 2x plus h and
this and that cancels out again and you see here we have the absolute value inside this
2x plus h and this right here is minus 2x and you might notice that this and that
also cancels out this is very nice and in the end we get the absolute value of just h
that's what we have after the simplification now what
now we look back now you see this is what we suppose and h minus zero of course is just
h right in the absolute value we have this to use the absolute value of h is less than delta
well we haven't found out what delta is yet but you see in the end we hope to show that this
right here is less than epsilon we already know that the absolute value of h is less than delta
and wouldn't it be nice if delta is the same as epsilon yes and remember this right here is
given already right you have that used already the delta will be based on epsilon and let me
tell you we are going to choose delta to be the same as the given epsilon right the given epsilon
have a look first the absolute value of h and i'm actually going to put down a star right here
this right here is less than delta why because we have this assumption that we can use
and then next i'm going to put down double star right here
delta as we said already right if we just kind of go back because you have to read it from here to
here so you said it already this right here by double star or star star just to be cool
this right here is the same as epsilon so have a look we have shown that if this inequality is true
then this part will be less than epsilon and that's exactly what we
need right here and that means we are done so of course put the box and shade the box
if you prefer you can also put on a qed but of course i like the box
all right so what do you think about this
total 15 points how many points do you think that a student can get out of this anyway that's it