this is a very classic calculus tangent line question check this out first off we are
considering the function 1/x and let's just look at it in the first quadrant so we have
a picture like this and we have y=1/x and of course we need a tangent line but where
though let me tell you it doesn't matter i'm just going to draw a tangent line like this
okay and as we can see we have a triangle formed by the tangent line and also the x and y axis
like so and let me tell you the secret it doesn't matter where the tangent line is
the triangle that we are going to get will always have area being equal to should i tell you guys
the answer now okay i'll tell you guys the answer now 2. always if you don't believe me
we'll just continue watching and this video is sponsored by brilliant if you want to know more
about calculus go ahead and check that out but more about them later all right so how
can we stop this i will tell you we don't need to use integration i know don't leave though
you can but we are not going to because this is just the area of the triangle
we just need to find out the base and also the height multiplied and divided by 2 right
this is just a line if we have an equation of the line we can just work with it so here's the deal
because this is just a random point i'm just going to label that as a and in order to find
the equation of the line we need the slope and the point the slope is exactly the derivative so let's
see this is our y well this is the same as x plus negative 1 bring the power to the front minus 1
and we will get y prime being equal to negative x to a negative 2 which is negative 1 over x
squared cool and with that said we know that the slope is just going to be denoted by m which is
going to be put the a in here so we get negative 1 over a squared like so all right and the point
is just that we put the a back to the y equation so here is the point we just have to plug that in
so we get the x value being equal to a but the y value is going to be 1 over a so that's okay cool
and let's use our point slope of the line and to do so i'll just write it down just
put it down we're going to use the points of 4 which is y minus y1 equals m times x minus x1
well y is just y y one is this so we have one over a and then m is this
so we have this being equal to negative one over a squared and then times x minus x one which is a
and now you can actually just find the x and y intercept from here but perhaps i will clean
this up a little bit i'm going to multiply this out so we get negative 1 over a squared times x
and the a and a will cancel negative t times negative we get the positive i get 1 over
a to the first power on the bottom and then we have y minus 1 over a
add 1 over a on both sides so we get y equals negative 1 over a squared x plus 2 over a so in
fact that's the equation of the tangent line and now we just have to find the x-intercept
and also the y-intercept to find the x-intercept let me just put it down right here for you guys
well we just have to make y equal to zero so we put y is equal to zero and then solve for x so
this is equal to negative one over a squared times x plus two over a and move this well actually move
this to the other side we get positive one over a squared x and that's equal to two over a and i'm
just going to multiply both sides by a squared this and that will cancel we get x by itself
and then this and that will cancel but i still have one more a so we have 2a here
so as we can see the x-intercept right here is 2 a and then for the y intercept whoops
but the y-intercept
well for this one we just have to make y for this one we just have to make x equal to zero so we get
y it's equal to well you know the answer is down already but i just want to show you
guys all the work so multiply by zero and then plus 2 over a and just get y equals 2 over a
so that means this right here is 2 over a and now the moment of truth what is the
area of that triangle as we all know area is just going to be one half base times height
the base is 2a and the height is 2 over a so you just have to multiply this by 2 over 8. and ladies
and gentlemen as we can see one-half and the two cancel and this a and i cancel congratulations the
area is just always going to be equal to two a-ha so that's why i told you it doesn't matter where
we put the tension like earlier we will always end up with two and i think this is super super cool
but unfortunately i do not know if there's any other function that has similar property like
this maybe you guys can try y is equal to 1 over x to the second power and let me know how that goes
what do you guys think and now i want to thank today's sponsor brilliant as
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