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hi everyone today we're going to talk about how to find the equation of the normal line
which is the line perpendicular to the tangent line to complete this problem we'll find the
equation of the tangent line at the given point then use the point and the negative
reciprocal of the slope of the tangent line to find the normal line let's take a look
in this particular problem we've been asked to find the equation of the normal line to
the curve y equal x to the fourth plus two e to the x at the point zero two so whenever
we're asked to find the normal line the first thing we want to do is actually find the equation
of the tangent line and the reason is because we can easily find the tangent line by taking
the derivative of our function and then plugging in the point zero two and the normal line
will be through the same point but with the negative reciprocal of the slope of the tangent
line so the tangent line is kind of like our gateway to the normal line so the way that
we're going to find the tangent line is by first taking the derivative of our function
so our function is y equals x to the fourth plus two e to the x we're going to take the
derivative y prime and we'll get using the power rule four x cubed plus two e to the
x remember that the derivative of e to the x is just e to the x and when you have a constant
coefficient it just stays with it so here is the equation of our derivative now we just
want to plug our point into the derivative function so we'll plug in zero for x because
we're at the point zero two and when we do that we'll get four times zero cubed plus
two e to the zero which will of course give us zero plus two times e to the zero or anything
raised to the zero power is just one so we get zero plus two which of course just gives
us two what this means is that the slope of our tangent line is equal to two now that
we have the slope two and a point zero two we can plug those two things into the point
slope formula for the equation of the line to find the equation of the tangent line so
the equation of the tangent line remember that the point slope form is y minus the y
coordinate in our point so in this case two is equal to m the slope and m our slope is
two so two times and then x minus and whatever the x coordinate is in our coordinate point
which in this case is zero so we get x minus zero now when we simplify this we'll get y
minus two equals two x if we want to change this into slope intercept form we'll get y
equals two x plus two and this is the equation of our tangent line now to find the equation
of the normal line we'll use the same point zero two and the negative reciprocal of the
slope that we found for the tangent line so we know the slope is two right here we found
it here so the negative reciprocal would be negative one half so our slope will negative
one half and our point will be zero two and we'll plug those two things into the point
slope form for the equation of the line which again will be y minus the y coordinate which
in this case is two is equal the negative reciprocal of our slope so in this case negative
one half again you just take the slope two and you put one over the slope so one over
two is one half and of course make it negative so negative one half that's our new slope
times x minus the x coordinate from our point in this case zero so now we just simplify
y minus two is equal to negative one half x and if we want to turn this into slope intercept
form we'll get y equals negative one half x plus two and that's that's the equation
of the normal line remember that the equation of the normal line is the line perpendicular
to the tangent line so if this is our curve here our function and our tangent line is
the point here we found the tangent line at the given point this would be the point here
zero two this would be the equation of the tangent line at that point and the equation
of the normal line or the normal would be the line perpendicular to the tangent line
which would look about like this this would be the normal line at that point so I hope
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