00:00Hi guys. So in this
00:02lesson, what we're going to
00:03do is find the equation
00:04of the tangent and the
00:06normal to a curve. Now
00:08there's a strong connection between
00:10tangents and derivatives because the
00:14derivative actually gives you the
00:17gradient of the tangent of
00:19the curve. So it would
00:20be nice to and certainly
00:21very useful to be able
00:23to find the equation of
00:25a given tangent. Now what
00:28is a tangent. Well, a
00:29tangent is a straight line.
00:32The touches the curve, well,
00:35that just touches the curve
00:36at a particular point like
00:38that. So it's just touching
00:40it there. It doesn't cross
00:41it. It just touches it
00:44just once. That is the
00:47tangent. The normal, the normal
00:53to the curve. So that
00:54is the tangent to the
00:56curve
00:56of y equals f of
00:56x at this point. The
00:59normal to the curve y
01:01equals f of x at
01:02that same point is this
01:05line here. Let me try
01:07and draw it there. Now,
01:13what have I drawn? At
01:15what I've drawn, a perpendicular
01:18line. So the normal is
01:21perpendicular to the tangent.
01:24this is the normal. Okay,
01:31that's it. That's the tangent.
01:33That's the normal. Now we're
01:34going to find the equation
01:35of both. So to find
01:37the equation of a line,
01:39look, I've given this straight
01:40line equations here, section 2
01:43.1. This is the formula
01:44that I am going to
01:46focus on. This is y
01:47minus y minus y minus
01:47y minus m to x.
01:49M into x minus x,
01:50one. This is my favorite
01:51formula.
01:52as I've said before, you
01:53just find the gradient and
01:55a point and then you
01:56sub it into the formula.
01:58So that's what we're going
01:59to do down here. We
02:00need to find the gradient,
02:07but we need a point,
02:09we need a point, a
02:10gradient and the formula. So
02:12let me write those three
02:13things. We need a point,
02:17we need a gradient,
02:20and we need the formula.
02:25So let's start with, let's
02:28start with the point, the
02:30point. So it says find
02:34the equation of the tangent
02:35and the normal to the
02:37curve y equals x squared
02:38at x equals three. So
02:41the point we have, we
02:43have the x coordinate, but
02:46the y coordinate I don't
02:48have,
02:48But I can easily sub
02:49it in here. So when
02:51x is 3, y is
02:533 squared, which is 9.
02:56So let's just, let's actually
02:57show that working. So at
03:00x equals 3, at x
03:05equals 3, y equals 3
03:08squared, which is 9. Therefore,
03:11the point is 3 9.
03:16easy. Next thing we need
03:19is the gradient. Now the
03:21gradient, how do I find
03:24the gradient? Well there's the
03:27gradient formula if I'm dealing
03:28with straight lines but when
03:29I'm dealing with curves the
03:31way we find the gradient
03:32is the derivative. Remember that
03:35the derivative gives you the
03:37gradient. So whenever you see
03:38the word gradient or slope
03:41it usually means you
03:44you need to differentiate unless
03:47you're dealing with actual straight
03:49line geometry. But when you
03:52see that we're gradient, think
03:54differentiate. So I'm going to
03:55differentiate this. So define the
03:57gradient, I need to differentiate
03:59y equals x squared. The
04:03derivative dy dx equals 2x.
04:08That's a redefrencie, 2 times
04:10x. Let me know in
04:11the power of the derivative.
04:12is 2x and the derivative
04:14at, I'm going to write
04:18it like this, 2idx at
04:20x equals 3 equals 2
04:23times 3, which is 6.
04:28So the gradient of this
04:30function when x equals 3
04:32equals 6. Now, be careful.
04:36That is the gradient of
04:38the tangent. So I'm going
04:40to write here.
04:40M, and a little capital
04:43T. If I can say
04:44that in a little capital
04:45T, the gradient of the
04:47tangent equals 6. If the
04:51gradient of the tangent equals
04:536, the gradient of the
04:55normal M with a little
04:56capital N equals negative 1
05:00over 6. I have definitely
05:02gone over that in previous
05:03lessons. The perpendicular line, you
05:06flip it and change the
05:07sign. I'll do one more
05:08For example, if your MT
05:11was 3 quarters, then your
05:15M normal is negative 4
05:18over 3. That is essential
05:23that you know how to
05:25do that. So now I
05:26have the gradient for the
05:27tangent and I have the
05:28gradient for the normal. Now,
05:31next thing I need to
05:32do is sub it into
05:34the formula.
05:36Normally, no point intended, you
05:40would be asked to find
05:42one of them, find the
05:43equation of the tangent or
05:45find the equation of the
05:46norm. It's rare that they'd
05:47actually say find the equation
05:48of the tangent and the
05:49equation of the norm. But
05:50I just wanted to show
05:51you how to do both.
05:53And both, you do both
05:55the exact same, you get
05:56the point, you get the
05:57gradient, and you sub it
05:58into the formula. The only
06:00difference is for the tangent,
06:01you use this gradient, and
06:03for the normal, you use
06:04this
06:04the negative reciprocal. So the
06:08formula is y minus y1
06:11equals m into x minus
06:14x1 straight from the formula
06:16booklet. And this is why
06:18I like it. You're subbing
06:20in your point. y minus,
06:24let's actually write over here.
06:27x1 y1. So it's y
06:32minus 9.
06:33equals M. So for the
06:37tangent, it's 6 into x
06:43minus 3. I'm just going
06:46to multiply this out. Give
06:47me y minus 9 equals
06:506x minus 18. And if
06:53I put it in this
06:54form y equals 6x plus,
06:59I'm sorry, minus
07:019, so it's negative 18
07:02plus 9. So this is
07:04the gradient of the tangent.
07:09Maybe I should have given
07:13it a little title here.
07:16So, not that. I'm going
07:19to make this tangent. Then
07:28this will be
07:29the normal. So the norm
07:33is the same thing. It's
07:35y minus y1, same point,
07:39equals m. This is the
07:40only thing I'm changing. Instead
07:42of 6, it's negative 1
07:44over 6, m into x
07:47minus x1. Do I want
07:53to leave it like this?
07:55Probably not. Let's put this
07:56in the other
07:57So I'm going to multiply
07:58across by 6. So this
07:59is 6y minus 54 equals
08:06negative x minus 3. 6y
08:11minus 54 equals negative x
08:16plus 3 because I'm doing
08:18minus x and then minus
08:19times minus this plus and
08:21I'm going to bring or
08:22add this x over here.
08:24So this is x.
08:25plus 6y, x plus 6y,
08:32and then minus 57 equals
08:370. So look, it didn't
08:41ask me to put it
08:42in any particular form. The
08:45tangent I put in the
08:46form ycosm x plus c,
08:47and the normal I put
08:48it in the form a
08:49x plus b y plus
08:50c equals 0 only to
08:52show you the
08:53two different forms. So just
08:57to recap. What is a
09:01tangent? It is a straight
09:02line that touches the curve
09:05at one particular point. And
09:07when I say touch, I
09:08mean it just touches and
09:11moves away. It doesn't go
09:12through it or anything like
09:12that. The normal is a
09:15straight line that is perpendicular
09:16to the tangent. How do
09:19we get the equations of
09:20these?
09:21Well, you do these three
09:24steps for the equation of
09:25the line. You need a
09:27point, a gradient and a
09:28formula. The point you find
09:30from, well, it'll give you
09:32part of the point, if
09:33not all of the point.
09:34The gradient you find from
09:36differentiating, that's the important bit,
09:38and then you sub into
09:39the formula, making sure to
09:42remember if it's the normal
09:43that you do this flip
09:45and change the sign.