Differentiation: Exercise 3 – Calculus Methods

00:00 So now that we've learned how to calculate the gradient on a curve.

00:05 We have also moved that forward, with the idea forward, to calculating the gradient at a particular point on the curve.

00:12 So all we did remember was put x values or later on you will see even y values into your gradient, into your dy by dx to find a numerical gradient at a particular point on the curve.

00:23 We are going to take this idea a little bit further now.

00:27 And we are going to start to look at gradients of tangents and normals.

00:31 And we are also going to move forward and actually try and find equations of tangents and normals.

00:36 Remember what I said that this is quite a useful concept specially when you are measuring the decay or growth of any substance.

00:43 Say perhaps in the field of medicine, you can actually work out with a gradient of a particular point is.

00:49 And you can work out on the equation of the tangent or the equation that runs next to that point and extrapolate it to predict further results.

00:57 Let's have a look at a curve.

01:00 So we've been dealing with these types of curves previously.

01:04 So we've been looking at a quadratic equation where we are given the function and we actually find the gradient of that function.

01:12 However, when we start to talk about tangents and normals, say for example, you are asked to find the gradient at a particular point.

01:21 So let's just call this point here point P.

01:23 You can find this point P or you can find the exact gradient of that point P by substituting your x values into there.

01:32 But another thing you can do from here, is also find the gradient of the tangent.

01:37 So the tangent runs right next to this point here.

01:41 So it's almost like it touches the point P or just touches the one point on the curve here.

01:47 So this is just one point.

01:50 And this here is the tangent.

01:54 So remember that tangent is any line that just touches the curve at one point.

02:01 So it may not look like it does in my picture.

02:04 But it is just suppose to be touching the point at one point.

02:08 And the amazing thing about this point here, is that the gradient of the curve and the gradient of the tangent are the same at this point.

02:15 So in this case you can say that the gradient at P of curve is the same as the gradient of tangent at P.

02:29 I don't particularly have to say that this point for this gradient of the tangent.

02:33 Because this is just a straight line.

02:35 And if I know the gradient here, I will know the gradient here and I will know the gradient here or at any point of that line.

02:42 Because the gradient of a line is constant, whereas the gradient of a curve is ever changing.

02:48 So in order to find the gradient of a curve at a particular point, we will have to substitute your x or y values into it.

02:55 However, if you are looking for the gradient of a straight line, you just need to find the gradient at one point.

03:01 And so you'll have the gradient of the entire line.

03:04 So let me just recap.

03:07 That if you want to know the gradient of a tangent that touches the curve at one point, you can find the gradient of the curve at that one point and these share the same gradient which makes a life lot easier for us.

03:19 Another thing that could come up in these types of cases or you might be interested is the gradient of the line that runs perpendicular to this.

03:29 So what if we want to know the gradient of this line here.

03:32 This line here meets at 90 degrees.

03:36 It's perpendicular to this tangent.

03:39 And we usually refer to this line as the normal.

03:42 Now remember that the normal at this point perhaps would have the same gradient but it isn't running along or it isn't running along the same.

03:54 It isn't touching that point in the curve and running along it.

03:57 So this gradient will be different to the gradient of the tangent.

04:01 And we will discuss the combination of the two shortly.

04:05 The normal is the line that goes 90 degrees to the tangent, it meets at 90 degrees.

04:11 And this is a special kind of line because it meets the 90 degrees.

04:15 And they hold a special relationship with their gradients.

04:18 Now say that this line or the tangent has gradient m1.

04:22 Let's just call this m1 for now.

04:24 And let's called the gradient of the normal m2.

04:29 If you think about this, if this gradient here, if this gradient here is positive, it's going up, you can almost see that this gradient here is going to be negative.

04:40 That's only true in this case.

04:43 Obviously your tangent could have a negative gradient and the normal could have a positive gradient.

04:47 But it's quite important to notice that there is a change in sign that's fairly evident here.

04:52 So you can see that it's either positive or negative.

04:55 And they share that opposite sign relationship.

04:59 The definition for this is that if two lines are perpendicular, then m1 multiplied by m2 is going to give you an answer of minus 1.

05:10 And this is only true for perpendicular lines.

05:17 And that's how we can link tangents and normals together.

05:20 So if you remember this little rule here on the side.

05:23 If we know the gradient m1, we can find here work out the gradient too, just by using this relationship that the two gradients should multiply to give you an answer of minus 1.

05:37 Similarly, you can think about two parallel lines.

05:41 So if I just do that here on this side here, so if I have two parallel lines here, it doesn't matter what length they are.

05:47 If say this gradient is called m1 and this gradient is called m2.

05:51 We know that if two lines are parallel, they must have the same gradient.

05:56 They don't at any point meet or intersect.

05:58 So if m1 is equal to m2, then these lines are parallel.

06:06 And these are some of the basics that you may have already covered in your previous studies of maths.

06:12 But this can just be a recap for us.

06:14 And it's important for us to remember this.

06:16 Specially this little rule here.

06:20 Because you will see that we're going to use a lot of this as we find equations of tangents and normals.

06:25 So let me just go over this quickly.

06:27 The equation of the tangent you can find by finding the gradient of a point on the curve here.

06:33 And that gradient is shared.

06:37 So it's exactly the same gradient at the point P as it is of the line here or the tangent.

06:42 And if from here you want to calculate the gradient of the normal, you could use this formula that m1 multiplied by the m2 will give you an answer of minus 1.

06:51 If in a scenario your two lines are parallel.

06:55 So if they run parallel to each other, you know in that case the gradients must be the same.