Equations of Tangents and Lines

00:01 Now that we know how to calculate the gradients of tangents and normals.

00:06 We can move at one step further.

00:08 We can now start to look at equations of tangents and normals.

00:12 So we are extending our ideas or concepts of differentiation to gradients of tangents and normals.

00:19 And now we are going to learn how to find the equations of tangents and normals.

00:23 Now there's a little bit of background that we are going to discuss before.

00:27 And then we'll move forward to actually doing some examples and giving you a chance to work some of them out yourself.

00:33 So we've previously discussed what a tangent and a normal is.

00:37 So a tangent if I'm looking for a gradient at a point P here, the tangent runs along it.

00:45 So this here is your tangent.

00:47 And the normal meets at 90 degrees there.

00:55 Once you know the gradient of the tangent.

00:57 So if we just focus for a moment on our tangents, once you know the gradient of this, you can from here calculate the actual equation of the line of the tangent.

01:10 Now you may have already studied this before.

01:14 But the equation of a straight line ...

01:23 follows the basic structure of y = mx + c.

01:29 Let me just remind you everything here.

01:34 y and x are two points that the line goes through.

01:38 So any points.

01:39 Okay, so the line passes through these two points.

01:42 Passes through.

01:45 And here m is your gradient.

01:49 So that's your gradient.

01:51 So that's quite important for us to find out.

01:53 And c here is your y intercept.

01:55 So when the line crosses the y intercept.

01:58 That is your c point.

02:00 And again that's really important if you are extrapolating any types of line to get results on curves describing decay or growth within any type of medical field.

02:11 So if I drew an x-y axis here.

02:18 Let's call this x.

02:20 Let's call this y.

02:21 And if I extended this line, this point here would be your c.

02:26 So make sure you remember this.

02:29 And you familiarize yourself with this little idea here of y = mx + c.

02:35 It's really quite important not just in mathematics but in physics and engineering.

02:39 It's almost a prerequisite for any type of mathematical field that you understand what a general equation of a straight line is.

02:46 And the tangents and the normals obviously follow the same format.

02:51 Now there is a different way of writing it that some of you may prefer.

02:55 And when we do the examples, you can have a look at both.

02:57 And that is this, you got y - y1 = m(x-x1).

03:04 It takes some practice of doing these questions to actually start appreciating this method here.

03:09 Because this equation actually includes the calculation for your gradient here when you are dealing with a straight line.

03:17 So you can actually just calculate what the m is.

03:20 Whereas, in this equation you have to this equation here.

03:23 You have to calculate your m first.

03:25 Then you put it into this equation and then find c.

03:29 But both methods are fine.

03:31 Both methods are valid.

03:32 This might be, this method here might be just a few seconds faster than the other.

03:38 But both of them are fine.

03:39 But the most important thing is that you understand the importance of this equation or the format of this equation.

03:46 Because often you're given questions in this form.

03:49 And it's important you understand where the m is and where the c is.

03:53 And with this equation it usually is a lot more difficult to do that.

03:56 And you often have to rearrange this equation in order to write this in y = mx + c.

04:03 So you can manipulate where you gradient is and where your intercept is.