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Equations of Tangents and Lines

by Batool Akmal
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    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.


    About the Lecture

    The lecture Equations of Tangents and Lines by Batool Akmal is from the course Calculus Methods: Differentiation.


    Included Quiz Questions

    1. The gradient
    2. Where the line crosses the y-axis
    3. Where the line crosses the x-axis

    Author of lecture Equations of Tangents and Lines

     Batool Akmal

    Batool Akmal


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