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3-19 AIM3 7.5 Secants and Tangents

Added Mar 19, 2024 •

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and hello again so let's build on our knowledge of how we find equations of secant with respect to graphs and built how we could find the equation of a tangent but to do that let's just back up remind ourselves of something alright so here's part of a graph distance equals t squared and so let us do some calculations what's the number one over here average speed Saga call by that we mean average rate of change we discuss how to find that so from one too well and let CT squared so it would be 2 squared or 4 - 1 / 2 - 1 and so we would get three let's see what happens at 1 we're here and it to were here and if we were to draw a line through there let me see if I can do this the fancy will make it look semi-decent let's see if I can pull this off not bad I suppose but if I count I can see the slope here knows we go up 123 and over 170 the slope is 3 well what if now works so what if we go from 1 to 1.1 is Lil bit harder to see on the graph I can sketch it a bit weekly sowon and then let's say 1 at 1 is there and so that would be there until we have a secant line with two points and the key is what we're looking at today is the two points getting closer and closer so in our example we're going to keep the one fixed and I'm just clean that up a little bit keep the one fixed and get the second part coming closer and closer as hope exactly are we gone from 2 to 1.1 but we do our calculation well one point squares we get 1.21 - 1 / 1.1 1.1 - 1 so we get Point 21 / 0.1 until we get 2.1 not quite as deep well next time I can't draw it all but if we were to go from one 1.01 and look at Jerry's average Speedwell 1.01 Square turns out to be one point 0201 - 1 over 1.1 having a tough time again one point 01 - and that turns out to be 2.01 okay now I hear something interesting really happens in this case what happens it is instantaneous speed another word when we want to have one pointed one and the next point one they're both at one place we did the calculation 1 squared - 1 squared 1 / 1 - 1 - 0 / 0 and that gets us sad face cuz I don't know what that means I can't calculate it and that it works brings us to the point of day we surely could draw tangent we could draw on and try and do it free hand I could draw a line I'm going to try and draw line it's going to go through in don't touch the graph where TV in this tasty is wanted that's my tensions as terrible as that looks I surely can draw it and it's a line so it has a slow and it has a point so I should be able to find it Asian but using this kind of approach I guess just isn't going to work so what does that mean when you have to modify something when you come up something new with Portia Lee Frost we're going to buy something we've already used and as usual it just won't be back bigger deal will be able to find equations of secant and tangent which is you'll discover down the road in your math career is something you'll be great important Okay so let's go back to the equations you can by the way what he knows about this function I'm just it's missing it's it's Square, that's okay well I change the colored again so we got point to Ephrata to send other words we got as it turns out 25 bus to save you. Come on out 25 and free 22 I believe 27 - 4 + my 6 okay so 25 and 3:22 so we do the calculations using the a rock approach 22 - 5 you should be going along with me or even being ahead of him here free money too and so we just get 17 okay so that means 9 - 5 17 * x - 2 and by the way in point slope form this is fine there's nothing wrong with this is an answer but if it was multiple choice maybe this the answer is maybe they put it in point slope where they're already I already put in slope intercept play put it in equal 0 form and so we will just as well do it so why equals 17 x when it's 34 and I would need to add 5 so we get y equals 17 x - 29 okay so that's fine I certainly could come up with that there's something cool that we can do that we had talked about before so let's look at this so here's our function X cubed minus 2x Plus one and I might as well put in the 0x squared and now what is this on here this ex-guard -5x + 6 where did that come from how to pause and think about it for a second and perhaps you discovered that this is in fact x - 2 x - 3 the two points they were tomboy directions to nxs3 joke we take those two points and write them the multivitamin together and do long division to go through the process but here's the wonderful thing look at what a remainder is rrr of x it is in fact equation of the secant pretty cool didn't mention that earlier so just to sum up SQ - 2x + 1 is x - 2 x - 3 * something + it looks plus the remainder and that's what we're going to use cuz this remainder is linear because we're dividing by a quadratic by second-degree so we know we're done when I remainder is linear and then your means line and that gives us our you can and this is the inside that's going to help us find equation of the tangent since we can't do it the other way so let's take a look here's our staff Frozen we have asked you can we got two points A and B if we do x minus a x minus B and we were to do it through long division we get some quotient and some room for manger let me just remind you of this bit here this is a quotient so we have our divisor * quotient we have a remainder standard stuff so the divisor X closer look at the remainder all right and again I guess I should emphasize RX and your function since we're dividing by a quadratic so what does that do for us for the tangent won't remember the tangent is what happens as our second Point gets closer and closer to the first point that's true instead of x - 8 + x - B we're getting closer to having x - 8 + x - A visually what I'm saying is let's see here know if we have some value a and its associated with some point men over here we have some value B and it's okay with some point but stay here then as big as closer and closer to a cup of this point the way to active so this would be our she can't buy let me get a little bit go to the A and I find this point here then I'd have a difference to can't if I let Bebe come here then it's associated with this point but if he gets really close to a hey man it's value is getting real close to FAA and eventually they got as close as I can tell I couldn't see any any better if it's too hard to see but I hope that you understand the idea of getting closer Macy's points are merging to each other which means instead of having x- a x - b x - 8 * x - AF simply x minus 8 squared with some clothes and some remainder which is still in and it's give me the graph of the tangent same time for us to go see can't we're just going to do it how to find the tangent stuff for example here let's see what we're looking at the point 25 Ashley find equation of the tangent of 2.25 and so it's x -2 squared and how we can find it well let's do some long division so Hopewell x -2 squared course in the x squared - 4x + 4 + x squared - 4x + 4 I want to be divided into x cubed + 0 x squared minus 2x plus one you might want to apologize video and do the lawn bring yourself before you see my answer just for the practice of it alright well let's see what we got here x squared x Cube logo x x so that X cubed - Maybe or x squared + 4x change it x squared - 6X bring down the one that again x-grain of those four times so there's are closing for C * 4 x squared - 16x + 16 + - track -6 - x 10 x -15 I have found a linear remainder and this means raisin I've always been very hard to write on this equation of the tangent don't let the pain way high off the screen when you're trying to move it reads it an upgrade that is not an upgrade I'm so wow can you believe this stop laughing we get y equals 10 10 x -, Camas easier to walk over to Gwen's house and do this why keep walls we can do it no we can't I'm giving up on that one you went through that you're a trooper can't give up I got to try one more one more time maybe I'm a little bit thicker why close - 15 good enough holy cow anyway show by the national YouTube long division we can find the equation of the tangent in a similar way we could find the equation of secant the last thing I want to show you play Young huh I have no idea what's happening over here boy in the red we have the cubic and the blue line is 10x - 15 can't see the stop making 15 and it's tough to see tangents cuz they're the two grass seems to blend into each other but there we go there's the point of tangency and just a visual of it if you can forgive the writing I hope you understood what was happening there and I wish you the best of luck

Uploaded by Brian Macky.

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3-19 AIM3 7.5 Secants and Tangents

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