Use calculus, NOT calculators!

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Күн бұрын

Use calculus, NOT calculators! We will use a tangent line approximation and differentials to approximate sqrt(8.7). This is called local linear approximation which is an application of derivatives in Calculus 1. Subscribe for more calculus tutorials 👉 bit.ly/just_calc
0:00 use tangent line approximation for sqrt(8.7)
4:49 use differentials to approximate sqrt(8.7)
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"Just Calculus" is dedicated to helping students who are taking precalculus, AP calculus, GCSE, A-Level, year 12 maths, college calculus, or high school calculus. Topics include functions, limits, indeterminate forms, derivatives, and their applications, integration techniques and their applications, separable differential equations, sequences, series convergence test, power series a lot more. Feel free to leave calculus questions in the comment section and subscribe for future videos 👉 bit.ly/just_calc
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Пікірлер: 507

@bprpcalculusbasics 2 жыл бұрын

Too slow? Try the fast version: kzfaq.info/get/bejne/jd5nn7F0s7SVqWw.html

@Zetsuke4 2 жыл бұрын

Nice

@leonardobarrera2816 Жыл бұрын

Day 32 wishing you that you can pass calculus!!! Thanks a lot How are you billant at maths??? I would like to know!!!! (Serious question)

@Mdead26 Жыл бұрын

Thank u sir U tought what 12 years of school didn't

@leonardobarrera2816 Жыл бұрын

@@Mdead26 I don’t understand you =(

@knochentrocken96 2 жыл бұрын

Just here to mention that the square root function is a function whose slope gets lower lower and (albeit always bigger than 0), so this means, this method works incredibly well for larger x

@Graknorke 2 жыл бұрын

you're also less likely to be near a square root at higher values though

@trangium 2 жыл бұрын

@@Graknorke even the worst case gets better and better

@salihoyundaaa2014 2 жыл бұрын

Aaaa seni biyerde daha görmüştümm

@Graknorke 2 жыл бұрын

@@trangium fair enough I didn't actually work it out, just pointing out it's a bit more complex than just being more accurate because the second derivative gets smaller at higher x

@piratebs 2 жыл бұрын

Yep also 2nd derivative is always negative (when x>0) as well so every time you use linearization for sqrtx it’s gonna be an overestimation

@33_minhtai_b93 2 жыл бұрын

The way he switches his markers is so smooth

@thaibul1580 2 жыл бұрын

yea man his writing while holding 2 markers is even better than mine writing normally

@Willy_Wanka 2 жыл бұрын

Focus on his points not how he writes. Or can't you understand the content

@William_Webber 2 жыл бұрын

i didn’t even spot that he was changing markers

@fatsquirrel75 2 жыл бұрын

Should name the channel after the pens. 😄

@TheWtfareyoulooking 2 жыл бұрын

@@fatsquirrel75 Naa there's already another youtuber that did that already

@anshumanagrawal346 2 жыл бұрын

It's worth mentioning here that the two methods are completely equivalent

@learpcss9569 2 жыл бұрын

looking at the answers, it seems like we could say so, but still I don't see why these methods are equivalent

@anshumanagrawal346 2 жыл бұрын

@@learpcss9569 Because the method of differentials isn't really rigorous, and to make it rigorous is really complicated, if you look closely, all we're claiming in the method is, in a sufficiently small neighborhood, delta(y)/delta(x)≈ dy/dx at that point. And that's the same thing as saying the curve is close to it's tangent near that point

@OnFireByte 2 жыл бұрын

@@learpcss9569 it's come form linear equation itself Y2-Y1 = m(X2-X1) Y2 = Y1 + m(X2-X1) We know that Y1 = f(x), m = f`(x), and X2-X1 = dx Hence Y2 = f(x) + f`(x)dx

@volodymyrgandzhuk361 2 жыл бұрын

Yes. And the actual value of the square root will always be smaller than the one they give.

@ladyravendale1 2 жыл бұрын

@@volodymyrgandzhuk361 True, but it depends on the slope. When using linear approximation, whether the approximate value is over or under the true value depends on the slope of the function. In this case, because the second derivative of Sqrt(x) is always positive, that means it will always be below the tangent line, which is equivalent to saying the estimate will always be bigger. Given a function with a second derivative that is always positive, like e^x, the tangent line will always be below the function, and the result will always be lower.

@alberteinstein3612 2 жыл бұрын

This differential method was very interesting to see. I’ve never seen a differential used like this before, thanks for showing this to us!!!

@Invisible12345ful 2 жыл бұрын

I'm pretty sure you already knew this stuff, who are you pranking?

@AlexEEZ 2 жыл бұрын

ok "albert einstein" sure you haven't (ꐦ○_○）

@skylardeslypere9909 2 жыл бұрын

If you work out the differential method for a general function y=f(x), you eventually get the exact same result, namely that f(x) ≈ f(x*)+f'(x*)(x-x*)

@nathanielnotbandy991 2 жыл бұрын

What do the asterisks mean?

@slender1892 2 жыл бұрын

@@nathanielnotbandy991 It is the point around which you consider the derivative, and make a linear approximation. To go a bit more in depth, it also is basically taking the first term of the taylor series of the function.

@skylardeslypere9909 2 жыл бұрын

@@nathanielnotbandy991 i used x* to denote an arbitrary point, just like x_0. But x* was a bit less messy in my opinion

@NZ-fo8tp 2 жыл бұрын

Ahh what a lovely first order Taylor series expansion you got there

@skylardeslypere9909 2 жыл бұрын

@@NZ-fo8tp jep LOL. Every approximation formula is basically the same

@bollyfan1330 2 жыл бұрын

I used Algebra, which is much much easier 3^2 = 9 > 8.7 (3 - x)^2 = 8.7 3^2 - 2 * 3 * x + x^2 = 8.7 9 - 6 x + x^2 = 8.7 6 x - x^2 = 9 - 8.7 6 x - x^2 = 0.3 Since x is very small, x^2

@Ok-fu5yi 2 жыл бұрын

Try using this formula a^1/2 = (a+b)/(2*b^1/2). Where a is the number you want to square and b is the closest perfect square

@alial-khalili9232 2 жыл бұрын

@@Ok-fu5yi why does this work?

@Ok-fu5yi 2 жыл бұрын

@@alial-khalili9232 it assumes that the square root of ab is the same as the average of a and b ((a+b)/2). This is not true normally but becomes more accurate as a and b are approaching each other. That is why we need b to be the closest perfect square

@phiefer3 2 жыл бұрын

I wouldn't really call that using algebra. What you just did was basically calculus. The act of ignoring the x^2 term because it's "very small" is the basis of limits. And your whole method is literally just the differential method from the video just rearranged. The algebraic method, would be to do what you did, but then to NOT ignore the x^2 and instead solve the quadratic equation you came to, but that would have still left you with a radical in the solution.

@ScienceNerd3336 2 жыл бұрын

@@phiefer3 It's calculus, but it's calculus for babies. Lol.

@robertbach9376 2 жыл бұрын

I'm gonna put this in excel and iterate it quickly, that way I don't have to use a calculator

@liambecker558 2 жыл бұрын

Can we appreciate how smoothly he switches markers?

@sabyasachichoudhury2920 2 жыл бұрын

Wait. You could improve this method by using it recursively, right? So, for example, to find the root of 8.7, instead of just using the derivative at 9, you could use the derivative at 9 to find the root of 8.9. Then using use root 8.9, find root 8.8, and then finally reach 8.7. That should, in theory, provide a closer approximation.

@hockeypro3728 2 жыл бұрын

That is correct. Its the same as eulers method with a step size of -0.1 where you "track" the function with tangent lines.

@Schaex1 2 жыл бұрын

Exactly. This can be used for numerical solutions of differential equations, given a starting point. If I remember correctly this is called "Euler's Method".

@bprpcalculusbasics 2 жыл бұрын

Yes!

@trapccountant 2 жыл бұрын

man i love the maths community on youtube fr

@jorritmorrit 2 жыл бұрын

You should add the second derivative devided by 2 factorial times (X1-X2)^2 , the third derivative devided by three factorial (X1-X2)^3 etc. etc. Where the derivative is filled in like X1=9. And X2 being 8,7. Keep doing it and you will add up with square root of 8,7. It's called the Taylor series. In this video only the first derivative of the linerisation of the function is used. Which is enough most of the times.

@stapler942 2 жыл бұрын

Differentials are strange objects. In their original meaning they are infinitesimals, which mathematicians came to frown on. In modern mathematics they mean a lot of different things, including linear approximations. In first-year derivatives they pretend to be a ratio, but with limited algebraic properties. In integral notation they just kind of sit there as a reminder that we're dealing with limits, and otherwise just provide some mental shortcuts for manipulating differential equations. And then it becomes the Jacobian, and so on...

@sniperwolf50 2 жыл бұрын

Then comes vector calculus with gradient, divergences and curls, and now you have vectors made of differentials

@stapler942 2 жыл бұрын

@@sniperwolf50 Then tensors are like: we heard you like vectors so we put vectors in your vectors so you can map while you map...

@sniperwolf50 2 жыл бұрын

@@stapler942 Finally, Clifford algebra comes along saying: I've heard you like dimensions, so I brought all of them

@aesir1ases64 2 жыл бұрын

I have no clue what you are talking about lol

@logiciananimal 2 жыл бұрын

There's also some versions of nonstandard analysis where they are taken to be infinitesimals again.

@martinepstein9826 2 жыл бұрын

How I think about it: The derivative of sqrt(x) is 1/(2sqrt(x)) so the derivative at x = 9 is 1/6. So by definition of the derivative sqrt(9 + h) = sqrt(9) + h/6 + o(h) ~ 3 + h/6 Now plug in h = -0.3 to get sqrt(8.7) ~ 3 - 0.3/6 = 2.95

@anshumanagrawal346 2 жыл бұрын

That's actually a great way to think about it

@scj8863 2 жыл бұрын

That's the underlying theory behind the calculation of derivatives, but it's really unnecessary

@anshumanagrawal346 2 жыл бұрын

@@scj8863 Why is unnecessary, it's literally what approximation by differentials is saying

@martinepstein9826 2 жыл бұрын

@@scj8863 In my mind f'(x) is, by definition, the number satisfying f(x+h) = f(x) + f'(x)h + o(h). So it's all the other methods that would need to prove themselves "necessary".

@user-ne5ij8sw8d 2 жыл бұрын

√8.7 = 3√(1-0.3/9) ~ 3(1-1/2×0.3/9)=2.95

@kradius1 2 жыл бұрын

Blew my mind, love to see the many methods out there that are typically overlooked

@JesusMartinez-zu3xl 2 жыл бұрын

This is so awesome! Finished Calculus 1 and just love watching calculus videos now😅

@seapanda384 2 жыл бұрын

Thank you, I'm currently trying to learn calculus by myself and these videos help increase my understanding about the topic

@Its_Me17 2 жыл бұрын

Good luck

@HappyGardenOfLife Жыл бұрын

Watch professor leonard's calculus lectures. he has all his lectures for calc 1, 2 and 3 on youtube.

@jee2736 Жыл бұрын

In your country... at what age do they start teaching calculus? I'm indian and we start learning at 16 years of age... (My age was zero the day I was born... mentioning because in some countries the age is 1 when child is born)

@ygalel 2 жыл бұрын

I really loved this concept. I actually like to use quadratic approximation for one step closer.

@let716 2 жыл бұрын

i just learnt this in my calc class last week but you made me understand it way more, tqsm

@user-lm7yx7wj5l 2 жыл бұрын

You can take this approximation even further by using more and more terms of the taylor expansion of sqrt(x) around the point 9... Or by using itteration on the answer you got (newton's method exactly).

@davegoodo3603 2 жыл бұрын

Thank you! You have a very refreshing style. Using the differential was very insightful.

@shreenathkamble5862 2 жыл бұрын

Thanks for this intuitive explaination. Though I used numerical techniques numerous times, This is is a new perspective I learnt from you. Thanks again.

@theblackherald 2 жыл бұрын

Just to add that this is a straightforward extension with limits of the secant line method where m = (3 - 2)/(9 - 4). This method gives you the decent approximation sqrt(8.7) = 2.94

@GabrielLima-gh2we 2 жыл бұрын

very cool! I loved the video man, keep it up!

@rohaansahu2924 2 жыл бұрын

Such simple explanation.... Thank you so much

@rob876 2 жыл бұрын

What about using the second iteration to get an even more accurate approximation? The second iteration uses sqrt(8.7025) = 2.95 You're essentially using the Newton-Raphson method.

@carultch 2 жыл бұрын

Or the first order Taylor Series.

@aryaman4068 2 жыл бұрын

You just earn a new subscriber bro👍🏻

@gpn962 9 ай бұрын

It's great how you showcased both methods to the one problem in a single video. It would also be good to know whether the two methods are always interchangeable, or whether there are problems where only one of the two methods can be applied. Thanks!

@GRBtutorials 2 жыл бұрын

Yet another equivalent method is to use the first order Taylor series at 9: f(x) ≈ f(a) + f'(a)(x-a) sqrt(8.7) ≈ sqrt(9) + (8.7-9)/(2*sqrt(9)) = 3 - 0.3/6 = 3 - 0.05 = 2.95 It’s equivalent to the two methods shown in the video, but more direct.

@vanessamagnano6375 Жыл бұрын

I never thought to use local linearity (or even differentials) to approximate square roots before (outside of calculus class, that is). This is a very handy technique for those who have studied calculus but are, for whatever reason, solving arithmetic or algebra problems on a tight time constraint and without a calculator. Thank you so much for sharing!

@RealLifeArchitecture 2 жыл бұрын

watching this brings me right back to high school maths class; heart racing, clammy hands, rising sense of panic as I completely loose track of what is going on. I don’t know why KZfaq suggested this video, it took me over 20 years to make my peace with mathematics but now I’m an Architect. If this video make you panic don’t worry, it is possible to get on in the world without fully understanding calculus.

@ThePiMan0903 Жыл бұрын

Nice video just calculus!

@ronaldrosete4086 2 жыл бұрын

New channel? I'm not hesitating to subscribe.

@volodymyrgandzhuk361 2 жыл бұрын

Both of these methods give the same approximation, which can be expressed as √x≈(x+a²)/(2a), where x is the number whose square root we want to find and a² is the nearest perfect square to x (so a is an integer), x>0, a>0. Btw, the value that is obtained will always be greater than the actual value of the square root.

@fizixx 2 жыл бұрын

I've done bunches of these, and yes, it is very, very kewl!

@WMTeWu 2 жыл бұрын

It would be nice to specify error range, otherwise I can say sqrt(8.7) ~= 3.

@abdoonyt9049 6 ай бұрын

We can also approach this with the (a+b)² method, write the sqrt value as a sum/difference (depending on the closest square) of the decimal component (set as unknown value 'b') and a whole number component and square it, remove the b² because it is negligible (square of a decimal component ≈0). Solve for b and then substract/add with the whole number and you get the same value

@bobingstern4448 2 жыл бұрын

This is so cool!

@ThomasHaberkorn 2 жыл бұрын

diff variant is way more practicable - love it

@theguy1580 2 жыл бұрын

Man i wish you were my math teacher, really wanted a teacher who’d tell me why im learning something and to actually use what im learning

@m3rify 2 жыл бұрын

Loved it!

@omegahaxors3306 2 жыл бұрын

You can do something similar to do 6/7/8 and 9 during multiplication. Normally they're hard numbers to deal with by by splitting it into 5+1, 5+2, 10-1 and 10-2 you can solve them much quicker.

@KhenksMarr 2 жыл бұрын

Double marker use is so cool

@petereziagor4604 2 жыл бұрын

Nice video. Thank you 😊

@nasdpmlima6248 2 жыл бұрын

Thought you were holding your coffee for the first 5 mins 😂

@max-yasgur 2 жыл бұрын

It may be first order, it’s still really cool how close the approximation is. But I’m biased as an engineering student (approximation go brrrrr)

@swaroop529 2 жыл бұрын

That's pretty cool! My approach was 8.7 x 10 i.e. 870 which comes almost halfway between 841 (29^2) and 900 (30^2) Therefore its sqrt would be halfway between 29 & 30 i.e. 29.5; dividing by 10 we'd get 2.95 [sqrt 8.7]

@AK-pq3cw 2 жыл бұрын

Great video!

@jacobmacdonald223 2 жыл бұрын

This is freaking awesome

@jmadratz 2 жыл бұрын

Good job sir! Very nice tutorial for first or second year college students.

@sebasromero2505 2 жыл бұрын

I understood everything, good explanation

@tillybillyboyboy 2 жыл бұрын

Oh my goodness I love this content 💖💖💖

@atrumluminarium 2 жыл бұрын

There's another approximation (but very crude) for very large numbers that's used in computer science as a "first guess" before using Newton's method. Say we have an N digit number (call it A), then your first approximation for sqrt is the first N/2 digits (call it a) and then you average a and A/a

@bobh6728 2 жыл бұрын

In the time it took to do the long division of 8.7 by 6, you could have computed the first three digits by the digit by digit method.

@MridulSharma21 2 жыл бұрын

You can use binomial expansion as well.

@thanosaias2717 2 жыл бұрын

Does this work if the square root is not close to a rounded value like the square root of 9?

@zendruoflynstin8275 2 жыл бұрын

Just a question, why did you changed the general axis notations?

@HeyKevinYT 2 жыл бұрын

The differential method is taught in AP Calculus AB

@rocketeer9065 2 жыл бұрын

What about Taylor series to calculate this using derivatives?

@ddr3629 2 жыл бұрын

One can use Pade approximation for square root. rewrite sqrt(8.7)=3*sqrt(1-1/30) 1st term is the same as in Taylor series sqrt(1-x)~a1=1-0.5*x next use recurrence a_{n}=1-x/(1+a_{n-1}) so for x=1/30 a1=59/60 ; a2=117/119 let's try 2nd term sqrt(8.7)~3*117/119~2.94958 correct up to 5th sign

@U014B 2 жыл бұрын

Now can you do it via integration?

@LegacyLens343 2 жыл бұрын

That's greatly explained

@wesleyblack8302 2 жыл бұрын

What happened to the 3 in the tangent line step?

@mmh1922 2 жыл бұрын

Well done!

@beyondnature6511 2 жыл бұрын

Is there any other method available to calculate degree values at any point without calculator?. Kindly do video on this sir.

@isk3804 2 жыл бұрын

What if the given value was sqrt(6) ? It's not so close to 3, almost in the middle. How would you determine reference point to determine tangent line?

@siddharthkorukonda7662 2 жыл бұрын

I could be wrong but IVT is a possible way to solve this right?

@ian-hm6cx 2 жыл бұрын

L(x)=f(a)+f'(a)(x-a) it helps me to think about a as the number you're using to approximate and x as the actual value that you're finding

@faisalrio4433 2 жыл бұрын

I just wondering what if I have to find out an approximate root value with calculus where the nearest square number is far away. For instance if we have to approximate root over 598.6 where the nearest square number are not that close lowest square number 576 and highest square number is 625. Now my question is, will it yield as accurate results as it did for root over 8.7??

@dhrubajyotidaityari9240 2 жыл бұрын

Nice and beautiful idea

@mikevilters6237 2 жыл бұрын

Is the first method not basically taylor aproximation of the first order?

@shashankprasad4227 2 жыл бұрын

Thankyou!!

@qazizarifulislam6568 2 жыл бұрын

Important note. The reason the first Method worked was because 9 is a number that is very close to 8.7. so, in the square root curve, the slope at f(9) can be considered equal to the slope at f(8.7). PS im describing the square root curve as the function f(x) here.

@qazizarifulislam6568 2 жыл бұрын

So if you had a problem where you had to find sq root of 46, use the slope at 49 to approximate the answer at 46.

@chessematics 2 жыл бұрын

Whenever I have time I use Taylor Series, Geometric Series, etc to evaluate stuff.

@idjles 2 жыл бұрын

Yep. Sqrt(8.7)=sqrt(9-3/10)=3*(1-1/30)^0.5 and Taylor expand it to as many terms as you like.

@ikeetkroketjes8431 2 жыл бұрын

4:58 thats the piece im playing on my piano rn XD (entertainer, scott joplin )

@sebastiancabrera7986 2 жыл бұрын

I'm a spanish native speaker and your accent helps me to understand better English i mean it's more difficult to understand... so i keep learning

@templa6590 2 жыл бұрын

In Japan, people(especially, high school students in entrance exam) often use "Kaiheihou" to calculate squared numbers.

@templa6590 2 жыл бұрын

Sorry, not "to calculate squared numbers" but "to calculate the square root of numbers".

@abdulrhmanalsedairi8022 2 жыл бұрын

Great!! I think if we just use the lever rule, it would be much easier 👌🏻

@WizardOfArc 2 жыл бұрын

I like the differential approach

@spongebob3643 2 жыл бұрын

Can we determine how big is the error we have made by using this method?

@trainwreck8219 2 жыл бұрын

I love this guy's accent. And he's teaching amazingly well!

@pkphd 2 жыл бұрын

Numerical differentiation, similar to interpolation. Assuming slope in the vicinity of point approximately same.

@9machine4you 2 жыл бұрын

7:05 why can it only be the "new - original" and not the other way around?

@larry9210 2 жыл бұрын

Thanks!

@daquack._ 6 ай бұрын

I wasn't expecting to understand, but I understood. Dang.

@s888r 2 жыл бұрын

There's an other way to approximate square roots, more accurate for larger numbers: 4 is the square of 2, 9 is the square of 3. 9 - 4 = 5, 3 - 2 = 1 8.7 - 4 = 4.7 Let d be the difference between √8.7 and √4. 4.7/5 is approximately equal to d/1. After calculation, we get d to be 0.94. 2 + 0.94 = 2.94 Therefore, √8.7 ~ 2.94. Basically, the two square numbers, between which is the number (let this be x) whose square root is to be found, are taken, with their square roots, which are integers. (x - Smaller square number/The difference between the two square numbers) is approximately equal to (√x - Smaller square root/The difference between the square roots). Eg: Let x be 3. 1 and 4 are the required square numbers, and 1 and 2 are their square roots. According to the formula, (3 - 1/4 - 1) ~ (√3 - 1/2 - 1) -> (2/3) ~ (√3 - 1/1) -> √3 - 1 ~ 0.666... -> √3 ~ 1.666... (~1.732) As I said, this method is more accurate for larger numbers.

@kozokosa9289 2 жыл бұрын

tbh I think that the simple (a-b)^2 = a^2-2ab+b^2 approach is a bit easier, a^2 being 9, so a is 3, and you have b(a-2b) = 0.3, a being 3 means that so 6b-b^2 = 0.3, and you can say that b^2 is very small, so 6b(slightly b= 0.5- change. this isn't a calculus approach but more of an algebric one, another way is 2.9^2 is 841, and then by (a+b)^2 you can to a bit of trial and error and come around to approximately 2.94 (2.95 ^ is 8.7025 I think).

@binaryparrot3352 2 жыл бұрын

Could you also use a Taylor series to approximate 8.7? taking 9 as the center.

@rodrigoelias1987 2 жыл бұрын

Why does the derivative of sqrt(x) is 1/2sqrt(x) ? I was thinking that derivatives was the exponent times x, what should be x/2

@seanfaherty 2 жыл бұрын

That is a fun way to do it . You ever hear about Ed Marlo ?

@FinalMiro 2 жыл бұрын

I love watching stuff that I can understand at the beginning, and that becomes super hard after 30 seconds LOL

@rhoddryice5412 2 жыл бұрын

Calculating an approximation without pen and paper on my head 3^2-2*3*x+x^2 = 8.7, 3^2>>x^2, 2*3*x = 0.3, x=0.5, sqrt(8.7) ~2.95

@namanshah2990 Жыл бұрын

sir mad respect

@rasyidmystery6891 2 жыл бұрын

in elementary school i learn to approximate square root using linear interpolation, even without knowing what it is called. in this case 8.7 is between 4 and 9 so the square root has to be more than 2, for the decimal it is (8.7-4)/(9-4) which is equal to 0.94. combine both and you will get 2.94. the problem with this method is that the estimate will always be underestimated because instead of using the tangent line at x=9 this method use the line that cross the curve at (4,2) and (9,3), but it certainly easier cuz even an elementary student can understand them.

@ahnafsakib 2 жыл бұрын

The differential method is pure beauty

@nds2k1409 2 жыл бұрын

I’m not even taking Math right now but I was intrigued to watch lol

@dimitris17 2 жыл бұрын

If x is a number with known root and y a number with irrational root we can assume that sqrt(y)=(x+y)/2sqrt(x)

@GeeztJeez 2 жыл бұрын

It's kinda hard for me to understand at first but now I get it, still, great methods, I guess it's useful to explain such mathematics question on some of exercise our mad math teacher one gave us lol

@Edsalahr 2 жыл бұрын

this is just beautiful

@rioferzz 2 жыл бұрын

love this

@itsmeagain1415 2 жыл бұрын

what if you take the two square roots before and after the number you are approximating, find the equation of the straight line of each tangent, and consider the two points of tangency and the point of their intersection, and then figure out the equation of a straight line that is equidistant from these three point, and use that new straight line for a much better approximation :D