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## Multivariable calculus

### Course: Multivariable calculus>Unit 3

Lesson 2: Quadratic approximations

# Quadratic approximation formula, part 2

A continuation from the previous video, leading to the full formula for the quadratic approximation of a two-variable function. Created by Grant Sanderson.

## Want to join the conversation?

• Ok, now I have a serious question... which song did Grant start singing? :P
• he was just saying line things up right here it isn't a real song
(1 vote)
• okay i got a question at this point i am not sure if it is just me or what but i am starting to mix this up with the taylor series do these have some kind of connection? is this like some kind of multi dimensional equivalent of taylor series
• That is an exceptional point. This is in fact very closely related to the Taylor Series.
Just as functions can be multidimensional, so too can the Taylor Series.

How are they related?
Well, the Taylor Series is a means to represent some function (can be multidimensional) as a polynomial. I.e., of the form a+bx+cx^2+dx^3+...
Now, the Taylor Series can have infinite terms. The more terms the series has, the closer it is to the original function. But, if we cut the Taylor Series short, say, by only including the terms up to x^1, we have ourselves a linear approximation (or a local linearisation) of the function. However, if we include all the terms in the Taylor Series up to x^2, we have ourselves a quadratic approximation to the original function.

So, to summarise, *approximations are just the Taylor Series cut short.*
-If you cut it at x, you've got a linear approximation
-If you cut it at x^2, you've got a quadratic approximation
-If you cut it at x^3, you've got a cubic approximation
-If you cut it at x^4, you've got a quartic approximation
.
.
.
and so on.

Hope this helps.
• At , why we want the 2nd partial derivative? I feel dull.. but I miss the point.
• My understanding of this topic is still growing but I feel I have an understanding sufficient to give you the intuition for an appropriate answer to your question..what I'm trying to say is, I might be wrong but I think I'm right.

The reason you take the second derivative is because the second derivative tells you what direction the curve around that region is going to be i.e. either positive or negative (think back to single variable quadratic equations and what second derivatives do there - it tells you whether the point you're looking at is curving up or down (max or min point)). These second derivative terms acts as the control knobs that make your graph curve and therefore hug your original function more closely allowing for closer approximations. In other words, the second derivative is what turns the flat tangent plane into the curvy sheet that Grant showed at . If you're wondering why the first derivative wasn't used, its because the first derivative gives you the tangent line which is just a straight line which is what the linear terms already do. Quadratic terms are necessary for the second derivative to happen and its also what we want to make the second derivative a constant as mentioned at .

Hope that helps:)
• I think it's very similar to taylor expansion except it's multivariable.
• Why do we still need the linear part of the quadratic approximation function? Why can't we just throw it out and keep the ax^2 + bxy + c^2 term?
• wondering where the article on quadratic approximations would be found ? - mentioned at
• aren't you just doing taylor expansions except instead of approximating curves you're approximating surfaces?
• They both have some similarities, but we have to use slightly different methods because we are working in 3D.

But yes, the Taylor/Macluarin Expansion creates a quadratic that approximates a 2D graph and now we are creating a quadratic equation to approximate 3D graphs, so we have the same ideas in mind.

Hope this helps,
- Convenient Colleague
(1 vote)
• Why do they call it a quadratic approximation and not a local quadratic approximation, you are still talking about a specific point on the curve.
(1 vote)
• I think the linear approximation is only around that point, but the quadratic approximation is able to approximate more than just the point. It is still local but it covers a lot more area and therefore should be differentiated from the local aspect!
(1 vote)
• Like Dave already asked, is there a reason we started out trying to create this formula with the linear part of the quadratic approximation function? Why can't we just throw it out and keep the ax^2 + bxy + c^2 term?

That is what I would have expected, and Grant doesn't really explain why.

Thanks for your time. :)
(1 vote)
• In which practical problems would we use quadratic approximations?
(1 vote)