Talk:Linear approximation

Clarification needed

• The crucial point of this approximation is that it is local around some chosen point. This does not become clear sufficienty.
• More or less, linear approximation make sense only for functions that are differentiable at the point of consideration. This should be made clear explicitly.
• The function $f(x)=ax+b$ isn't linear in the sense of the reference "linear transformation".

Hottiger 16:29, 15 April 2006 (UTC)

I did some changes. Feel free to clarify more where you think is not clear. Oleg Alexandrov (talk) 20:35, 15 April 2006 (UTC)

Merging with linearization

I disagree with a merger. Linearization is about much more than a linear approximation, it is the study of the differential equation obtained after linearization, and how it relates to the original equation. As such, linear approximation is an elementary calculus property, while linearization ends up a rather involved topic in theory of PDEs, and while the subjects of the articles are definitely related, I think they better be kept separate. Comments? Oleg Alexandrov (talk) 02:21, 17 September 2006 (UTC)

I don't see the difference. Can you give an example where replacing one term with the other would lead to an error? —Keenan Pepper 02:42, 17 September 2006 (UTC)

You linearize a differential equation, while you do a linear approximation to a function, as far as I am aware.

Ideally, linearization (of a differential equation) should be much expanded with results from stability theory, and that would be overkill for the linear approximation article I think.

I must confess I don't know the topic of linearization in good detail however. If you are willing (and able) to do the merger, then I would be for it. Oleg Alexandrov (talk) 03:48, 17 September 2006 (UTC)

Note also the linerization article on PlanetMath. Oleg Alexandrov (talk) 03:48, 17 September 2006 (UTC)

functions of 2 variables????

someone needs to add local linear approximation for functions of multivariables (2 vars and up)

Another form of the equation

I was taught a different equation of linear approximation: f(x+h)≈f(x)+hf'(x). Is it worth adding this to the article as well?Baggers89 (talk) 08:48, 16 May 2008 (UTC) That form is the same, if you let h equal x-a. —Preceding unsigned comment added by 99.255.57.120 (talk) 06:17, 1 October 2009 (UTC)

Remainder requires continuous second derivative

First few sentences under DEFINITION should read: Given a twice continuously differentiable function f of one real variable, Taylor's theorem for n=1 states that.... —Preceding unsigned comment added by 99.255.57.120 (talk) 05:59, 1 October 2009 (UTC)

Applications

There are several applications of linear approximations that could be mentioned here. The first to come in mind is the concept of marginal cost/revenue. 313.kohler (talk) 05:07, 31 January 2011 (UTC)

Why is the tangent line better than any other line?

Something extremely crucial is being left out in this article. I was browsing the main article on differentiation and it also seemed to sidestep the issue.

Take f(x) = x^2. Then yes, the tangent line L(x) = 2x - 1 is a good approximation for f(x) near the point (1,1). But so is the line L(x) = 3x - 2! That is, both lines have the property that "limit as h goes to zero of f(1+h) - L(1+h) is zero".

The fact that L(x) - f(x) approaches zero as x approaches 1 is NOT what makes the tangent line special among all other lines passing through the point (1,1). What makes it special is that, if you take a smaller and smaller viewing window, f(x) becomes "indistinguishable" from its tangent line, which is not the case with any other line. This is a statement about RATIOS of values, not DIFFERENCES of values. I really think someone with more of a real analysis background than me needs to address this. —Preceding unsigned comment added by 99.34.211.79 (talk) 01:27, 27 February 2011 (UTC)

Here is a rough explanation of what I mean. Take a viewing window x = 1-h .. 1+h, and y = (1-h)^2 .. (1+h)^2 for small h. No matter how small h is, you will always be able to "tell the difference" between f(x)=x^2 and the non-tangent line L(x)=3x-2. That's because the difference between f(1+h) and L(1+h) is "large" (i.e. clearly non-zero) compared to h.

Of course, this is not just a statement about the tangent LINE to f(x) at (1,1). Take any other differentiable function g(x) such that g(1) = f(1) and g'(1) = f'(1) (e.g. g(x) = (2/3)x^3 + 1/3). Then it is also the case that g(x) becomes "indistinguishable" from f(x) for smaller and smaller viewing windows. So, this article should contain some kind of theorem, or illustration of a theorem, along the lines of

"if f(x), g(x) are differentiable functions at a, then the limit as h --> 0 of

(f(a+h)-g(a+h))/h

goes to zero if and only f(a) = g(a), and f'(a) = g'(a)"

which is pretty straightforward to prove. —Preceding unsigned comment added by 99.34.211.79 (talk) 01:48, 27 February 2011 (UTC)

Not twice differentiable function

For not twice differentiable function, how to define linear approximation, without using Taylor's theorem? -- Doyoon1995 (talk) 17:43, 10 November 2015 (UTC)

Does the definition using Taylor's theorem with Peano remainder require the function to be twice differentiable? -- Doyoon1995 (talk) 18:11, 10 November 2015 (UTC)