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Calculus/Linear approximation

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Expert: - 3/25/2011


Question
Hi,

I'm trying to linear the following eq. about x=0 up to the 3rd term:

x / (L^2 - x^2)

I know the formula is:

f(x) = f(0) + f'(0)x

If I keep my original f(x) as it is, I wind up after the 3rd expansion at 1 / L^2, which I don't think is correct.  I THINK the correct answer should be x/L

Then I thought that I should approximate the original function as x / (L^2 - x ), for small x, but that doesn't seem to work.

Can you help?

Answer

Linearize
Questioner: Mike
Country: United States
Category: Calculus
Private: No
Subject: linearize a function
Question: Hi,

I'm trying to linear the following eq. about x=0 up to the 3rd term:

x / (L^2 - x^2)

I know the formula is:

f(x) = f(0) + f'(0)x

If I keep my original f(x) as it is, I wind up after the 3rd expansion at 1 / L^2, which I don't think is correct.  I THINK the correct answer should be x/L

Then I thought that I should approximate the original function as x / (L^2 - x ), for small x, but that doesn't seem to work.

Can you help?
.........................................
f(0) = 0, obviously.
       L^2 - x^2 - (-2x)
f'(x) = -------------------
         (L^2 - x^2)^2

       L^2 - x^2 + 2x
f'(x) = ----------------
        (L^2 - x^2)^2

Now

         L^2      1
f'(0) = ------ = -----
         L^4     L^2

So your 'linearization' would be:

f(x) ~~ 0 + x/L^2

(I attached a graph of these functions, with L = 3.)
.......................
I am not sure what you mean by 'up to the third term'.  But I can suggest that you might want to look into the Taylor(Maclaurin?) polynomial.  You can get it by either:

1. Doing long division:

L^2 - x^2 INTO x.  (Yes, this is allowed, even though your fourth grade teacher would throw up.)

2. Use a binomial expansion of:

(L^2 - x^2)^-1,  then times x.

Let me know what you did.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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