Local Linearization Question

[Pikachu!]

Hi! I feel like the error in this question that I'm making is absurdly basic, but I've been stuck on it for awhile now.

So my question is asking for the approximation of sqrt(103).

So, easy. Pick the closest number with an easy square root, that's 100 and it's square root of 10. Gives us the point (100,10)
Know the derivative of the square root function is 1/2(sqrt (x)). Sub in 100 here, get a slope for the question of 1/20. Great, we have all the components to create the equation of our line we will need to find our approximation.

So, to set this up like any other line, y-10=1/20(x-100) Thus y= 1/20(x-100)+10

So, y= (x/20)-5+10, right? Uh, wrong, apparently. Somehow, my book and solutions manual are both telling me that to find the approximation of the square root of 103, I should be doing this:


Then L(103) = 10 + 3/20

Why would it be 10 +3/20? I understand why we are entering 3 in to the "x" spot, because the value of our original (103) minus our close number (100) is 3. But why is it 10 + that? Am I making an easy mistake, or is my book and solutions guide simply wrong? (I really hesitate to say that, I usually assume I've made a basic error)

Thank you!

 

[tkhunny]

(103/20)-5+10 = 103/20 + 5 = 10.15

L(103) = 10 + 3/20 = 10.15

These are just slightly different implementations of the same idea. Your definition abandons the original curve and lives on the tangent line. The "L" definition tries to remind us that there is an original curve sitting around, somewhere.

I would do it this way:

\(\displaystyle y = \sqrt{x}\)

\(\displaystyle dy = \frac{dx}{2\cdot\sqrt{x}}\)

Then add those two expressions.

\(\displaystyle y + dy = \sqrt{x} + \frac{dx}{2\cdot\sqrt{x}}\)

That looks a little more like the "L" version, but it's a little different. Don't become discouraged by a slightly different appearance.

 

[Pikachu!]

(103/20)-5+10 = 103/20 + 5 = 10.15

L(103) = 10 + 3/20 = 10.15

But where did the negative 100 go? On all the other examples I did the number in that position worked out just fine with all of the other numbers.

EDIT: Ohhhhhhhh. Nevermind. Got it. Stock, easy mistake. I just don't know why they put 3 in for X instead of 103 then.