February 22nd, 2019, 02:28 PM  #1 
Newbie Joined: Feb 2019 From: East Lansing, MI Posts: 3 Thanks: 0  Linearization problem https://ibb.co/BcY1dxJ Find the best function f(x) and value a so that the linearization of f(x) at x=a can be used to estimate √16.2. Find the linearization of f(x) at a. Use the linear approximation from (b) to estimate √16.2. I think I can figure out the last two, but I'm not sure how to get the function. Last edited by skipjack; February 22nd, 2019 at 11:14 PM. 
February 22nd, 2019, 02:56 PM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 2,373 Thanks: 1276 
Taking the first 2 terms of the Taylor series of the square root function about $x_0$ we have $\sqrt{x} \approx \sqrt{x_0} + \dfrac{xx_0}{2\sqrt{x_0}}$ hopefully clearly to find $\sqrt{16.2}$ we'll choose $x_0 = 16$ as it is the closest perfect square. We then end up with $\sqrt{16.2} \approx \sqrt{16} + \dfrac{16.216}{2 \cdot 4} = $ $4 + 0.025 = 4.025$ which is a pretty good approximation. 
February 22nd, 2019, 03:20 PM  #3  
Newbie Joined: Feb 2019 From: East Lansing, MI Posts: 3 Thanks: 0  Quote:
 
February 22nd, 2019, 03:58 PM  #4  
Senior Member Joined: Sep 2016 From: USA Posts: 578 Thanks: 345 Math Focus: Dynamical systems, analytic function theory, numerics  Quote:
Here is a linear approximation which is better: $y = \sqrt{16.2}$. Here is another one which is less trivial: $y =\sqrt{16.2} + \frac{1}{2 \sqrt{16.2}}(x  \sqrt{16.2})$. Whoever assigned this question should be slapped.  

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