My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

LinkBack Thread Tools Display Modes
May 20th, 2015, 09:30 AM   #1
Joined: Sep 2013

Posts: 58
Thanks: 0

Linear Approximation Clarification

Hey Guys,

So I have 2 homework questions I didn't really understand. They are question 34 C and 35 B.

I attached the question and solution to the two questions below, but I didn't really understand the solution. Here: Gyazo - faf58e848ac1eef80ed3a0488741a11c.png

What I understood is:

34.C How I understood it is that the vertex of 1-x^2 at x=0 is lying directly on top of the function 1/(1+x^2). In essence, both functions intersect at (0,1). Hence, the slope of the tangent of the vertex 1-x^2 is the same as the slope of the tangent at 1/(1+x^2) (ie, 0). Is that true?

With local linearization, we've been creating lines tangent to a point on a function in which we use to estimate values around to it. The answer in A is linear, yet B is a quadratic, meaning the local linearization we've been looking at isn't so linear anymore. Is the idea behind this that this quadratic is a much better approximation of points close to x=0 (than a line)?

Also, why is it that if g(x)=f(x^2) than we can get a quadratic g(x) that can approximate f(x) as 1-2(x^2) just by subbing in (x^2)?

I did it by hand and got a very different answer with very different approximations





l(x)=0*(x-0)+1 --> l(x)=1 which is very different than the answer 1-2(x^2) if -0.25>x or x>0.25.

(But a good approximation for values very close around x=0) (Gyazo - 75becba1945f9fe397b8e33bd4df4d05.png)

Is the idea behind this question that the real linear approximation is just a horizontal line y=1 but if you didn't want to do all the work 1-x^2 (which is g(x)=f(x^2)) would be a reasonable estimate?


Last edited by skipjack; May 20th, 2015 at 12:57 PM.
Tzad is offline  

  My Math Forum > College Math Forum > Calculus

approximation, clarification, linear

Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
Linear approximation of Non linear system by Taylor series RGNIT Calculus 0 March 20th, 2014 11:21 PM
Need Help! Linear approximation! Luckyy Calculus 2 February 24th, 2011 05:43 PM
Derivatives: Linear Approximation Oranges'n'Lemons Calculus 1 February 20th, 2011 09:31 PM
Linear approximation shango Calculus 1 October 27th, 2009 02:45 PM
integration and linear approximation help vesparados150 Calculus 4 May 19th, 2009 12:04 PM

Copyright © 2019 My Math Forum. All rights reserved.