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 Calculus Calculus Math Forum

 May 20th, 2015, 09:30 AM #1 Member   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? 35.B 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)=g'(x)(x-a)+g(x) g'(x)=-4x/(2x^2+1)^2 g'(0)=0 g(0)=1 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? From, Tzad Last edited by skipjack; May 20th, 2015 at 12:57 PM. Tags approximation, clarification, linear Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post RGNIT Calculus 0 March 20th, 2014 11:21 PM Luckyy Calculus 2 February 24th, 2011 05:43 PM Oranges'n'Lemons Calculus 1 February 20th, 2011 09:31 PM shango Calculus 1 October 27th, 2009 02:45 PM vesparados150 Calculus 4 May 19th, 2009 12:04 PM

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