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Linear approximation Find the linear approximation to the following functions at the given point a.$$f(x)=4 x^{2}+x ; a=1$$
$\quad L(x)=9 x-4$
Calculus 1 / AB
Chapter 4
Applications of the Derivative
Section 6
Linear Approximation and Differentials
Derivatives
Differentiation
Campbell University
Oregon State University
University of Michigan - Ann Arbor
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in order to solve this problem, all we need to do is we take our affects, derive it or sorry, take the derivative of it to find f prime of X. Substituting are known value or substitute in one for F of X and F prime of X and put plugs at into our elevates. And that will be Mel B. It. So let's start with finding F prime affects or the derivative of F X, using a simple power rule in the four X squares when to give us eight X and then the derivative of a single X is just going to be one so that so f prime of X is just eight x plus more. Now let's substitute a into both of these equations. Death of A or F of one, since a Zeke with one is going to be four times one squared plus one. My goal small plus one, which is just five one square, is 11 Times Force for four plus one is five. Now. A prime of one is is equal to eight times one plus one, which I think most of you can probably just disease. Obviously not now. We have all the needed information to find our linear approximation. So we're going to find Ella backs. And so all of X equals f of A or F of one you already know F one. So let's just put in five plus f prime of A or F one with Dessau villages into five. Close nine and nine is being multiplied by X minus a, which in this case is one. So it's X minus one. Distributing out the nine by the X minus one. We're going to get l've X equals five plus nine x minus nine, which is going to simplify down pre simply and give us a final elevates to be nine x well nine x minus four and that is our final answer.
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