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Find the linear approximation to $f(x)$ at $x=x_{0}$ Graph the function and its linear approximation.$$f(x)=\sqrt{2 x+9}, x_{0}=0$$

$$3+\frac{1}{3} x$$

Calculus 1 / AB

Chapter 3

Applications of Differentiation

Section 1

Linear Approximations and Newton's Method

Derivatives

Differentiation

Applications of the Derivative

Campbell University

University of Nottingham

Idaho State University

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Okay, So the first thing that we need to do to solve their linear approximation is to find f, uh, x sub zero. So to get that would just plug in zero into this function. So we with that we will get to time zero plus mine, and that will give us three. So then, from there, we need to find the derivative of the function of So what? That would be news, the change, or when you use a channel, um, we will get that the derivatives gonna be won over the square of two X plus nine. Okay, so then, from there we want also plug in Xerox or when a pawn zero into the derivative. So it will be one over slurred of two time zero plus nine. And with that, we'll get 1/3. Okay, some here. We just need a plug in. Now what we have found in toil in your approximation formula so we have l affects is approximately equal to half of axe is equal to so f prime of X subzero, which is 1/3 then times X minus X here, which is zero and then plus f of X subzero, which is three. So here, if we just simplify a little bit, this is gonna be equal to 1/3 X plus three. So this is gonna be the equation for the linear approximation of the function square toe to X plus nine at X, equal to zero. So let's see what that looks like you grab for. Okay, so this is what it looks like. So it's the tangent line to this curve of this function squared of two X plus nine at X, equal to zero, um, where it touches the graph. So this is where the linear approximation takes place at X equal to zero.

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