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Find the linear approximation to $f(x)$ at $x=x_{0}$ Graph the function and its linear approximation.$$f(x)=\sin 3 x, x_{0}=0$$
$$3 x$$
Calculus 1 / AB
Chapter 3
Applications of Differentiation
Section 1
Linear Approximations and Newton's Method
Derivatives
Differentiation
Applications of the Derivative
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Okay, so two panel on the approximation for this, we're going to plug zero into this f of X function, so that'll be this part of our formula, So f of X of zero is going to be signed of three times zero. And this will just be zero then if they take the derivative so f prime of X. So it's gonna be the German tive of Sign the Reacts, which will then be three co sign of three X. And if we go ahead and plug zero and here, that will give us this value here. So when we do that, we get three co sign of three times zero so co sign of zero is one. So one times three will give us three. Now, if we go ahead and plug in these two values into our formula, you have l of X is approximately equal to effort vax, which is equal to so f prime of X sub zeros three and then times x minus X sub zero, which is zero which were given and then plus f of X of zero, which is up here just also zero. From here, we can just distribute so out of X is gonna be able to three X and this will be our linear approximation for the function sine three x at X equal to zero. So let's see what that looks like in a giraffe. So in a graph this ah sine wave, which is this soon you saw it'll wave, uh, is not perfect, um, graphing, but it's basically just a simple wave of the function sine three x Where, um, the tangent line off the graph at X equal to zero. So the equation of this line is three X, which we found here.
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