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Find the linear approximation to $f(x)$ at $x=x_{0}$ Graph the function and its linear approximation.$$f(x)=(x+1)^{1 / 3}, x_{0}=0$$

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

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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in problem two. We want to give that inner approximation of the function of X at X equals explode, which is here is zero. The first system is to get the differentiation off the function of X. When we differentiate dysfunction, we right the power third multiplied by the inner function X plus one and then we differentiate the inter function. The inner functional differentiation is one. Multiply it by own. The second step is to get if this off explode 30 equals we substitute here by X equals zero third multiplied by one equals one third. The third step is to get than in your ization it of x which equals if explode from zero plus f dash off extra note dash zero multiplied by x minus x a node X minus zero We can get f of zero by substituting in the main function by X equals zero equals zero plus one. So the bar off one third equals one plus f dash of zero. We've got evidence of zero from this step which is third multiplied by X and this is the notarization off all of X. That's a graph f of X, along with aerobics Mm, It's we have here. 1234 every year. 123 Let's start by f of X We got here every of X on here. Alot of X We got many points to draw F of X, for example. Let's put X equals zero we get every X equals one we what? Ever x equals one, for example. Mm old ah p one. We've got X equals to the ward off two thirds. It's about f of X equals minus one. A point before exclude way we bought minus one. We got X equals zero and get out of X. When we substitute by X equals zero, we got all of X equals one. They must be the same at X node. And when we put X equals one or let's put it three to get hell of X two and appointed before it snowed, for example, minus three toe. Get the love X equals zero. Now let's start by every weeks. Zero on one. The second point is one on two toe about off. One third is not 11 3rd not to search. 1 30 year equals 111.26 one 0.26 Almost here. And when we have X equals minus one have the point equals zero. It's something like that. Now let's grab all of X with another color. We have zero on one. The same point here. Three and two. We have a point here. Um, minus three and zero is minus three and zero. Here, you can find that this delay notarization is the tangent of every X at X unload, which is you.

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