# Local Linearization: Finding the Formula of a Graph

• UrbanXrisis
In summary, the conversation revolves around finding the formula for a given graph and determining where a line should go through. The individual asking the question received a different answer than what was provided in the book and is trying to figure out where they went wrong. They also applied a formula incorrectly and received help from another individual, eventually arriving at the correct answer.
UrbanXrisis

I got a different answer than what the book says...

so I need to find the formula of the graph.

$$H'(3)=f(3)=2$$
$$m=\frac{\Delta y}{\Delta x}$$

$$2=\frac{\Delta y}{x- \int^3_0 f(t)dt}$$
$$y=2(x+2)$$
$$y=2x+4$$

where did I go wrong?

Last edited by a moderator:
Through what point did you want your line to go through? It looks like you used (-2, 0)...

you mean I should do:
$$2=\frac{y-2}{x-3}$$
$$y=2x-4$$
??

what I did was...

$$H'(x)=\frac{y-\int_{-2}^yf(t)dt}{x-\int_0^xf(t)dt}$$

Why do you want your line to go through the point (3, 2)?

What you need to do is stop guessing and think it through. Working through a simpler problem might help.

What is the local linearization of the function f(x) = x2 near x = -1? First tell me what that means geometrically, then work out the answer algebraically.

f'(x)= (y2-y1) / (x2-x1)
-2= (y2-1) / (x+1)
y=-2x-1

I notice you didn't try a geometric explanation.

Anyways, that's exactly right. Now, why did you pick the point (x1, y1) = (-1, 1)? Apply the same reasoning to your problem.

## 1. What is local linearization?

Local linearization is a mathematical method used to approximate the behavior of a nonlinear function at a specific point by creating a linear approximation of the function at that point.

## 2. How is local linearization used to find the formula of a graph?

By using local linearization, we can find the slope and y-intercept of the linear approximation at a specific point on the graph. This information can then be used to create an equation in the form of y = mx + b, where m is the slope and b is the y-intercept.

## 3. Can local linearization be used to find the formula of any graph?

No, local linearization can only be used for functions that are differentiable at the specific point we are trying to approximate. This means that the function must have a well-defined slope at that point.

## 4. How accurate is local linearization in finding the formula of a graph?

The accuracy of local linearization depends on how closely the linear approximation matches the behavior of the actual function at the specific point. The closer the approximation, the more accurate the formula will be.

## 5. What are the limitations of using local linearization to find the formula of a graph?

Local linearization is only accurate at the specific point we are approximating and may not accurately represent the overall behavior of the function. Additionally, it can only be used for differentiable functions and may not work for all types of nonlinear functions.

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