How to find the linearization of a function?
Understand the Problem
The question is asking how to determine the linearization of a given function, which involves finding the linear approximation of the function at a specific point. This typically includes using the derivative of the function at that point to create a linear equation that closely resembles the function near that point.
Answer
$L(x) = f(a) + f'(a)(x - a)$
Answer for screen readers
The linearization of the function $f(x)$ at the point $a$ is given by:
$$ L(x) = f(a) + f'(a)(x - a) $$
Steps to Solve
- Define the function and point of interest
Assume we have a function $f(x)$ that we want to linearize around a point $a$.
- Find the derivative of the function
Calculate the derivative of the function $f'(x)$. This represents the slope of the function at any point $x$.
- Evaluate the function and its derivative at the point
Find the value of the function and its derivative at the point $a$.
- Evaluate the function: $f(a)$
- Evaluate the derivative: $f'(a)$
- Construct the linearization equation
The linearization of the function around the point $a$ can be expressed using the formula: $$ L(x) = f(a) + f'(a)(x - a) $$
- Simplify the linearization equation
Substitute the values obtained in step 3 into the linearization equation to obtain the final linear approximation, $L(x)$.
The linearization of the function $f(x)$ at the point $a$ is given by:
$$ L(x) = f(a) + f'(a)(x - a) $$
More Information
The linearization of a function provides a way to approximate the values of the function near a specific point using a tangent line. This is particularly useful when the function is complex or difficult to evaluate at certain points.
Tips
- Forgetting to evaluate both the function and its derivative at the point $a$.
- Not applying the linearization formula correctly by mixing up the terms.
- Using an incorrect derivative, which can lead to inaccurate approximations.
