Linear Approximation | An Efficient Method to Estimate Function Values Near Known Points

how do you do linear approximation/tangent line approximation?

Linear approximation, also known as tangent line approximation, is a method used to approximate the value of a function at a certain point using the equation of the tangent line at that point

Linear approximation, also known as tangent line approximation, is a method used to approximate the value of a function at a certain point using the equation of the tangent line at that point. It is particularly useful when you need to estimate the value of a function near a known point without using complex calculations.

To perform linear approximation or tangent line approximation, follow these steps:

1. Choose a point on the function where you know the coordinates: (x₀, f(x₀)). This point will serve as the base for the approximation.

2. Find the derivative of the function, which represents the slope of the tangent line at any given point.

3. Evaluate the derivative at x₀ to find the slope of the tangent line at x₀, denoted by m.

4. Write the equation of the tangent line using the point-slope form, which is given by:

y – f(x₀) = m(x – x₀)

This equation represents the tangent line at x₀ with slope m.

5. Depending on the problem, you may need to simplify the equation further.

6. Now, substitute the x-value at which you want to approximate the function, denoted by x₁, into the equation obtained in step 4.

y = f(x₁) ≈ f(x₀) + m(x₁ – x₀)

This equation provides an approximation of the value of f(x) at x = x₁.

By using this method, you can estimate the value of the function at a particular point, even if the function itself is complex or involves difficult calculations. The accuracy of the approximation depends on how close x₁ is to x₀.

It’s worth noting that linear approximation is just an approximation and may introduce some error. However, for small changes in x or when x₁ is close to x₀, the error tends to be negligible.

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