Question

How can linear approximation be used to approximate the change in \(y=f(x)\) given a change in \(x ?\)

Short answer

Answer: The equation used to approximate the change in y for a given change in x using linear approximation is ∆y = f'(x₀)∆x, where f'(x₀) is the derivative of the function evaluated at the point of interest x₀, and ∆x is the given change in x.

Step-by-step solution

Determine the function and the point of interest

First, we need to know the function \(f(x)\) and the point of interest \((x_0, y_0)\) around which we want to approximate the change in y. Since the exercise does not provide a specific function or point, we'll use a general function \(f(x)\) and a point \((x_0, f(x_0))\) for our calculations.

Calculate the derivative of the function at the point of interest

Next, we'll find the derivative \(f'(x)\), which gives us the slope of the tangent line at any point on the function. Then, we will evaluate the derivative at the point of interest \(x_0\) to find the slope of the tangent line at this specific point. \(f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} \)

Find the equation of the tangent line at the point of interest

Now having the coordinates of the point of interest \((x_0, f(x_0))\) and the slope of the tangent line at this point \(f'(x_0)\), we'll find the equation of the tangent line using the point-slope form: \(y - f(x_0) = f'(x_0)(x - x_0)\)

Rewrite the tangent line equation with the change in x notation

Given a change in x, \(\Delta x\), the new x value will be \(x_0 + \Delta x\). To approximate the corresponding change in y, we substitute \(x = x_0 + \Delta x\) into the tangent line equation: \(y - f(x_0) = f'(x_0)((x_0 + \Delta x) - x_0)\) This simplifies to: \(\Delta y = f'(x_0) \Delta x\) where \(\Delta y = y - f(x_0)\), represents the approximate change in y corresponding to the change in x.

Use the tangent line equation to approximate the change in y

Finally, use the tangent line equation \(\Delta y = f'(x_0) \Delta x\) to find the approximate change in y for a given change in x. It's important to note that this approximation is more accurate for smaller changes in x, as the tangent line is a better approximation of the function near the point of interest.