Final answer:
To estimate the derivative of the given function, f(x) = 2 - 8x, at x = 2, we can calculate the slope of the function's tangent line at that point. By choosing two points very close to each other on the curve of the function, we can approximate the derivative. In this case, the derivative of the function at x = 2 is approximately -16.
Step-by-step explanation:
To estimate the derivative of the given function, f(x) = 2 - 8x, at the point x = 2, we can use the concept of instantaneous rate of change. The derivative of a function represents the rate at which the function is changing at a specific point. To approximate the derivative, we can calculate the slope of the function's tangent line at x = 2.
The slope of a tangent line can be approximated by finding the slope between two points that are very close to each other on the curve. In this case, we can choose two points, x = 2 and x = 2 + δx, where δx is a very small number.
Using the equation for a straight line, y = mx + c, where m is the slope and c is the y-intercept, we can calculate the slope, or the derivative, of the function at x = 2.
Substituting x = 2 and δx = 0.001 into the equation y = 2 - 8x, we can find the approximate value of the function at two points that are very close together. Let's say the value of the function at x = 2 is y1 and the value at x = 2 + δx is y2. By calculating the slope between these two points, we can estimate the derivative of the function at x = 2.
For example, if y1 = 2 - 8(2) = -14 and y2 = 2 - 8(2 + 0.001) = -14.016, we can calculate the slope as (y2 - y1) / (2 + δx - 2) = (-14.016 - (-14)) / (2 + 0.001 - 2) ≈ -0.016 / 0.001 = -16.
Therefore, the derivative of the function f(x) = 2 - 8x at x = 2 is approximately -16.