Understanding Instantaneous Rate of Change: Calculating the Slope of Tangent Lines to Functions

Instantenous Rate of Change

The instantaneous rate of change measures the rate at which a function is changing at a specific point

The instantaneous rate of change measures the rate at which a function is changing at a specific point. It gives you the slope of the tangent line to the graph of the function at that point.

To find the instantaneous rate of change, you need to find the derivative of the function and evaluate it at the desired point. The derivative represents the rate of change of the function at any given point.

For example, let’s say we have the function f(x) = 3x^2 + 2x – 1. To find the instantaneous rate of change at a specific point, let’s say x = 2, we would follow these steps:

Step 1: Find the derivative of the function f(x). In this case, the derivative of f(x) is given by f'(x) = 6x + 2. The derivative represents the slope of the tangent line to the graph of f(x) at any given point.

Step 2: Evaluate the derivative at the desired point. Plug in x = 2 into the derivative: f'(2) = 6(2) + 2 = 12 + 2 = 14.

Therefore, the instantaneous rate of change of the function f(x) = 3x^2 + 2x – 1 at x = 2 is 14. This means that at the point (2, f(2)), the function is changing at a rate of 14 units per unit change in x.

It is important to note that the instantaneous rate of change can vary at different points along the function, and it can be positive, negative, or zero depending on the function and the point in question. The derivative provides a powerful tool for calculating the instantaneous rate of change and analyzing the behavior of functions.

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