Problem 2 - Calculus Early Transcendentals

Question

Explain why the slope of a secant line can be interpreted as an average rate of change.

Step-by-Step Solution

Answer

Answer: The slope of a secant line can be interpreted as an average rate of change because it represents the change in output (y-values) per unit change in input (x-values) between two distinct points on a curve. This change in output per unit change in input is essentially the average rate of change for the function between the two points.

Step 1: Define a secant line

A secant line is a straight line that intersects a curve at two distinct points, P and Q. In the context of a function, the two points represent two different inputs (x-values) and their respective outputs (y-values).

Step 2: Calculate the slope of the secant line

To calculate the slope of the secant line, we'll use the formula for the slope of a line, which is given by: slope = (change in y) / (change in x) or m = (y2 - y1) / (x2 - x1) Here, (x1, y1) and (x2, y2) are the coordinates of points P and Q, respectively.

Step 3: Relate the slope to the average rate of change

The slope of the secant line represents the change in y (output) per unit change in x (input) between the two points P and Q. This change in y per unit change in x is essentially the average rate of change for the function between the two points. To see this more clearly, let's denote the function as f(x), the input as x, and the output as y = f(x). Then, the average rate of change between points P and Q would be: average rate of change = (f(x2) - f(x1)) / (x2 - x1) This expression is the same as the formula for the slope of a secant line, which shows that the slope of a secant line can be interpreted as an average rate of change.

Step 4: Provide an example

To illustrate this concept, let's use a simple linear function: f(x) = 2x. We will choose two points on the function's graph: P(1, 2) and Q(3, 6). Now, let's compute the slope of the secant line and the average rate of change between these two points: slope = (y2 - y1) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2 average rate of change = (f(x2) - f(x1)) / (x2 - x1) = (6 - 2) / (3 - 1) = 4 / 2 = 2 As seen in this example, the slope of the secant line is the same as the average rate of change in the function between points P and Q.