Final Answer:
The slope of the secant line between x₁ = 8 and x₂ = 72 for the function f(x) = √x is 1/6.
Step-by-step explanation:
To find the slope of the secant line between two points, x₁ and x₂, on a function, we use the formula for slope: Δy/Δx, where Δy is the change in y-values and Δx is the change in x-values. For this function f(x) = √x, when x₁ = 8 and x₂ = 72, the corresponding y-values are f(8) = √8 and f(72) = √72. Therefore, the change in y-values (Δy) is √72 - √8 and the change in x-values (Δx) is 72 - 8 = 64.
Applying the formula for slope, Δy/Δx = (√72 - √8) / (72 - 8) = (√72 - √8) / 64 = (6√2 - 2√2) / 64 = 4√2 / 64 = √2 / 16 = 1/6.
This simplifies to 1/6 as the slope of the secant line between x₁ = 8 and x₂ = 72 for the function f(x) = √x. This value represents the average rate of change of the function between these two x-values, indicating the steepness of the secant line over this interval.