Question

what is the estimated value of f '(30), based on the values below? x f(x) 10 13 20 16 30 18 40 23 a. 0.2 b. 0.35 c. 0.5 d. 1.25

Answer 1

The estimated value of f '(30) is the average rate of change between the points (20, 16) and (40, 23), which is calculated to be 0.35.

Step-by-step explanation:

The student's question asks for the estimated value of f '(30), which suggests finding the derivative of f(x) at x = 30 based on the given discrete values. We use the nearby points to estimate the slope at x = 30. The points given are (20, 16) and (40, 23). To estimate the derivative, one can calculate the average rate of change between these points:

Δy / Δx = (f(40) - f(20)) / (40 - 20) = (23 - 16) / (40 - 20) = 7 / 20 = 0.35

Therefore, the estimated value of f '(30) is 0.35, which corresponds to option b.

Answer 2

The estimated value of f'(30), using the slope between the points (20, 16) and (40, 23), is 0.35, corresponding to option b.

Step-by-step explanation:

To approximate the value of f'(30) based on the given values, we can use the two points closest to x=30, which are (20, 16) and (40, 23).

The derivative at a certain point is essentially the slope of the tangent line to the curve at that point, which we can estimate using the slope of the secant line between these two points.

The formula for the slope is:

Slope (m) = (f(x₂) - f(x₁)) / (x₂ - x₁)

Therefore, the slope between the points (20, 16) and (40, 23) is:

m = (23 - 16) / (40 - 20) = 7 / 20 = 0.35

This means the estimated value of f'(30) is 0.35, which corresponds to option b.