Approximation for Definite Integrals Question on a Table

Question

Question:

Let function f be continuous and decreasing, with values as shown in the table below

x: 2.5 3.2 3.5 4.0 4.6 5.2

f(x): 7.6 5.2 4.5 4.0 2.6 1.6

(c) Find the average rate of change of the function on the interval [2.5, 5.2].

(d) Find an approximation for the instantaneous rate of change of the function at x = 4.6

Can someone help me with these 2 questions based on the graph? Thank you

Answer 1

Average rate of change = [f(5.2) - f(2.5)] / [5.2 - 2.5] = -6 / 2.7 = -2.222... = -20/9

Instantaneous rate of change at x = 4.6 is the slope of the tangent at that point, f'(4.6). We can approximate the slope of the tangent by using the slope of the secant line at nearby points.

f'(4.6) ≈ [f(5.2) - f(4.0)] / [5.2 - 4.0] = [1.6 - 4.0] / 1.2 = -2.4 / 1.2 = -2

Notice that the slope of the secant is the same as the average rate of change on the interval [4.0, 5.2].