Question

5. Let f(x) = a. Find the average rate of change of f(x) over [-4,-2]. Show how you got your answer.

b. Find the average rate of change of f(x) over _a,b,a, b> 0. Use this formula to determine the average rate of change if b is always twice a.
c. An approximation to the instantaneous rate of change of f, accurate to 2 decimal places, at x = 2 can be found in two ways. Show the difference in approximations using each method.

Answer 1

The average rate of change of f(x) over the interval [-4,-2] is 0. If b is always twice a, then the average rate of change of f(x) over the interval [a,b] is a / (b - a). The instantaneous rate of change of f at x = 2 is 0.

Step-by-step explanation:

To find the average rate of change of f(x) over the interval [-4,-2], we need to calculate the difference in the values of f(x) at the endpoints and divide it by the difference in x-values. Since f(x) = a, the average rate of change is given by:

(f(-2) - f(-4)) / (-2 - (-4))

Since f(x) = a for all values of x, the numerator f(-2) - f(-4) will always be 0, regardless of the value of a. Therefore, the average rate of change of f(x) over the interval [-4,-2] is 0.

b. If b is always twice a, then f(x) = 2a. To calculate the average rate of change of f(x) over the interval [a,b], we use the same formula as in part a, but substitute f(x) = 2a:

(2a - a) / (b - a) = a / (b - a)

c. Since the function f(x) = a is constant, the rate of change of f(x) is always 0. Therefore, the instantaneous rate of change of f at x = 2 is also 0.