Question

Let f(x) = 6√x. What is the average rate of change of f over the interval from x = 3 to x = 4.5?

Answer 1

To find the average rate of change of f(x) = 6√x from x = 3 to x = 4.5, calculate the function values at these points, find their difference, and then divide by the difference in x values. The average rate of change is approximately 2.0161.

Step-by-step explanation:

To find the average rate of change of the function f(x) = 6√x over the interval from x = 3 to x = 4.5, we calculate the difference in the function values at these points and divide by the difference in x values. This gives:

Average rate of change = √[f(4.5) - f(3)] / [4.5 - 3]

First, calculate the function values:

• f(4.5) = 6√4.5
• f(3) = 6√3

Then, find the difference and average rate:

• Difference in function values = 6√4.5 - 6√3
• Average rate of change = (6√4.5 - 6√3) / (4.5 - 3)

After performing the calculations:

• f(4.5) ≈ 13.4164
• f(3) ≈ 10.3923
• Average rate of change ≈ (13.4164 - 10.3923) / 1.5
• Average rate of change ≈ 3.0241 / 1.5
• Average rate of change ≈ 2.0161

The average rate of change of f(x) = 6√x from x = 3 to x = 4.5 is approximately 2.0161.