Question

Given a function f (x) = 2x^2 + 3, what is the average rate of change of f on the interval [2, 2 + h]

11
2h+8
2h^2 + 8h
2h^2 + 8h + 11

Answer 1

The average rate of change of the function f(x) = 2x² + 3 on the interval [2, 2 + h] is 2h + 8.

Step-by-step explanation:

To find the average rate of change of a function f(x) on an interval [a, b], we use the formula:

The average rate of change = ∂(∂f(x))/∂(∂x) = (f(b) - f(a)) / (b - a).

For the function f(x) = 2x² + 3 on the interval [2, 2 + h], we have:

• 
  
• a = 2
• 
  
• b = 2 + h
• 
  
• f(a) = f(2) = 2(2)² + 3 = 8 + 3 = 11
• 
  
• f(b) = f(2 + h) = 2(2 + h)² + 3 = 2(4 + 4h + h²) + 3 = 8 + 8h + 2h² + 3
• 
  
• f(b) - f(a) = (8 + 8h + 2h² + 3) - 11 = 2h² + 8h

Dividing this by (b - a) which is h, gives:

Average rate of change = (2h² + 8h) / h = 2h + 8. Therefore, the average rate of change of f on the interval [2, 2 + h] is 2h + 8.

Answer 2

The average rate of change of the function f(x) = 2x^2 + 3 on the interval [2, 2 + h] is 2h + 8.

Step-by-step explanation:

The student has asked to find the average rate of change of the function f(x) = 2x^2 + 3 on the interval [2, 2 + h]. We can find this rate by calculating the difference in function values at the ends of this interval, which are f(2 + h) and f(2), and then dividing by the length of the interval, h.

The function value at x = 2 + h is f(2 + h) = 2(2 + h)^2 + 3, which expands to 2(4 + 4h + h^2) + 3 = 8 + 8h + 2h^2 + 3. Simplifying, we obtain f(2 + h) = 2h^2 + 8h + 11. The function value at x = 2 is f(2) = 2(2)^2 + 3 = 8 + 3 = 11.

Now, the average rate of change is (f(2 + h) - f(2)) / h. Substituting the function values we found gives us ((2h^2 + 8h + 11) - 11) / h = (2h^2 + 8h) / h. We can cancel the h from the numerator and denominator to get 2h + 8, which is the correct expression for the average rate of change.