A function is given below. Determine the average rate of change of the function between

t = 9
and
t = 9 + h.
f(t) =

Square root of 4t

f(t) = √4t

f(t+h) = √4(t+h)

f(9) = √36 = 6
√(9+h) = √(36t+4h)

avg rate of change is (√(36t+4h) - 6)/h

Well, well, well, we're talking about some math here! To find the average rate of change of the function between t = 9 and t = 9 + h, we need to calculate the difference in the function values at those two points and divide it by the difference in the input values.

Let's start by plugging in t = 9 into the function f(t):

f(9) = √(4 * 9) = √36 = 6.

Now, let's move on to evaluating the function at t = 9 + h:

f(9 + h) = √(4 * (9 + h)) = √(36 + 4h).

To find the average rate of change, we need to calculate the difference in the function values and divide it by the corresponding difference in input values:

Average rate of change = (f(9 + h) - f(9)) / ((9 + h) - 9).

Now, let's substitute the values we found earlier:

Average rate of change = ( (√(36 + 4h)) - 6 ) / h.

And there you go! That's the average rate of change of the function between t = 9 and t = 9 + h, considering the square root of 4t.

To determine the average rate of change of the function between t = 9 and t = 9 + h, we need to find the difference between the function values at these two points and divide it by h.

Let's calculate it step by step:

Step 1: Substitute t = 9 into the function:
f(9) = √(4 * 9)

Step 2: Simplify the expression:
f(9) = √36
= 6

Step 3: Substitute t = 9 + h into the function:
f(9 + h) = √(4 * (9 + h))

Step 4: Simplify the expression:
f(9 + h) = √(36 + 4h)

Step 5: Calculate the difference in function values:
f(9 + h) - f(9) = √(36 + 4h) - 6

Step 6: Divide the difference by h:
Average rate of change = (f(9 + h) - f(9)) / h
= (√(36 + 4h) - 6) / h

Therefore, the average rate of change of the function between t = 9 and t = 9 + h is (√(36 + 4h) - 6) / h.

To determine the average rate of change of the function between t = 9 and t = 9 + h, we need to calculate the difference in the function's values at these two points and divide it by the difference in the input values.

Step 1: Find the value of the function at t = 9.
To do this, substitute t = 9 into the given function:
f(9) = √(4 * 9) = √36 = 6

Step 2: Find the value of the function at t = 9 + h.
Substitute t = 9 + h into the function:
f(9 + h) = √(4 * (9 + h)) = √(36 + 4h)

Step 3: Calculate the difference in the function's values.
Subtract the value of the function at t = 9 from the value at t = 9 + h:
f(9 + h) - f(9) = √(36 + 4h) - 6

Step 4: Calculate the difference in the input values.
The difference in the input values is t = 9 + h - t = 9 = h.

Step 5: Calculate the average rate of change.
Divide the difference in the function values by the difference in the input values to get the average rate of change:
Average rate of change = (f(9 + h) - f(9)) / (9 + h - 9) = (√(36 + 4h) - 6) / h

So, the average rate of change of the function between t = 9 and t = 9 + h is (√(36 + 4h) - 6) / h.