Find the average rate of change of g(x)=9x^4+2/x^3 on the interval [-4,4].
nice try, but
(2304 1/32) - (2303 31/32) = 2/32 = 1/16
To find the average rate of change of a function on an interval, we can use the formula:
Average rate of change = (f(b) - f(a)) / (b - a)
In this case, the function is g(x) = 9x^4 + 2/x^3, and the interval is [-4, 4]. Therefore, a = -4 and b = 4.
Step 1: Find g(-4)
To find g(-4), substitute -4 into the function:
g(-4) = 9(-4)^4 + 2/(-4)^3
= 9(256) + 1/(-64)
= 2304 - 1/64
= 2304 - 1/64
= 2304 - 0.015625
= 2303.984375
Step 2: Find g(4)
To find g(4), substitute 4 into the function:
g(4) = 9(4)^4 + 2/(4)^3
= 9(256) + 1/(64)
= 2304 + 1/64
= 2304 + 0.015625
= 2304.015625
Step 3: Calculate the average rate of change
Using the formula mentioned above,
Average rate of change = (g(4) - g(-4)) / (4 - (-4))
= (2304.015625 - 2303.984375) / (4 - (-4))
= 0.03125 / 8
= 0.00390625
Therefore, the average rate of change of g(x) = 9x^4 + 2/x^3 on the interval [-4, 4] is approximately 0.00390625.
To find the average rate of change of a function over an interval, follow these steps:
1. Calculate the function values at the endpoints of the interval.
Let's start with the function g(x) = 9x^4 + 2/x^3.
Evaluating g(x) at the left endpoint (x = -4):
g(-4) = 9(-4)^4 + 2/(-4)^3 = 9(256) - 2/(-64) = 2304 + 1/32 = 2304 + 0.03125 = 2304.03125.
Evaluating g(x) at the right endpoint (x = 4):
g(4) = 9(4)^4 + 2/(4)^3 = 9(256) + 2/(64) = 2304 + 0.03125 = 2304.03125.
So, g(-4) = 2304.03125 and g(4) = 2304.03125.
2. Calculate the difference between the function values at the endpoints.
In this case, the difference is: g(4) - g(-4) = 2304.03125 - 2304.03125 = 0.
3. Calculate the difference in x-values (the interval length).
In this case, the difference in x-values is: 4 - (-4) = 4 + 4 = 8.
4. Divide the difference in function values by the difference in x-values.
So, the average rate of change of g(x) on the interval [-4, 4] is: 0 / 8 = 0.
To find the average rate of change, we need to find the change in the value of the function divided by the change in x over the given interval [-4,4].
First, we find the value of the function at the endpoints of the interval:
g(-4) = 9(-4)^4 + 2/(-4)^3 = 9(256) - 2/64 = 2304 - 1/32 = 2303 31/32
g(4) = 9(4)^4 + 2/(4)^3 = 9(256) + 2/64 = 2304 + 1/32 = 2304 1/32
Next, we find the change in the value of the function over the interval:
Δg = g(4) - g(-4) = (2304 1/32) - (2303 31/32) = 1/32
Finally, we find the change in x over the interval:
Δx = 4 - (-4) = 8
The average rate of change is given by Δg/Δx:
Average rate of change = (1/32)/8 = 1/256
Apologies for the mistake. You are correct.
The correct calculation is:
(2304 1/32) - (2303 31/32) = 2304 - 2303 = 1
Therefore, the change in the value of the function over the interval is 1.
The average rate of change is given by Δg/Δx:
Average rate of change = (1)/(8) = 1/8