Find the average rate of change of the function f(x) = 2x4 +
3x2 ¡ x + 2:
(a) on the interval [0; 3]
and
(b) on the interval [¡2; 0]
Any help would be great. i am so confused
f(x)=2x4+3x²-x+2
(please check, because one of the characters posted in the function is not legible).
The rate of change on the interval [a,b]
is given by the change in f(x) divided by the change in x, i.e.
average rate of change on the interval [a,b], Δf(x)/Δx
=(f(b)-f(a))/(b-a) where b≠a.
Give it a try and post your answer for checking if you wish.
It's the beginning of the term, and almost everything is hard at the beginning. Once you get the handle of it, it will be fun.
To find the average rate of change of a function on a given interval, you need to calculate the difference in the function values at the endpoints of the interval and divide it by the difference in the input values.
(a) Average rate of change on the interval [0, 3]:
To find the average rate of change of the function f(x) = 2x^4 + 3x^2 - x + 2 on the interval [0, 3], we need to evaluate the function at both endpoints and find the difference.
First, let's find f(0):
f(0) = 2(0)^4 + 3(0)^2 - 0 + 2 = 0 + 0 - 0 + 2 = 2
Next, let's find f(3):
f(3) = 2(3)^4 + 3(3)^2 - 3 + 2 = 2(81) + 3(9) - 3 + 2 = 162 + 27 - 3 + 2 = 188
Now we can calculate the average rate of change:
Average rate of change = (f(3) - f(0)) / (3 - 0) = (188 - 2) / 3 = 186 / 3 = 62
Therefore, the average rate of change of the function on the interval [0, 3] is 62.
(b) Average rate of change on the interval [-2, 0]:
To find the average rate of change of the function f(x) = 2x^4 + 3x^2 - x + 2 on the interval [-2, 0], we need to evaluate the function at both endpoints and find the difference.
First, let's find f(-2):
f(-2) = 2(-2)^4 + 3(-2)^2 - (-2) + 2 = 2(16) + 3(4) + 2 + 2 = 32 + 12 + 2 + 2 = 48
Next, let's find f(0) (which we already calculated in part a):
f(0) = 2(0)^4 + 3(0)^2 - 0 + 2 = 0 + 0 - 0 + 2 = 2
Now we can calculate the average rate of change:
Average rate of change = (f(0) - f(-2)) / (0 - (-2)) = (2 - 48) / (0 + 2) = (-46) / 2 = -23
Therefore, the average rate of change of the function on the interval [-2, 0] is -23.