For the function f(x) = 5/x+4 find the average rate of change of f from 1 to x:
f(1) = 5/1 + 4 = 9
f(x) = 5/x + 4
avg rate of change from x=1 to x
= (5/x + 4 - 9)/(x-1)
= (5/x - 5)/(x-1)
= (5 - 5x)/(x(x-1))
= 5(1-x)/(x(x-1))
= -5/x
What is the average rate of change of the function f(x) on the interval -5≤x≤−4?
To find the average rate of change of the function f(x) = 5/(x+4) from 1 to x, we need to calculate the difference in the function values at the two endpoints and divide it by the difference in the x-values.
Step 1: Calculate f(1)
Plugging x = 1 into the function, f(x), we get:
f(1) = 5/(1+4) = 5/5 = 1
Step 2: Calculate f(x)
The function itself is f(x) = 5/(x+4), so we don't need to make any calculations.
Step 3: Calculate the difference in function values
The difference in function values between f(1) and f(x) is f(x) - f(1).
So, f(x) - f(1) = 5/(x+4) - 1
Step 4: Calculate the difference in x-values
The difference in x-values between 1 and x is x - 1.
Step 5: Calculate the average rate of change
The average rate of change is the difference in function values divided by the difference in x-values.
So, the average rate of change of f from 1 to x is:
(f(x) - f(1)) / (x - 1) = (5/(x+4) - 1) / (x - 1)
To find the average rate of change of a function from one point to another, we need to calculate the difference in the function values at those points and divide it by the difference in the x-values.
In this case, we want to find the average rate of change of the function f(x) = 5/(x+4) from 1 to x.
Step 1: Find the value of f(x) at x.
To find the value of f(x) at any given x, substitute x into the function:
f(x) = 5/(x+4)
Step 2: Find the value of f(1).
Substitute x = 1 into the function:
f(1) = 5/(1+4)
Simplifying further, we have:
f(1) = 5/5
f(1) = 1
Step 3: Calculate the difference in function values.
The difference in function values is found by subtracting the value of f(1) from the value of f(x):
f(x) - f(1)
Step 4: Calculate the difference in x-values.
The difference in x-values is calculated by subtracting the starting x-value (1) from the ending x-value(x):
x - 1
Step 5: Calculate the average rate of change.
To find the average rate of change of f from 1 to x, we divide the difference in function values by the difference in x-values:
(f(x) - f(1))/(x - 1)
In this case, it will be:
(5/x+4 - 1)/(x - 1)
So, the average rate of change of f from 1 to x is (5/x+4 - 1)/(x - 1).