Let f be the function defined by f(x) = x + ln x. What is the value of c for which the instantaneous rate of
change of f at x = c is the same as the average rate of change of f over [1,4]?
To find the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over the interval [1, 4], we need to follow these steps:
Step 1: Find the average rate of change of f over the interval [1, 4].
To calculate the average rate of change, we use the formula:
Average rate of change = (f(4) - f(1)) / (4 - 1)
Substituting f(x) = x + ln(x), we have:
Average rate of change = (4 + ln(4) - (1 + ln(1))) / (4 - 1)
Note that ln(1) = 0, so the expression simplifies to:
Average rate of change = (4 + ln(4)) / 3
Step 2: Find the derivative of f(x).
To find the instantaneous rate of change, we need the derivative of f(x).
Taking the derivative of f(x) = x + ln(x) with respect to x:
f'(x) = 1 + (1/x) = (x + 1) / x
Step 3: Set the derivative equal to the average rate of change and solve for c.
Setting f'(c) = Average rate of change:
(x + 1) / x = (4 + ln(4)) / 3
Cross-multiplying and solving for x:
3(x + 1) = x(4 + ln(4))
3x + 3 = 4x + x ln(4)
Rearranging the terms:
3 = x + x ln(4) - 4x
3 = -3x + x ln(4)
Combining like terms:
x ln(4) - 3x = 3
Factoring out x:
x(ln(4) - 3) = 3
Solving for x:
x = 3 / (ln(4) - 3)
Therefore, the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1, 4] is c = 3 / (ln(4) - 3).
To find the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4], we can follow these steps:
Step 1: Find the average rate of change of f over [1,4].
The average rate of change of f over [1,4] can be found using the formula:
Average rate of change = (f(4) - f(1)) / (4 - 1)
Substituting the values into the formula gives:
Average rate of change = (f(4) - f(1)) / 3
Step 2: Find f(4) and f(1).
Using the function f(x) = x + ln(x), we can evaluate f(4) and f(1) as follows:
f(4) = 4 + ln(4)
f(1) = 1 + ln(1)
Since ln(1) = 0, we have:
f(4) = 4 + ln(4)
f(1) = 1
Step 3: Substitute the values into the equation for the average rate of change:
Average rate of change = (f(4) - f(1)) / 3
Average rate of change = (4 + ln(4) - 1) / 3
Step 4: Find the derivative of f(x).
The derivative of f(x) = x + ln(x) can be found by differentiating each term separately. Using the chain rule for the second term gives:
f'(x) = 1 + (1/x)
Step 5: Set the derivative equal to the average rate of change and solve for c.
Setting f'(c) = (4 + ln(4) - 1) / 3, we have:
1 + (1/c) = (4 + ln(4) - 1) / 3
Simplifying the equation gives:
1/c = (4 + ln(4) - 1) / 3 - 1
1/c = (4 + ln(4) - 1 - 3) / 3
1/c = (4 + ln(4) - 4) / 3
1/c = ln(4) / 3
c = 3 / ln(4)
Therefore, the value of c for which the instantaneous rate of change of f at x = c is the same as the average rate of change of f over [1,4] is c = 3 / ln(4).
f'(x) = 1 + 1/x
so you want c such that
1 + 1/c = (f(4)-f(1))/(4-1)