AP Calculus derivatives?
Let f be the funtion defined by f(x) = x + lnx. 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]?
A. 0.456
B. 1.244
C. 2.164
D. 2.342
E. 2.452
Please explain how you arrived to the answer please! The first correct answer and explanation will receive ten points : ) Thanks!
Update:Thank you Gerry! I totally understood it, that was a big help.
I accidentally increased the question duration for a week, but you will get the 10 points as soon as its time to vote for best answer!
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f '(x) = (1 + (1/x)) - I will set this equal to the average rate of change later to determine what x is correct in this case
f(4) - f(1) = 4 -1 + ln(4) - ln(1) = 3 + ln(4) - ln(1) = 3 + ln(4/1) = 3 + ln(4) = 4.38629
(4.38629 / 3) = 1.46210
So the average rate of change is 1.46210.
(1 + (1/x)) = 1.46210...........(Instantaneous = average here)
(1/x) = 0.46210
x = (1 / 0.46210) = 2.164 <---------------- The answer is (C)
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