Wednesday, 10 September 2014

calculus - Mean value theorem with ln(x)



I understand how to do mean value theorum but I'm not sure how to apply it with ln(x).



f(x)=ln(x), [1,8]



How can I find a c that satisfies the conclusion of the Mean Value theorem by using ln(x)?



I know its f(b)f(a)ba, then take derivative and fill in the slope.




But how do I solve this with ln? I only did this with quadratic.


Answer



The mean value theorem states that if f(x) is continuous on an interval [a,b] and differentiable on (a,b), then there exists a c(a,b) such that f(c)=f(b)f(a)ba
In your case, the function f(x)=ln(x) is continuous on an interval [1,8] and differentiable on (1,8). The derivative of ln(x) is 1x in the interval (1,8). Hence, by mean value theorem, c(1,8) such that f(c)=1c=f(8)f(1)81=ln(8)ln(1)81=3ln(2)07=3ln(2)7
Hence, the desired point c is 73ln(2).


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