Final answer:
The average value of the function f(x)=ln(x) on the interval [1,16] is approximately calculated using the integral of ln(x) adjusted by the interval length, resulting in an average value of approximately 2.95.
Step-by-step explanation:
To find the average value of the function f(x)=lnx on the interval [1,16], we need to use the formula for the average value of a function over an interval [a,b]:
Average value = Ø(1/(b-a))×∫_a^b f(x)dx
For our function, a=1, b=16, and f(x)=ln(x). So we calculate:
Average value = (1/(16-1))×∫_1^16 ln(x)dx
To compute the integral, we use integration by parts or look up the antiderivative of ln(x), which is xln(x) - x. After applying the limits of integration, we find:
Average value = (1/15)×[(16ln(16)-16)-(1ln(1)-1)]
Since ln(1)=0, the expression simplifies to:
Average value = (1/15)×(16ln(16)-16)
Now, we use a calculator to find the numerical value of ln(16), which is approximately 2.77 (since 16 is 2 to the 4th power, and ln(2^4) = 4ln(2), and ln(2) is approximately 0.693147). Thus:
Average value = (1/15)×(16×2.77-16) = (1/15)×(44.32) ≈ 2.95
So the average value of f(x)=ln(x) on the interval [1,16] is approximately 2.95.