Video: Finding the Average Value of a Reciprocal Function in a Given Interval

Find the average value of 𝑓(𝑥) = − 1/5𝑥 on the interval [−5, −1].

02:21

Video Transcript

Find the average value of the function 𝑓 of 𝑥 is equal to negative one divided by five 𝑥 on the closed interval from negative five to negative one.

The question wants us to find the average value of our function 𝑓 of 𝑥 on the closed interval from negative five to negative one. And we recall for a function 𝑓 of 𝑥 which is continuous on the closed interval from 𝑎 to 𝑏, the average value of 𝑓 of 𝑥 on this interval is given by one divided by 𝑏 minus 𝑎 multiplied by the integral from 𝑎 to 𝑏 of 𝑓 of 𝑥 with respect to 𝑥. And we see that our function 𝑓 of 𝑥 is a rational function. This means it must be continuous anywhere its denominator is not equal to zero. So, in particular, it must be continuous on the closed interval from negative five to negative one.

Since we want the average value on the closed interval from negative five to negative one, we’ll set 𝑎 equal to negative five and 𝑏 equal to negative one. So this gives us that the average value of our function 𝑓 of 𝑥 on the closed interval from negative five to negative one is. One divided by negative one minus negative five multiplied by the integral from negative five to negative one of negative one divided by five 𝑥 with respect to 𝑥. We can simplify negative one minus negative five to give us four. And we know, for any constant 𝑎, the integral of 𝑎 divided by 𝑥 with respect 𝑥 is equal to 𝑎 multiplied by the natural logarithm of the absolute value of 𝑥 plus the constant of integration 𝐶.

So we’ll just take the constant factor of negative one-fifth outside of our integral. This gives us negative the natural logarithm of the absolute value of 𝑥 divided by five evaluated at the limits of our integral, negative five and negative one. Evaluating this at the bounds of our integral gives us one-quarter multiplied by negative the natural logarithm of the absolute value of negative one divided by five plus the natural logarithm of the absolute value of negative five divided by five.

And We know the absolute value of negative one is equal to one and the natural logarithm of one is equal to zero. Then, since the absolute value of negative five is just equal to five, we can simplify our answer to the natural logarithm of five divided by 20.

Therefore, we’ve shown the average value of the function negative one divided by five 𝑥 on the closed interval from negative five to negative one is equal to the natural logarithm of five divided by 20.

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