Given the function 𝑔 of 𝑥 is equal to two 𝑥 cubed over 𝑥 squared, find the average rate of change on the closed interval one to five.
We might recall that given a continuous function 𝑓, we find the average rate of change on the closed interval from 𝑎 to 𝑏 by calculating 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎. Now, of course our function is 𝑔 of 𝑥. So it would be 𝑔 of 𝑏 minus 𝑔 of 𝑎 over 𝑏 minus 𝑎. But of course we said the function needed to be continuous. So let’s check for continuity of our function 𝑔 of 𝑥. 𝑔 of 𝑥 is a quotient. It’s a fraction of two polynomials, in fact monomials, which are single-termed polynomial functions.
Now, we know polynomials themselves are continuous over their entire domain. And we know that if we divide two continuous functions, we can only preserve continuity if the denominator is not equal to zero. So 𝑔 of 𝑥 will be continuous except where 𝑥 squared is equal to zero. Now, if 𝑥 squared is equal to zero, we can say that 𝑥 itself must be equal to zero. But our closed interval is the interval from one to five. And so 𝑥 is equal to zero is outside of this interval. And we could say that 𝑔 of 𝑥 must be continuous on the interval we need.
This means then that the average rate of change will be 𝑔 of five minus 𝑔 of one over five minus one. Remember, 𝑎 is the lower limit of our interval and 𝑏 will be the upper limit. So let’s calculate 𝑔 of five and 𝑔 of one. Since our function is given by two 𝑥 cubed over 𝑥 squared, 𝑔 of five is two times five cubed divided by five squared.
Now, in fact, we can simplify by dividing through by five squared. And so 𝑔 of five is two times five, which is just equal to 10. We can do a similar process with 𝑔 of one. We get two times one cubed over one squared. And then when we divide through by one squared, we get two times one, which is just two. Five minus one is equal to four. And so the average rate of change becomes 10 minus two over four, but 10 minus two is eight. So we get eight over four, which is simply equal to two.
The average rate of change of our function 𝑔 of 𝑥 on the closed interval from one to five is two.