Consider the function 𝑓 of 𝑥 equals 𝑥 cubed minus 75𝑥 over 50. What is the average rate of change in 𝑓 of 𝑥 over the closed interval from two to four?
The average rate of change of a function is a measure of how much that function changed per unit on average over that interval. The formula that we use to find the average rate of change of a function 𝑓 of 𝑥 over the closed interval 𝑎 to 𝑏 is 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎. Now, we’re told that 𝑓 of 𝑥 in this case is 𝑥 cubed minus 75𝑥 over 50, and our closed interval is from two to four. So, we’re going to let 𝑎 be equal to two and 𝑏 be equal to four. And so, the average rate of change will be 𝑓 of four minus 𝑓 of two over four minus two.
Now, we’ll simplify that denominator to two. And our next job, of course, is to evaluate 𝑓 of four and 𝑓 of two. Now, of course, 𝑓 of four is found by replacing 𝑥 with four in our original function. It’s four cubed minus 75 times four over 50. Four cubed minus 75 times four is negative 236. So, we get negative 236 over 50, which can be simplified by dividing the numerator and the denominator of the fraction by two. And we get negative 118 over 25. Let’s repeat this process to find the value of 𝑓 of two. This time, we substitute 𝑥 with two, and we get two cubed minus 75 times two over 50. That’s negative 142 over 50, which simplifies to negative 71 over 25.
Our last job is to substitute these back into the average rate of change function. That’s negative 118 over 25 minus negative 71 over 25 over two. Now, of course, subtracting a negative is the same as adding a positive. And so, the numerator here becomes negative 118 over 25 plus 71 over 25. Now, since the denominator of this pair of fractions is equal, we simply add the numerators. That gives us negative 47 over 25. And so, we have negative 47 over 25 all over two. Dividing our fraction by two is the same as multiplying it by one-half. And so, we find the average rate of change is negative 47 over 25 times one-half, which is negative 47 over 50.
The average rate of change in 𝑓 of 𝑥 over the closed interval from two to four is negative 47 over 50.