Let 𝑓 of 𝑥 be equal to negative three 𝑥 squared plus seven 𝑥 minus two. Compute the average rate of change function of 𝑓 as 𝑥 varies from 5 to 5.1.
Remember, we can find the average rate of change of a function 𝑦 equals 𝑓 of 𝑥 between the 𝑥-values of 𝑎 and 𝑏, by using the formula 𝑓 of 𝑏 minus 𝑓 of 𝑎 over 𝑏 minus 𝑎. And whilst this formula might look a little strange, it’s essentially just a rewrite of the slope function 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. In this question, 𝑓 of 𝑥 is equal to negative three 𝑥 squared plus seven 𝑥 minus two. And we’re looking to find the average rate of change of 𝑓 as it varies from 5 to 5.1. So we’ll let 𝑎 be equal to five and 𝑏 be equal to 5.1. We’ll begin then by calculating the value of 𝑓 of five and 𝑓 of 5.1. 𝑓 of five is negative three times five squared plus seven multiplied by five minus two. That’s negative 75 plus 35 minus two, which is negative 42.
Repeating this process for 𝑓 of 5.1. And we get negative three times 5.1 squared plus seven times 5.1 minus two, which is equal to negative 78.03 plus 35.7 minus two, which is negative 44.33. We’re now going to substitute this into the average rate of change function. We obtain 𝑓 of 5.1 minus 𝑓 of five to be negative 44.33 minus negative 42. And then this is over 𝑏 minus 𝑎, which is 5.1 minus five. When we subtract negative 42, that’s the same as adding 42. So we get negative 2.33 divided by 0.1. And of course, dividing by 0.1 is the same as multiplying by 10. So this becomes negative 23.3.
And we found that the average rate of change of our function 𝑓 as 𝑥 varies from five to 5.1 is negative 23.3.