Question Video: Finding the Average Rate of Change of Polynomial Functions at a Point Mathematics • Higher Education

Video Transcript

Determine the average rate of change function 𝐴 of ℎ for 𝑓 of 𝑥 is equal to six 𝑥 squared minus three at 𝑥 is equal to one.

The average rate of change function 𝐴 of ℎ for a function 𝑓 of 𝑥 at 𝑥 is equal to 𝑥 one is given by 𝐴 of ℎ is equal to 𝑓 at 𝑥 is equal to 𝑥 one plus ℎ minus 𝑓 at 𝑥 is equal to 𝑥 one over ℎ, where ℎ is a small change in 𝑥. We’ve been given the function 𝑓 of 𝑥 is six 𝑥 squared minus three and asked to determine the rate of change function 𝐴 of ℎ at 𝑥 is equal to one. This means that in our average rate of change function 𝐴 of ℎ, 𝑥 one is equal to one. So, we have the function 𝑓 of 𝑥 is six 𝑥 squared minus three and 𝑥 one is equal to one. And our average rate of change function 𝐴 of ℎ is equal to 𝑓 at 𝑥 is equal to one plus ℎ minus 𝑓 at 𝑥 is equal to one all over ℎ.

Of course, we haven’t finished yet. We need to evaluate our function 𝑓 at 𝑥 is equal to one plus ℎ and 𝑓 at 𝑥 is equal to one. So, let’s do this. Replacing 𝑥 with one plus ℎ in our function 𝑓 gives us six times one plus ℎ squared minus three. That is six times one plus two ℎ plus ℎ squared all minus three, which is six plus 12ℎ plus six ℎ squared minus three. And collecting like terms and rearranging gives us six ℎ squared plus 12ℎ plus three.

Now, we evaluate 𝑓 at 𝑥 is equal to one. We have six times one squared minus three, which is six minus three, and that’s equal to three. We have 𝑓 at 𝑥 is equal to one plus ℎ is six ℎ squared plus 12ℎ plus three and 𝑓 at 𝑥 is equal to one is equal to three, which we can now substitute into our average rate of change function 𝐴 of ℎ. This gives us 𝐴 of ℎ is equal to six ℎ squared plus 12ℎ plus three minus three all over ℎ.

The positive three cancels with the negative three so that we have six ℎ squared plus 12ℎ over ℎ. We have a common factor of ℎ in the numerator. If we rewrite this as ℎ times six ℎ plus 12 over ℎ, we can cancel this ℎ in the numerator with the ℎ in the denominator, and we’re left with six ℎ plus 12.

The average rate of change function 𝐴 of ℎ for 𝑓 of 𝑥 is equal to six 𝑥 squared minus three at 𝑥 is equal to one is six ℎ plus 12.