Average Rate of Change of a Quadratic equation but with variables

Question

I received a question that gives the standard form of a quadratic: $y=ax^2+bx+c$ and I am to prove that within the interval $r \le x \le s$, the average rate of change is: $a(s+r) + b$.

I was thinking of using this equation: $\frac{f(s)-f(r)}{s-r}$ and plug in $r$ and $s$ into the original equation to find $f(r)$ and $f(s)$. However, my math does not seem to be working out as I am not getting the correct AROC. Does anyone know the correct way to go about solving this problem?

Thank you!

Answer 1

Your equation is correct, with AROC you'll usually find a factor of $(s-r)$ in the numerator that will cancel with the denominator to get to the result you're looking for (and this becomes a very useful algebraic trick when you get to instantaneous rates of change):

$$ \text{AROC} = \frac{f(s)-f(r)}{s-r} $$ $$ = \frac{(as^2+bs+c)-(ar^2+br+c)}{s-r} $$ $$ = \frac{as^2+bs-ar^2-br}{s-r} $$ $$ = \frac{a(s^2-r^2)+b(s-r)}{s-r} $$ $$ = \frac{a(s-r)(s+r)+b(s-r)}{s-r} $$ $$ = \frac{(s-r)\left(a(s+r)+b\right)}{s-r} $$ $$ = a(s+r)+b, s\neq r .$$

Answer 2

Depending on the class, you may be being asked to prove that somewhere on the interval the average rate of change is the expression you gave. In a calculus class, you may be talking about the mean value theorem...proving that the average rateof change is equal to the average rate of change.