Average rate of change? How would I figure the following problem out?

Find the average rate of change of $g(x)=x^2+3x+7$ from $x=5$ to $x=9$

My thought is that I would plug in 5 and 9 for the x values to get the y values. And the use the slope formula $\frac{y_2-y_1}{x_2-x_1}$.
 A: Yes, your solution is correct.
Differentiating first is a detour.
A: The Fundamental Theorem of Calculus says that $\int_a^b f(t)\ dt=F(b)-F(a)$ where the rate of change of $F(t)$ with respect to $t$ is $f(t)$. The only thing that matters is where you start and where you end.  Slight observation: if you have to make sure if you apply this to something like average speed as long as you used distance traveled rather than distance from start to end because speed is always a non-negative absolute value of velocity. 
In the case of this question, this says the average rate of change of $g(x)$  (as opposed to the average change of $g$), is
$$\mathrm{Average\ rate\ of\ change} ={{g(b)-g(a)} \over {b-a}}={ {g(9)-g(5)}\over {9-5}}= {{9^2+3\cdot 9+7)-(5^2+3\cdot 5+7)}\over {9-5}}={{115-47}\over {47}}=17$$ and this is exactly what a precalculus student should do. 
A: The good thing here is that you have a quadratic equation, so differentiating w.r.t. $x$ , you get a linear equation...which makes it easy to find average rate of change.
Consider your equation $$g(x)=x^2+3x+7$$
differentiate it w.r.t. $x$ $$g'(x)=2x+3$$
The naive way of doing this question is finding $g'(5), g'(6),g'(7), g'(8),g'(9)$ and then finding its average. Alternatively, just find the average of $g'(5)$ and $g'(9)$. You will get 17.
