Rates of change of sums

On the second homework assignment you were asked to compute the rate of change at x=2 for functions with rules x², x³, and then x² + x³. I hope you noticed that not only did the rates of change for the x² function and the x³ function add to give you the rate of change for the x² + x³ function, but, more importantly, that at every stage of the computation, you were adding the expressions from your earlier computations. The result is common sense: if you have two buckets of water, one leaking at two gallons per second and the other at 3 gallons per second, then water is accumulating on the floor at the rate of 5 gallons per second. Nothing strange here, you add the rates to get the total rate. In mathematical terms, if I is an interval of real numbers containing the real number a, and $f:I\rightarrow(-\infty,\infty)$ and $g:I\rightarrow(-\infty,\infty)$ then the function f+g is said to have domain I as well, with rule (f+g)(x) = f(x) +g(x). So the get the rate of change of f+g at x=a we compute as follows:

$$\begin{array} {rcl} {\displaystyle \frac{(f+g)(a+h)-(f+g)(a)... ...ystyle \frac{f(a+h)-f(a)}{h}+\frac{g(a+h)-g(a)}{h} }\end{array}$$

which tells us that if we can compute the rates of change of f and g separately, then they will add to give us the rate of change of f+g, just as we want.

If we combine this new insight with our power rule for finding rates of change we discover that we can compute rates of change for functions whose rules are polynomials. We are now really on our way. For example, if $g:(-\infty,\infty)\rightarrow(-\infty,\infty)$, g(x) = 5x⁴ -9x² + 2x + 3, then g'(a) = 5(4)a^4-1 + (-9)(2)a^2-1 + 2 + 0 = 20a³ - 18a+ 2. Note that the rate of change of a polynomial has a rule which is also a polynomial, but of one lower degree.