Chapter 2 - Derivatives - 2.7 Rates of Change in the Natural and Social Sciences - 2.7 Exercises - Page 180: 31

(a) $A(x)$ = $\frac{p(x)}{x}$ $A'(x)$ = $\frac{xp'(x)-p(x)}{x^{2}}$ $A'(x)$ $\gt$ $0$ then $A(x)$ is increasing; that is, the average productivity increases as the size of the workforce increases. (b) $p'(x)$ is greater than the average productivity then $p'(x)$ $\gt$ $A(x)$ $p'(x)$ $\gt$ $\frac{p(x)}{x}$ $xp'(x)$ $\gt$ $p(x)$ $xp'(x)-p(x)$ $\gt$ $0$ $\frac{xp'(x)-p(x)}{x^{2}}$ $\gt$ $0$ $A'(x)$ $\gt$ $0$

(a) $A(x)$ = $\frac{p(x)}{x}$ $A'(x)$ = $\frac{xp'(x)-p(x)}{x^{2}}$ $A'(x)$ $\gt$ $0$ then $A(x)$ is increasing; that is, the average productivity increases as the size of the workforce increases. (b) $p'(x)$ is greater than the average productivity then $p'(x)$ $\gt$ $A(x)$ $p'(x)$ $\gt$ $\frac{p(x)}{x}$ $xp'(x)$ $\gt$ $p(x)$ $xp'(x)-p(x)$ $\gt$ $0$ $\frac{xp'(x)-p(x)}{x^{2}}$ $\gt$ $0$ $A'(x)$ $\gt$ $0$