SOLUTION:
Case: Equation of per capita concumption.
Required: To compare the rate of consumption in 2005 [as in C'(15)] and 2009 [as in C'(19)].
C'(t) represents the first derivative of the expression
First we differentiate the Per capita expression
[tex]\begin{gathered} C(t)=-0.0032t^3+0.099t^2-\text{ 0.351t -12.1} \\ \text{Differentiating} \\ C^{\prime}(t)=-3(0.0032)t^2+2(0.099)t^{}-\text{ 0.351} \\ \text{Then we find C'(15) and C'(19)} \\ \text{getting C'(15)} \\ C^{\prime}(15)=-3(0.0032)(15)^2+2(0.099)(15)^{}-\text{ 0.351} \\ C^{\prime}(15)=0.459 \\ \text{getting C'(19)} \\ C^{\prime}(19)=-3(0.0032)(19)^2+2(0.099)(19)^{}-\text{ 0.351} \\ C^{\prime}(19)=-0.0546 \end{gathered}[/tex]
Final answer:
C'(15) = 0.459
C'(19) = -0.0546
From observation, the rate of change was more rapid in 2005 than in 2009
