13.7 Applications of Differentiation

13.7 Applications of Differentiation

Derivatives model marginal profit, velocity/acceleration, investment growth rates, and rates of change of stress.

Example 12 (Marginal profit)

$P(x) = 100x - 5x^2$: $P'(x) = 100 - 10x$; $P'(8) = 20$.

Example 13 (Particle motion)

$s(t) = 4t^3 - 3t^2 + 2t$:

• $v(t) = 12t^2 - 6t + 2$; $v(2) = 38$ m/s
• $a(t) = 24t - 6$; $a(2) = 42$ m/s$^2$

Example 14 (Investment)

$V(t) = 5000(1+0.04t)^2$: $V'(t) = 400(1+0.04t)$; $V'(10) = 560$ (Rs/year).

Example 15 (Structural stress)

$S(x) = 500x - 2x^3$: $S'(x) = 500 - 6x^2$; $S'(5) = 350$ Pa/m.