8 Indefinite Integrals
8.1 Summary Applied of Derivatives
Applied derivatives illustrate how differentiation is used to solve real-world problems across engineering, economics, physics, and other fields. By examining rates of change, extrema, and concavity, derivatives provide tools for optimizing processes, predicting behavior, and supporting decision-making. Key concepts, descriptions, and applications are summarized in Table 8.1.
| KeyConcept | Description | ExampleApplication |
|---|---|---|
| Optimization | Find maxima or minima by solving \(f'(x)=0\) \ and checking \(f''(x)\) | Maximize profit: \(P'(x)=0\) |
| Rate of Change | Quantifies how one variable changes \ with respect to another | Velocity: \(v(t)=s'(t)\) |
| Critical Points | Points where \(f'(x)=0\) or undefined; \ used to find maxima, minima, or inflection points | Minimizing cost: \(C'(x)=0\); \ analyzing structure stress |
| Motion Analysis | Derivatives of position give \ velocity and acceleration | \(v(t)=s'(t)\), \(a(t)=s''(t)\) |