7 Appllied of Derivatives
7.1 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 7.1.
| 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 | \(v(t)=s'(t)\), \(a(t)=s''(t)\) |
7.2 Indefinite Integrals
Indefinite integrals, or antiderivatives, undo the process of differentiation, enabling us to recover the original function from its derivative. They represent accumulated quantities such as displacement, total growth, or total charge. The Table 7.2 below summarizes key concepts, descriptions, and example applications see also [1], [2].
| KeyConcept | Description | ExampleApplication |
|---|---|---|
| Definition | \(F'(x) = f(x)\); \(\int f(x) dx = F(x) + C\) | Displacement: \(s(t) = \int v(t) dt\); e.g., \(v(t)=3t^2 \implies s(t)=t^3+C\) |
| Power Rule | \(\int x^n dx = \frac{x^{n+1}}{n+1}+C, n\neq -1\) | Integration of \(x^2\) gives \(\frac{x^3}{3}+C\) |
| Constant Multiple | \(\int c f(x) dx = c \int f(x) dx\) | Multiply constant with integral |
| Sum Rule | \(\int [f(x)+g(x)] dx = \int f(x) dx + \int g(x) dx\) | \(\int (x^2+2x) dx = \frac{x^3}{3}+x^2+C\) |
| Total Accumulation | Accumulation of quantity over time | Revenue: \(\int R(t) dt\) |
[1]
Stewart, J., Calculus: Early transcendentals, Cengage Learning, 2016
[2]
Apostol, T. M., Calculus, volume i: One-variable calculus with an introduction to linear algebra, Wiley, 1967