Applications of the Derivative

Applications of the Derivative

Calculus is not just about computing derivatives; it's about using them to understand how quantities change.

Rates of Change and Rectilinear Motion

If $s(t)$ represents the position of an object moving along a straight line at time $t$:

• Velocity: $v(t) = s'(t)$ (Instantaneous rate of change of position)
• Acceleration: $a(t) = v'(t) = s''(t)$ (Rate of change of velocity)
• Speed: $|v(t)|$

Optimization: Maxima and Minima

One of the most powerful applications is finding optimal values (maximum profit, minimum cost, strongest beam).

1. Critical Points: Values of $c$ where $f'(c) = 0$ or $f'(c)$ is undefined.
2. First Derivative Test: Determines if a critical point is a relative maximum or minimum by checking sign changes of $f'(x)$.
   • Positive to Negative $\to$ Maximum
   • Negative to Positive $\to$ Minimum
3. Second Derivative Test: Uses concavity.
   • $f''(c) > 0$ (Concave Up) $\to$ Minimum
   • $f''(c) < 0$ (Concave Down) $\to$ Maximum

Curve Sketching

Using derivatives, we can determine the shape of a graph without plotting points manually.

• Increasing/Decreasing: Sign of $f'(x)$.
• Concavity: Sign of $f''(x)$.
• Inflection Points: Points where concavity changes ($f''(x) = 0$ or undefined).

L'Hopital's Rule

Used to evaluate limits of indeterminate forms $0/0$ or $\infty/\infty$.

$$\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}$$

Provided the limit on the right exists.