Calculus - Applications of Derivatives

Applications of Derivatives

Tangent and Normal Lines

Tangent: y - f(a) = f’(a)(x - a) Normal: y - f(a) = -1/f’(a)(x - a)

Linear Approximation

$f(x) \approx f(a) + f'(a)(x-a)$

The linearization L(x) = f(a) + f’(a)(x-a).

Related Rates

Given variables related by equation, differentiate with respect to time.

Steps:

1. Identify related quantities
2. Find relationship equation
3. Differentiate implicitly with respect to time
4. Substitute known values and solve

Critical Points

Critical point: where f’(c) = 0 or f’(c) doesn’t exist

Fermat’s Theorem

If f has local extremum at c and f’(c) exists, then f’(c) = 0.

First Derivative Test

• f’(x) > 0 on (a,c): increasing → local maximum at c
• f’(x) < 0 on (a,c): decreasing → local minimum at c

Second Derivative Test

• f”(c) > 0: concave up → local minimum
• f”(c) < 0: concave down → local maximum
• f”(c) = 0: test inconclusive

Concavity and Inflection Points

• f”(x) > 0: concave up
• f”(x) < 0: concave down
• Inflection point: where concavity changes

Mean Value Theorem

If f is continuous on [a,b] and differentiable on (a,b), then there exists c ∈ (a,b) such that: $f'(c) = \frac{f(b) - f(a)}{b - a}$

Rolle’s Theorem

If f(a) = f(b), f continuous on [a,b], differentiable on (a,b), then there exists c such that f’(c) = 0.

Optimization

Closed interval: Evaluate f at critical points and endpoints. Absolute extrema: Largest/smallest values on entire domain.

Problems:

1. Define variables and objective function
2. Express constraints
3. Find critical points
4. Compare values at critical points and boundaries