Applications of Differentiation: Extrema, Concavity, Optimization, and Elasticity

Applications of Differentiation

Using First Derivatives: Extrema and Graph Sketching

The first derivative of a function provides critical information about the behavior of the function, including where it increases or decreases, and where it attains local maximum or minimum values. These concepts are essential for analyzing and sketching the graphs of functions in business calculus.

Increasing and Decreasing Functions

• Definition: A function f is increasing on an interval if, for any a < b, f(a) < f(b). It is decreasing if f(a) > f(b).

• The sign of the first derivative, f'(x), determines whether a function is increasing (f'(x) > 0) or decreasing (f'(x) < 0).

• The slope of the secant or tangent line can be used to determine intervals of increase or decrease.

Critical Values and Points

• Critical Value: A number c in the domain of f where f'(c) = 0 or f'(c) does not exist.

• Critical Point: The point (c, f(c)) corresponding to a critical value.

• Critical points are candidates for relative extrema (maximum or minimum values).

Relative (Local) Extrema

• Relative Maximum: A point where f(c) is higher than all nearby points.

• Relative Minimum: A point where f(c) is lower than all nearby points.

• Relative extrema occur at critical points, but not all critical points are extrema.

The First-Derivative Test

• Used to classify critical points as relative maxima, minima, or neither.

• If f'(x) changes from negative to positive at c, f(c) is a relative minimum.

• If f'(x) changes from positive to negative at c, f(c) is a relative maximum.

• If f'(x) does not change sign, f(c) is not a relative extremum.

Using Second Derivatives: Concavity and Inflection Points

The second derivative provides information about the concavity of a function and helps classify extrema further. It also identifies points of inflection, where the graph changes concavity.

Concavity

• Concave Up: The graph bends upward; f''(x) > 0.

• Concave Down: The graph bends downward; f''(x) < 0.

• Concavity is determined by the sign of the second derivative.

Second-Derivative Test for Relative Extrema

• If f'(c) = 0 and f''(c) > 0, f(c) is a relative minimum.

• If f'(c) = 0 and f''(c) < 0, f(c) is a relative maximum.

• If f''(c) = 0, use the First-Derivative Test.

Points of Inflection

• A point where the graph changes concavity (from up to down or down to up).

• Occurs where f''(x) = 0 or f''(x) does not exist, and concavity changes.

Strategy for Curve Sketching

• Find derivatives and domain.

• Identify critical values and endpoints.

• Classify extrema using the First- or Second-Derivative Test.

• Determine intervals of increase/decrease and concavity.

• Locate inflection points.

• Sketch the graph using all gathered information.

Optimization: Finding Absolute Extrema

Optimization involves finding the highest or lowest value (absolute maximum or minimum) of a function, often subject to constraints. This is crucial in business applications such as maximizing profit or minimizing cost.

Absolute Extrema

• Absolute Maximum: The highest value of f(x) over its domain.

• Absolute Minimum: The lowest value of f(x) over its domain.

The Extreme-Value Theorem

• A continuous function on a closed interval [a, b] has both an absolute maximum and minimum.

Finding Absolute Extrema: Maximum-Minimum Principles

• Find f'(x) and solve for critical values in the interval.

• Evaluate f(x) at critical values and endpoints.

• The largest value is the absolute maximum; the smallest is the absolute minimum.

Strategy for Solving Maximum–Minimum Problems

• Draw and label diagrams if possible.

• List variables, constraints, and units.

• Express the objective function in terms of one variable.

• Use calculus to find and classify extrema.

Optimization in Business and Economics

Optimization techniques are widely used in business to maximize profit, minimize cost, and allocate resources efficiently.

Profit Maximization

• Profit is maximized where the derivative of revenue equals the derivative of cost: $R'(x) = C'(x)$ and $R''(x) < C''(x)$.

Applications: Inventory and Revenue

• Inventory costs can be minimized by optimizing order size and frequency.

• Revenue is maximized by analyzing demand and price relationships.

Elasticity of Demand

Elasticity of demand measures how sensitive the quantity demanded is to changes in price. It is a key concept in economics for pricing strategies and revenue optimization.

Definition of Elasticity

• The elasticity of demand E(x) is given by $E(x) = -\frac{x D'(x)}{D(x)}$.

• If E(x) < 1, demand is inelastic; if E(x) > 1, demand is elastic; if E(x) = 1, demand has unit elasticity (revenue is maximized).

Elasticity and Revenue

• Total revenue increases when demand is inelastic and decreases when demand is elastic.

• Total revenue is maximized when elasticity equals 1.

Additional info: These notes synthesize the main theoretical results, definitions, and strategies for analyzing and optimizing functions using calculus, with a focus on business applications such as profit maximization, cost minimization, and elasticity of demand. The included images reinforce key concepts and provide visual summaries of theorems and strategies.