Applications of the Derivative

Chapter Outline

• Describing Graphs of Functions

• The First and Second Derivative Rules

• The First and Second Derivative Tests and Curve Sketching

• Curve Sketching (Conclusion)

• Optimization Problems

• Further Optimization Problems

• Applications of Derivatives to Business and Economics

Describing Graphs of Functions

Increasing and Decreasing Functions

Understanding whether a function is increasing or decreasing is fundamental in calculus and business applications. These concepts help describe how quantities change over time or with respect to other variables.

• Increasing Function: A function f(x) is increasing on an interval if, for any two points x1 and x2 in the interval with x1 < x2, then f(x1) < f(x2).

• Decreasing Function: A function f(x) is decreasing on an interval if, for any two points x1 and x2 in the interval with x1 < x2, then f(x1) > f(x2).

• Graphical Representation: An increasing function rises from left to right, while a decreasing function falls from left to right.

Example: The function f(x) = x is increasing for all x, while f(x) = -x is decreasing for all x.

Relative and Absolute Extrema

Extrema refer to the maximum and minimum values of a function, which are crucial in optimization problems in business calculus.

• Relative (Local) Maximum: A point where the function changes from increasing to decreasing.

• Relative (Local) Minimum: A point where the function changes from decreasing to increasing.

• Absolute Maximum: The largest value the function attains on its domain.

• Absolute Minimum: The smallest value the function attains on its domain.

Example: For f(x) = x^2, the absolute minimum is at x = 0 and there is no maximum as x increases.

Changing Slope

The slope of a function's graph indicates the rate of change. In business, this can represent growth rates, such as income or sales over time.

• Increasing Rate: If the slope itself increases, the function is rising at an increasing rate.

• Graphical Example: A curve that becomes steeper as x increases, such as exponential growth.

Example: If average annual income rises at an increasing rate, the graph of f(T) (income vs. year) will become steeper over time.

Concavity

Concavity describes the direction in which a curve bends and is determined by the second derivative.

• Concave Up: The graph lies above its tangent line; the slope increases as x increases. Mathematically, f''(x) > 0.

• Concave Down: The graph lies below its tangent line; the slope decreases as x increases. Mathematically, f''(x) < 0.

Example: f(x) = x^2 is concave up everywhere; f(x) = -x^2 is concave down everywhere.

Inflection Points

An inflection point is where the graph changes concavity, from concave up to concave down or vice versa.

• Definition: A point where f''(x) = 0 and the concavity changes.

• Continuity: The function must be continuous at the inflection point.

Example: f(x) = x^3 has an inflection point at x = 0.

x- and y-Intercepts

Intercepts are points where the graph crosses the axes.

• x-Intercept: Where f(x) = 0.

• y-Intercept: Where x = 0 and f(0) is the value.

Example: For f(x) = x^2 - 4, x-intercepts are at x = -2 and x = 2; y-intercept is at y = -4.

Asymptotes

Asymptotes are lines that the graph approaches but never touches.

• Horizontal Asymptote: As x approaches infinity, f(x) approaches a constant value.

• Vertical Asymptote: As x approaches a certain value, f(x) increases or decreases without bound.

Example: f(x) = 1/x has a vertical asymptote at x = 0 and a horizontal asymptote at y = 0.

Describing Graphs

To fully describe a graph, consider the following categories:

• Intervals of increase and decrease

• Relative and absolute extrema

• Intervals of concavity and inflection points

• x- and y-intercepts

• Undefined points

• Asymptotes

Example: A function may be increasing on -3 < x < -1 and 3 < x < 5.5, decreasing on -1 < x < 3, with relative maxima at x = -1 and x = 5.5, and a relative minimum at x = 3 and x = -3.

*Additional info: The images provided are textbook slides summarizing key concepts in describing graphs of functions, which are foundational for business calculus and optimization problems.*