Rates of Change and Derivatives 1.8

Slope and the Derivative

The derivative of a function f(x) is a fundamental concept in calculus that measures how the function changes as its input changes. The derivative at a point gives the slope of the graph of f(x) at that point, which is crucial for understanding rates of change in business and economics.

• Slope at a Point: The slope of the tangent line to the graph of f(x) at x is given by the derivative f'(x).

• Mathematical Definition: The derivative is defined as:

• Interpretation: The derivative tells us how fast f(x) is changing at any given value of x.

Rate of Change

The rate of change of a function describes how one quantity changes in relation to another. In business calculus, this often refers to how profit, cost, or revenue changes with respect to time, production level, or other variables.

• Instantaneous Rate of Change: Given by the derivative f'(x), representing the rate at a specific point.

• Average Rate of Change: Measures the change over an interval [a, b]: This is also written as .

• Application: Useful for estimating changes over time or production intervals.

Applications to Motion: Position, Velocity, and Acceleration

Position as a Function of Time

In calculus, the position of an object can be described as a function of time, s = f(t). This concept is widely used in business for modeling inventory, investment growth, and other time-dependent processes.

• Position Function:

• Velocity: The rate of change of position with respect to time is called velocity:

• Acceleration: The rate of change of velocity with respect to time is called acceleration:

Example: Falling Object

Suppose an object (e.g., glasses) is dropped from a window, and its height after t seconds is given by:

• Height Function:

• Velocity at Impact: To find how fast the glasses are falling when they hit the ground, solve for t, then compute at that value.

• Acceleration at Impact: Compute , which is constant for this quadratic function.

Example Solution:

• Find when (when the glasses hit the ground):

• Velocity:

• Acceleration: (constant)

Additional info: In business, similar models are used for depreciation, inventory depletion, or other processes with quadratic change.

Graphical Analysis of Functions

Interpreting Graphs of Position vs. Time

Graphs are essential tools for visualizing how a function changes over time. In business calculus, they help identify periods of stability, rapid growth, or decline.

• Stable Function: f(t) is stable when its derivative is zero (horizontal tangent).

• Increasing Most Rapidly: Occurs where the slope (derivative) is maximized.

• Decreasing Most Rapidly: Occurs where the slope (derivative) is minimized (most negative).

Example: Given a graph of f(t), identify:

• Points where the function is stable (flat regions).

• Points of maximum increase (steepest upward slope).

• Points of maximum decrease (steepest downward slope).

Types of Position-Time Graphs

Different shapes of position-time graphs represent different types of motion or change:

Additional info: In business, these graphs can represent sales growth, cost accumulation, or other time-dependent processes.

Summary Table: Derivatives and Their Applications