Limits and the Derivative

Introduction to Derivatives

The concept of limits is fundamental in calculus and helps solve two primary problems: finding the equation of the tangent line to a function at a specified point, and determining the instantaneous velocity of an object in motion. These problems are central to understanding how functions change and are foundational in business calculus.

• Tangent Line: The tangent line represents the instantaneous rate of change of a function at a specific point.

• Instantaneous Velocity: The velocity at a particular moment, as opposed to the average velocity over an interval.

Average Rate of Change

The average rate of change of a function over an interval is calculated by dividing the change in the function's value by the change in the input. In business applications, this often represents average velocity or average revenue per unit.

• Definition: The average rate of change is given by the difference quotient:

• Example: On a trip, if you pass mile marker 120 at 9 AM and mile marker 250 at 11 AM, the average velocity is:

mph

• Application: In business, the average rate of change can represent the average increase in revenue per unit produced.

Example: Revenue Analysis

Consider a company manufacturing plastic planter boxes. The revenue function is for . To find the change in revenue as production increases from 100 to 400 planters:

• Change in Revenue:

• Result: Revenue increases by $3,000 when production increases from 100 to 400 planters.

• Average Rate of Change: dollars per planter.

The secant line connecting the points and on the graph represents the average rate of change in revenue between these points.

Example: Velocity of a Falling Object

For a small steel ball dropped from a tower, the distance fallen in seconds is given by . The positions at seconds are:

• At : feet

• At : feet

• At : feet

• At : feet

The average velocity from to seconds is:

feet per second

For a general interval from to seconds, the average rate of change is:

As approaches zero, this expression approaches the instantaneous velocity at seconds.

Instantaneous Rate of Change and the Derivative

The instantaneous rate of change is the value the average rate of change approaches as the interval becomes infinitesimally small. In calculus, this is formalized as the derivative of a function.

• Definition: The derivative of at is:

• Interpretations:

  • Slope of the tangent line to the graph at

  • Instantaneous rate of change of with respect to

  • Velocity if represents position as a function of time

Secant and Tangent Lines

A secant line passes through two points on a function's graph and its slope represents the average rate of change. A tangent line touches the graph at a single point and its slope represents the instantaneous rate of change at that point.

• As the second point on the secant line approaches the first, the secant line becomes the tangent line.

• The slope of the tangent line is the derivative at that point.

Procedure: Four-Step Process for Finding the Derivative

To find the derivative of a function , follow these steps:

1. Find .

2. Compute .

3. Form the difference quotient .

4. Take the limit as .

Example: For :

• Step 1:

• Step 2:

• Step 3:

• Step 4:

Nonexistence of the Derivative

The derivative at exists only if the limit defining the derivative exists at that point. If the limit does not exist, the function is said to be nondifferentiable at .

• Key Point: Not all functions are differentiable everywhere; points of nondifferentiability include corners, cusps, and discontinuities.