An Application of Limits

Motion Along a Straight Line

In calculus, limits are used to analyze the motion of objects along a straight line. The position, velocity, and acceleration functions describe how an object's location changes over time.

• Position Function: s(t) gives the object's position at time t.

• Velocity Function: v(t) gives the object's velocity at time t.

• Acceleration Function: a(t) gives the object's acceleration at time t.

Average Velocity

• The average velocity over the interval is given by:

Instantaneous Velocity

• The instantaneous velocity at time t is:

• Alternatively, (the derivative of position).

• Similarly, acceleration is (the derivative of velocity).

Example

• If , then and .

The Definition of the Derivative and the Derivative as a Function

Limit Definition of the Derivative

The derivative measures how a function changes as its input changes. It is defined using limits.

• At a point:

• Or using increments:

• As a function:

Finding Derivatives Using the Limit Definition

• For :

• For :

Interpretations of the Derivative

• is the slope of the tangent line to at .

• is the instantaneous rate of change of at .

• is the slope of the secant line passing through and .

• is the average rate of change of on .

Derivative Rules

Basic Derivative Rules

Derivative rules allow us to compute derivatives efficiently for various types of functions.

• Sum/Difference Rule:

• Constant Multiple Rule:

• Product Rule:

• Quotient Rule:

• Chain Rule:

Derivatives of Famous Functions

Some functions have well-known derivatives that are frequently used in calculus.

• (where is a constant)

Example

• For , (using the power rule and sum rule).

• For , (using the product rule).