Position, Velocity, and Acceleration

Definitions and Relationships

In calculus, the concepts of position, velocity, and acceleration are fundamental for describing the motion of objects. These quantities are related through differentiation:

• Position function s(t): Describes the location of an object at time t.

• Velocity function v(t): The rate of change of position with respect to time. It is the first derivative of the position function.

• Acceleration function a(t): The rate of change of velocity with respect to time. It is the second derivative of the position function.

Formulas:

• Velocity:

• Speed:

• Acceleration:

Increasing and Decreasing Functions

Definitions

A function's behavior (increasing or decreasing) on an interval is determined by the sign of its derivative:

• Increasing function: On an interval , if for every in with , , then is increasing.

• Decreasing function: On an interval , if for every in with , , then is decreasing.

The derivative provides a test for increasing or decreasing behavior:

• If on , then is increasing on .

• If on , then is decreasing on .

Graphical Interpretation

• For an increasing function, the slope of the tangent line is positive.

• For a decreasing function, the slope of the tangent line is negative.

Example intervals:

• : is increasing,

• : is decreasing,

• : is increasing,

Examples and Applications

Projectile Motion Example

Example: A potato is launched vertically upward with an initial velocity of 100 ft/s from a potato gun at the top of an 85-foot building. The position function is:

To find when the potato hits the ground, set and solve for :

To find the velocity at impact, substitute the value of into the velocity function:

• At ,

Velocity and Acceleration Example

Example: Given , find velocity and acceleration at .

• Velocity:

• At ,

• Acceleration:

• At ,

A Polynomial Model: Application in Biology

Modeling Fetal Growth

Polynomial models are used in biology to estimate growth rates. For example, the femur length (in mm) of a fetus as a function of age (in weeks) can be modeled as:

To find the rate of growth, compute the derivative :

Evaluate at weeks to analyze how the rate of growth changes as time increases.

Summary Table: Increasing/Decreasing Functions and Derivatives

Key Takeaways:

• The sign of the derivative determines whether a function is increasing or decreasing.

• Position, velocity, and acceleration are related through differentiation.

• Polynomial models and their derivatives are useful for analyzing rates of change in real-world applications.