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.

The relationships can be written as:

• Velocity:

• Acceleration:

The speed of an object is the absolute value of its velocity: .

Example: Vertical Motion

• A potato is launched vertically upward with an initial velocity of 100 ft/s from a platform 85 ft above the ground. 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 .

Example: General Position Function

• Given , find velocity and acceleration at .

• Velocity:

• At :

• Acceleration:

• At :

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 , is increasing if for any in , . Equivalently, on .

• Decreasing function: On an interval , is decreasing if for any in , . Equivalently, on .

Graphical Interpretation

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

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

Mathematically, the slope between two points and is:

• If , the slope is positive (increasing).

• If , the slope is negative (decreasing).

Example: Piecewise Behavior

• Suppose behaves as follows:

• For , is increasing,

• For , is decreasing,

• For , is increasing,

Polynomial Model: Application to Biology

Modeling Growth with Polynomials

Polynomials are often used to model real-world phenomena, such as biological growth. For example, the femur length (in mm) of a fetus as a function of age (in weeks) can be modeled by:

• To find the rate of growth at a specific time, compute the derivative and evaluate at the desired values (e.g., weeks).

Example: Rate of Growth

• Differentiate with respect to :

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

Summary Table: Derivatives and Function Behavior

Key Takeaways

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

• Position, velocity, and acceleration are related through differentiation and are essential for modeling motion.

• Polynomial models and their derivatives can be used to analyze rates of change in real-world contexts, such as biological growth.