Problem 37 - College Algebra with Modeling and Visualization

Question

A car's distance from a service center along a straight highway can be described by a constant function of time. What can be said about the car's velocity?

Step-by-Step Solution

Answer

The car's velocity is zero; it is not moving.

Step 1: Understand the Problem

The problem states that the car's distance from a service center is described by a constant function of time. This means that the position of the car does not change over time and remains constant.

Step 2: Identify the Relation of Velocity and Constant Position

Velocity is determined by the rate of change of position over time. Mathematically, if a position function is constant, its derivative with respect to time, which represents velocity, is zero.

Step 3: Analyze the Function

Since the function describing the car's distance is constant, its value does not depend on time. For a constant function like \( f(t) = C \), where \( C \) is a constant, the derivative \( f'(t) \) equals zero.

Step 4: Conclude on the Car’s Velocity

Thus, if the position does not change over time, the car's velocity, being the derivative of the constant position function, is zero. This means the car is stationary with respect to the service center.

Key Concepts

Rates of Change

In mathematics, rates of change describe how one quantity changes with respect to another. This concept is vital in understanding motion and other dynamic processes. When we talk about a car's distance over time, the rate of change refers to how quickly the car's position changes.

• For motion, this rate of change is velocity, which is the derivative of the position function with respect to time.
• If the function is constant, like when the car's distance from a service center doesn't change, the rate of change of position is zero. It indicates no movement.

Understanding rates of change can help in many areas, from predicting motion to analyzing data trends. It allows us to quantify how quickly changes occur, providing insight into the dynamics of a situation.

Derivative

The derivative is a fundamental concept in calculus that measures how a function changes as its input changes. Let's connect this to our car example:

• The derivative of a position function with respect to time tells us the velocity of the car.
• If the car's distance is constant, the position function derivative is zero, meaning no velocity.

Mathematically, if a function is constant, written as \( f(t) = C \) where \( C \) is a constant, the derivative \( f'(t) \) is zero. This means no change is happening with respect to the variable, time, in this case.
The derivative helps identify whether an object is moving and if so, how fast. It's an essential tool in fields ranging from physics to economics, wherever change needs to be quantified and analyzed.

Position Function

A position function represents the location of an object at any given time. It maps each moment in time to the object's location on a path, like a highway. In our problem, the position function is constant.

• This implies the car remains at the same distance from the service center at all times.
• No matter when you check, the car appears to be in the same position.

When the position function is constant, it specifically means the graph of this function is a horizontal line.
In mathematical terms, if the position function is \( f(t) = C \), near-zero changes will not affect \( C \), indicating that there is no movement of the car. Position functions are crucial for modeling real-world scenarios, offering insights into the travel paths and circumstances affecting an object's movement.