Problem 16 - Calculus Single Variable

Question

Describes the position of a moving body at time \(t\). Determine whether, at time \(t=4,\) the body is moving forward, backward, or neither. $$ p(t)=1 / t $$

Step-by-Step Solution

Answer

At time \(t = 4\), the body is moving backward because the velocity is negative.

Step 1: Understand the Problem

We need to determine the direction of motion of the body at time \(t = 4\) using the position function \(p(t) = \frac{1}{t}\). Whether the body is moving forward, backward, or neither can be determined by examining the derivative of the position function, \(p'(t)\), which represents the velocity.

Step 2: Find the Derivative

Calculate the derivative of \(p(t)\). The derivative \(p'(t)\) is found using the rule for the derivative of \(t^{-1}\). The position function is: \[p(t) = t^{-1}\] The derivative is:\[p'(t) = \frac{d}{dt} (t^{-1}) = -t^{-2}.\]

Step 3: Evaluate the Derivative at t=4

Substitute \(t = 4\) into the derivative to find the velocity at that time: \[p'(4) = -4^{-2} = -\frac{1}{16}.\]

Step 4: Interpret the Result

Since \(p'(4) = -\frac{1}{16} < 0\), the negative velocity indicates that the body is moving backward at \(t = 4\).

Key Concepts

Understanding Velocity

The concept of velocity is crucial when analyzing motion. It not only indicates how fast an object is moving but also the direction in which it is moving.
In calculus, velocity is derived from the position function by taking its derivative.

• The position function, typically denoted as \(p(t)\), tells us the location of a moving object at any given time \(t\).
• The derivative of this position function, \(p'(t)\), is known as the velocity function.

Velocity gives insight into whether an object is speeding up, slowing down, or changing direction.
When the velocity is positive, the object moves forward. Conversely, when it is negative, the object is moving backward. If the velocity is zero, the body is momentarily at rest.

Exploring the Position Function

The heart of understanding the motion of a moving body is the position function. It expresses the location of an object over time.
In our example, the position function is \(p(t) = \frac{1}{t}\).

• This function represents how the position of a body changes as time passes.
• Each function has a characteristic shape that reveals the motion pattern of the body.

To assess the behavior of the moving body, you must first understand this function's derivative. By calculating the derivative, which is the velocity function, you can evaluate the movement at specific moments, such as at \(t = 4\), to see how it behaves. The derivative here, \(p'(t) = -t^{-2}\), reveals not only the speed but also the direction, which at \(t=4\) is backward, since \(p'(4) = -\frac{1}{16}\).

Analyzing a Moving Body's Behavior

When analyzing a moving body, understanding both the position and velocity functions is critical.
In our example, we use these functions to determine how the body is moving at a given time.

• First, the position function defines where the body is at any time \(t\).
• By taking the derivative, you calculate the velocity, which shows the rate and direction of change of the position.

With the step-by-step process, you capture how the body moves, whether it's moving forward (positive velocity) or backward (negative velocity).
Calculating the velocity at a specific time, such as \(t=4\), clarifies current motion trends. Since in this scenario the velocity \(p'(4) = -\frac{1}{16}\) is negative, it conclusively indicates backward movement. This analysis paints a clearer picture of how derivatives portray motion dynamics.