Problem 47 - Calculus Single Variable

Question

Describes the position of an object at time \(t\). Calculate the instantaneous velocity at time \(c\). $$ p(t)=t^{2}(3-2 / t) \quad c=3 $$

Step-by-Step Solution

Answer

The instantaneous velocity at time \( t = 3 \) is 16 units per time interval.

Step 1: Understand the position function

The position of the object is given by the function \( p(t) = t^2(3 - 2/t) \). This function describes how the position changes with time \( t \).

Step 2: Simplify the function

First, simplify the position function \( p(t) \). Begin by distributing \( t^2 \) in the expression: \( p(t) = t^2 \cdot 3 - t^2 \cdot \frac{2}{t} \) which simplifies to \( 3t^2 - 2t \).

Step 3: Find the derivative of the position function

The instantaneous velocity is the derivative of the position function with respect to \( t \). Differentiate \( p(t) = 3t^2 - 2t \) to find \( v(t) \). Using the power rule, \( v(t) = \frac{d}{dt}(3t^2) - \frac{d}{dt}(2t) = 6t - 2 \).

Step 4: Evaluate the derivative at \( t = c \)

To find the instantaneous velocity at \( t = 3 \), substitute \( t = 3 \) into the derivative \( v(t) \). So, \( v(3) = 6(3) - 2 = 18 - 2 = 16 \).

Step 5: Conclusion

The instantaneous velocity of the object at \( t = 3 \) is \( 16 \) units per time interval (using the units consistent with the position function).

Key Concepts

Position Function

In the realm of physics and calculus, the position function is crucial as it describes the location of an object in relation to time. For instance, consider the position function given by \( p(t) = t^2(3 - 2/t) \). This function expresses how the position of an object changes over time \( t \). By analyzing this function, we can understand how the motion of the object behaves at any point in time.

• Position functions are typically denoted as \( p(t) \) or \( s(t) \).
• They can take various forms depending on the motion being studied.
• Analyzing position functions helps determine velocity and acceleration.

Simplifying the position function is often the first step to uncovering more about the object's motion, paving the way to calculate the velocity.

Derivative

The derivative is a fundamental tool in calculus, used to analyze the rate at which quantities change. Specifically, the derivative of a position function provides the velocity of the object. Calculating the derivative involves differentiating the position function with respect to time, denoted as \( \frac{d}{dt} \). This reveals the instantaneous rate of change of the position, which is the instantaneous velocity.

• It provides a way to understand how fast or slow a position changes with time.
• It is denoted as \( p'(t) \) or \( v(t) \) in the context of velocity.
• Interpreting the derivative helps in understanding the behavior of objects in motion.

Derivatives form the basis of understanding dynamics in physics and are used extensively for motion analysis.

Power Rule

The power rule is a reliable technique in differentiation, simplifying the process of finding the derivative of functions involving powers of variables. When handling a function like \( p(t) = 3t^2 - 2t \), applying the power rule is straightforward and essential for finding the instantaneous velocity. Given a power function \( t^n \), the power rule states that the derivative is given by:\[ \frac{d}{dt} t^n = n t^{n-1} \]Using this rule, we differentiate each term:

• The derivative of \( 3t^2 \) becomes \( 6t \).
• The derivative of \( -2t \) is simply \( -2 \).

This efficient rule reduces complex differentiation problems into manageable steps, aiding in teaching and understanding calculus principles.

Evaluate Derivative

Once the derivative is determined, the next step is evaluating it at a specific point to find values such as instantaneous velocity. In our example, after differentiating the position function, we found the derivative \( v(t) = 6t - 2 \). To determine the instantaneous velocity at a particular time \( c \), substitute \( t = 3 \) into the expression:\[ v(3) = 6(3) - 2 = 18 - 2 = 16 \]By evaluating the derivative at \( t = 3 \), we calculate the instantaneous velocity to be 16 units per time interval.

• Substitute the point of interest into the derivative.
• Calculate the resulting expression for the desired value.
• Understand that this value represents how the object is moving at that specific instant.

This process solidifies the relationship between differentiation and its applications in real-world scenarios.