Problem 20 - Calculus

Question

The distance curve \(f=2 \cos 3 t\) yields the velocity curve \(v=-6 \sin 3 t .\) Explain the -6

Step-by-Step Solution

Answer

The \(-6\) results from applying the chain rule: derivative of \(3t\) (which is 3) times the coefficient 2, and the negative because the derivative of cosine is negative sine.

Step 1: Understanding the relationship between position and velocity

The given distance curve is expressed as the function \( f(t) = 2 \cos(3t) \). In calculus, the velocity function \( v(t) \) is the derivative of the distance function \( f(t) \) with respect to time \( t \). To explain the term \(-6\), we need to find this derivative.

Step 2: Calculating the derivative

To find the derivative of \( f(t) = 2 \cos(3t) \), we apply the chain rule. The chain rule states that the derivative of \( \cos(3t) \) is \(-\sin(3t)\) multiplied by the derivative of \(3t\), which is \(3\). Thus, the derivative of \( 2 \cos(3t) \) is \( 2 \times (-\sin(3t)) \times 3 = -6 \sin(3t) \).

Step 3: Explaining the coefficient -6

The coefficient \(-6\) in the velocity curve \( v(t) = -6 \sin(3t) \) comes from multiplying the coefficient of the original function, \(2\), by the derivative of the inside function of the cosine term, \(3\). This results from the application of the chain rule, producing \(-6\).

Key Concepts

Derivative

In calculus, the concept of a derivative is fundamentally about measuring how a function changes as its input changes. For a given function, the derivative tells us how quickly the output (or the value of the function) is changing at every point.
To compute the derivative of a function, we use differentiation—a process that involves various rules including power, product, and chain rules.
The distance function provided, \(f(t) = 2 \cos(3t)\), represents the position of an object over time. Derivatives play a key role in modeling motion, as they allow us to determine velocity and acceleration from position functions. For our specific function, the derivative is \(v(t) = -6 \sin(3t)\), which signifies the rate of change of position, giving us the velocity function.

Chain Rule

The Chain Rule is a powerful tool in calculus for finding the derivative of composite functions. A composite function is essentially a function within a function. When faced with such functions, the chain rule helps break down the task into more manageable pieces.
To apply the chain rule, identify the "outer" and "inner" functions. For our example, \(2 \cos(3t)\), the outer function is cosine, and the inner function is the linear expression \(3t\).

• Compute the derivative of the outer function: The derivative of \( \cos(u) \) is \(-\sin(u)\).
• Multiply this by the derivative of the inner function: The derivative of \(3t\) is \(3\).

Hence, the chain rule leads to computing \(-\sin(3t) \times 3\), giving the derivative as \(-6 \sin(3t)\). This step explains how the coefficient "-6" in the velocity function emerges.

Position and Velocity Relationship

The relationship between position and velocity is a cornerstone concept in physics and calculus. It explains how velocity, essentially the speed and direction of motion, is derived from the position—a body's location with respect to time.
In mathematical terms, velocity is the derivative of the position function with respect to time. Given the position function \(f(t) = 2 \cos(3t)\), the velocity function is \(v(t) = -6 \sin(3t)\). This reflects the law that velocity is the rate of change of position.

• The negative sign in the velocity function indicates directionality, suggesting that the position is decreasing over time when the sine function is positive.
• The magnitude of the coefficient determines how quickly the position is changing. Here, a coefficient of \(-6\) indicates a relatively rapid change in position.

Understanding this relationship allows us to predict future positions and analyze past motion in numerous practical applications.