Problem 145 - Calculus Volume 1

Question

A herring swimming along a straight line has traveled \(s(t)=\frac{t^{2}}{t^{2}+2}\) feet in \(t\) seconds. Determine the velocity of the herring when it has traveled 3 seconds.

Step-by-Step Solution

Answer

The velocity at 3 seconds is \( \frac{12}{121} \) feet per second.

Step 1: Understanding the Problem

The problem asks us to find the velocity of the herring at a specific time, \( t = 3 \) seconds. The velocity is the derivative of the position function with respect to time.

Step 2: Identify the Position Function

The position function for the herring is given by \( s(t) = \frac{t^2}{t^2 + 2} \). We need to differentiate this function to find the velocity.

Step 3: Use the Quotient Rule

To differentiate \( s(t) = \frac{t^2}{t^2 + 2} \), apply the quotient rule: if \( f(t) = \frac{u(t)}{v(t)} \), then \( f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2} \). Here, \( u(t) = t^2 \) and \( v(t) = t^2 + 2 \).

Step 4: Differentiate the Numerator and Denominator

Differentiate \( u(t) = t^2 \) to get \( u'(t) = 2t \). Differentiate \( v(t) = t^2 + 2 \) to get \( v'(t) = 2t \).

Step 5: Apply the Quotient Rule

Use the quotient rule formula: \[ s'(t) = \frac{2t(t^2 + 2) - t^2(2t)}{(t^2 + 2)^2} = \frac{2t^3 + 4t - 2t^3}{(t^2 + 2)^2} \]. Simplifying this will give \( s'(t) = \frac{4t}{(t^2 + 2)^2} \).

Step 6: Substitute the Given Time

Substitute \( t = 3 \) into the derived velocity function: \[ s'(3) = \frac{4(3)}{(3^2 + 2)^2} = \frac{12}{(9 + 2)^2} = \frac{12}{121} \].

Step 7: Simplify the Answer

No further simplification is needed, so the velocity of the herring at \( t = 3 \) seconds is \( \frac{12}{121} \) feet per second.

Key Concepts

Derivatives

Calculus often deals with how things change, and one of the core concepts to understand these changes is called a "derivative." Simply put, a derivative tells us how a function changes as its input changes. If you imagine a car driving along a path, the position of the car changes over time. Then, the derivative of the position function would tell us how the speed of the car, or its velocity, changes at any given moment.
Understanding derivatives is crucial because they form the backbone of many calculus problems. By finding the derivative, you can figure out rates of change, just like determining the car's speed from the position over time. The process of finding a derivative is known as differentiation. In our exercise, the position of the herring is described by the function \( s(t) = \frac{t^2}{t^2 + 2} \), and we differentiate this to get the velocity.

Quotient Rule

In calculus, when you have a function that is the ratio of two other functions, you use a specific technique to find its derivative called the "quotient rule." The quotient rule is essential when dealing with problems like ours, where the herring's position function is a fraction \( \frac{u(t)}{v(t)} \). Here, \( u(t) = t^2 \) and \( v(t) = t^2 + 2 \).
The quotient rule formula is given by:

• \( f'(t) = \frac{u'(t)v(t) - u(t)v'(t)}{[v(t)]^2} \)

This formula helps find the rate of change or derivative of the position function with respect to time, which ultimately gives us the velocity. You start by differentiating the numerator \( u(t) \) and the denominator \( v(t) \) individually:

• \( u'(t) = 2t \)
• \( v'(t) = 2t \)

Then apply these derivatives to the quotient rule formula, simplify the expression, and you get the derivative \( s'(t) = \frac{4t}{(t^2 + 2)^2} \), which represents the velocity function of the herring.

Velocity Calculation

Once you've found the derivative using the quotient rule, the next step is to calculate the velocity at a specific moment. Velocity tells us how fast an object's position changes over time, which is crucial for understanding the dynamics of motion. In this exercise, we found that the velocity function is \( s'(t) = \frac{4t}{(t^2 + 2)^2} \). To determine the velocity when the herring has traveled for 3 seconds, simply plug \( t = 3 \) into this function.
Substituting 3 into the velocity function gives:

• \( s'(3) = \frac{4(3)}{(3^2 + 2)^2} = \frac{12}{121} \)

This result, \( \frac{12}{121} \) feet per second, is the velocity of the herring at 3 seconds. This means that at that specific time, the herring is moving at a rate of \( \frac{12}{121} \) feet per second. Understanding how to calculate and interpret velocity using derivatives provides a deeper comprehension of motion in calculus.