Problem 69 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The weight of a small fish in grams after \(t\) weeks is modeled by $$W(t)=0.1 t^{2}$$. At what rate is the fish growing at time \(t=4 ?\)

Step-by-Step Solution

Answer

The fish is growing at a rate of 0.8 grams per week at \( t = 4 \).

Step 1: Understand the Problem

We need to find the rate of growth of the fish at a specific time, which means we need to determine the derivative of the weight function, \( W(t) \), and then evaluate it at \( t = 4 \).

Step 2: Find the Derivative

Given the function \( W(t) = 0.1t^2 \), find its derivative with respect to \( t \) using the power rule. The derivative of \( t^n \) is \( nt^{n-1} \). So, the derivative of \( W(t) \) is \( W'(t) = 0.2t \).

Step 3: Evaluate the Derivative at t = 4

Evaluate \( W'(t) \) at \( t = 4 \). Substitute 4 into the derivative: \( W'(4) = 0.2 \times 4 = 0.8 \).

Step 4: Interpret the Result

The value \( W'(4) = 0.8 \) represents the rate of growth of the fish's weight at \( t = 4 \) weeks. This means the fish is growing at a rate of 0.8 grams per week at that time.

Key Concepts

Rate of Change

The rate of change in mathematics often represents how one quantity changes with respect to another. In this exercise, it is all about determining how quickly the weight of the fish is increasing over time.
Understanding the rate of change helps to analyze growth patterns or trends. For instance, if you know the rate at which the fish's weight grows, you can predict its future weight given more time.

The rate of change is usually represented by the derivative in calculus, offering insight into instantaneous changes at specific points. In our problem, the rate of change tells us the growth rate of the fish's weight at exactly 4 weeks.

Power Rule

The power rule is a fundamental technique in calculus used for finding derivatives quickly and easily. If you have a power function of the form \( f(x) = ax^n \), the power rule states that the derivative \( f'(x) \) is given by \( nax^{n-1} \).

• Simplifies Complex Calculations: The power rule is especially handy when working with polynomials because it facilitates computation without needing lengthy algebra.
• Time-Saving: Instead of manually solving each term separately, the power rule allows easy application of differentiation to numerous terms simultaneously.

In our fish weight problem, applying the power rule to \( W(t) = 0.1t^2 \) yields \( W'(t) = 0.2t \). By decreasing the exponent by one and multiplying the preceding coefficient by the original exponent, we obtain the rate of change needed.

Differentiation

Differentiation is a cornerstone of calculus, encountering every time one needs to find the rate of change or the slope of a curve at a point.
This method allows us to derive the derivative of a function, providing essential insights into the behavior and nature of the original function.

Why is Differentiation Useful?

• Understanding Behavior: Differentiation helps understand how a function changes—whether it’s increasing or decreasing—and the rate of that change.
• Real-World Applications: From physics to biology and economics, differentiation aids in analyzing any system's dynamic behavior over time.

In our scenario of determining the fish's weight gain, differentiation transforms \( W(t) = 0.1t^2 \) into its rate, \( W'(t) \). By substituting \( t = 4 \), differentiation clearly outlines how much weight the fish gains per week at that moment, resulting in \( 0.8 \) grams per week.