Problem 51 - Applied Calculus

Question

The weight, \(w,\) in kilograms, of a baby is a function \(f(t)\) of her age, \(t\), in months. (a) What does \(f(2.5)=5.67\) tell you? (b) What does \(f^{\prime}(2.5) / f(2.5)=0.13\) tell you?

Step-by-Step Solution

Answer

(a) The baby weighs 5.67 kg at 2.5 months. (b) Her weight is increasing by 13% per month at this age.

Step 1: Interpret the Function Value

Given that \( f(2.5) = 5.67 \), this tells us that when the baby is 2.5 months old, her weight is 5.67 kilograms. The function \( f(t) \) gives the weight of the baby as a function of her age in months.

Step 2: Understand the Derivative

\( f^{\prime}(2.5) \) represents the rate of change of the baby's weight at 2.5 months. It indicates how quickly her weight is increasing at that exact moment in time. This value is essential in understanding the growth trend of the baby at that specific age.

Step 3: Calculate the Relative Growth Rate

The expression \( \frac{f^{\prime}(2.5)}{f(2.5)} = 0.13 \) indicates that the rate of change of the baby's weight is 13% of her current weight per month at this age. This is a relative growth rate, giving us an idea of how quickly she is growing compared to her size at 2.5 months.

Step 4: Conclude the Findings

Therefore, at 2.5 months, the baby weighs 5.67 kg, and her weight is increasing at a rate that is 13% of her current weight per month. This signifies a healthy weight increase relative to her size at this stage.

Key Concepts

Functions

In calculus, functions are essential for describing relationships between variables. A function like \( f(t) \) tells us how one variable, the baby's weight \( w \), changes with respect to another variable, in this case, time \( t \), measured in months.

• A function takes an input, which here is the age of the baby, and provides an output, the weight.
• Using \( f(t) \), we can determine the exact weight of the baby for any given month \( t \).
• For example, \( f(2.5) = 5.67 \) means that at 2.5 months, the baby weighs 5.67 kilograms.

Functions can be visualized by plotting points on a graph, where the input is on the horizontal axis and the output is on the vertical axis. Understanding functions is crucial for interpreting real-world phenomena through mathematical expressions.

Derivatives

Derivatives are a fundamental concept in calculus that help us understand the rate of change. The derivative of a function, symbolized as \( f'(t) \), tells us how fast or slow a function's output is changing in relation to the input.

• If \( f(t) \) represents a baby's weight over time, \( f'(t) \) represents how quickly the weight is changing at any given age.
• For instance, \( f'(2.5) \) would show the rate at which the baby's weight is increasing when she is 2.5 months old.
• This is useful in understanding trends or predicting future changes.

The derivative gives us critical insights into growth patterns, allowing us to quantify how one quantity changes with respect to another.

Relative Growth Rate

Relative growth rate is an important measure that shows how fast something is growing in proportion to its current size.
This is especially useful in biological and economic studies to compare growth rates as percentages.

• The relative growth rate is determined by dividing the rate of change (the derivative) by the current amount (the function value).
• In this scenario, \( \frac{f'(2.5)}{f(2.5)} = 0.13 \) conveys that at 2.5 months, the baby's weight is growing at a rate of 13% of her current weight each month.
• This helps us understand how significant the growth is relative to the baby's size.

Relative growth rates offer a way to compare different growth scenarios effectively, providing a frame of reference for how large a change is relative to what is already there.