Problem 44 - 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) Its weight is increasing at 13% per month at 2.5 months.

Step 1: Understanding the Function Value

The expression \(f(2.5) = 5.67\) indicates that when the baby is 2.5 months old, her weight is 5.67 kilograms. Here, \(f(t)\) represents the weight in kilograms as a function of age \(t\) in months.

Step 2: Interpreting the Derivative Ratio

The expression \( f^{\prime}(2.5) / f(2.5) = 0.13 \) represents the relative rate of change of the baby's weight at 2.5 months. It implies that at 2.5 months, the baby's weight is increasing at 13% of the baby's current weight per month. Here, \(f^{\prime}(t)\) represents the derivative of \(f(t)\), indicating the rate of change of weight.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus, focusing on how functions change. It involves calculating the derivative, which is a measure of how a function's output changes as its input changes. In the context of our problem, the baby’s weight function, denoted as \( f(t) \), describes the variation of the baby's weight over time.

The derivative, \( f'(t) \), signifies the rate at which the baby's weight changes with age. Essentially, finding \( f'(t) \) involves differentiating the function \( f(t) \). This derivative represents the instantaneous rate of change; for every tiny increase in age, how does the weight change?

For instance, the expression \( f'(2.5) \) tells us the growth rate of the baby's weight at exactly 2.5 months. Understanding this rate of change is crucial, as it offers insights into how rapidly or slowly an event, like weight gain during infancy, progresses at any given instant.

Functions

Functions are essential tools in mathematics to describe relationships between varying quantities. In the exercise, the weight of the baby is represented as a function \( f(t) \), where \( t \) indicates the age in months, and \( f(t) \) shows the corresponding weight in kilograms.

• A function allows us to forecast behavior: Knowing \( f(t) \) enables predictions like the baby's weight at a particular age.
• It simplifies complex relationships to manageable representations: Here, \( f(2.5) = 5.67 \) indicates the baby's weight at 2.5 months is 5.67 kg.

Functions help translate real-world situations into mathematical terms, enabling easier analysis and interpretation. In calculus, functions are often differentiated to understand how the relationships they describe evolve over time.

Growth Rate

The growth rate in this context refers to how quickly the baby's weight increases as she grows older. This rate is expressed through the concept \( \frac{f'(2.5)}{f(2.5)} = 0.13 \), which is the relative growth rate at 2.5 months.

• The term \( 0.13 \) indicates that the baby's weight increases by 13% of her current weight each month.
• This insight helps compare growth rates at different ages, quantifying not just growth, but the speed of growth.

Growth rate calculations, particularly relative growth rates like this, are integral in many fields, including biology and economics, offering a clear way to communicate how phenomena evolve over time relative to their current state.