Problem 19 - Applied Calculus

Question

The weight, \(W,\) in 1 bs, of a child is a function of its age, \(a,\) in years, so \(W=f(a)\) (a) Do you expect \(f^{\prime}(a)\) to be positive or negative? Why? (b) What does \(f(8)=45\) tell you? Give units for the numbers 8 and 45 (c) What are the units of \(f^{\prime}(a) ?\) Explain what \(f^{\prime}(a)\) tells you in terms of age and weight. (d) What does \(f^{\prime}(8)=4\) tell you about age and weight? (e) As \(a\) increases, do you expect \(f^{\prime}(a)\) to increase or decrease? Explain.

Step-by-Step Solution

Answer

(a) Positive; growth with age. (b) At 8 years, 45 lbs. (c) Pounds/year; weight change rate. (d) 8-year-old gains 4 lbs/year. (e) Likely decrease after certain age.

Step 1: Analyzing the Sign of the Derivative

The derivative \( f'(a) \) represents the rate of change in weight with respect to age. Since children generally gain weight as they grow older, \( f'(a) \) is expected to be positive. This indicates an increase in weight with increasing age.

Step 2: Understanding the Function Value

The notation \( f(8) = 45 \) means that when the child is 8 years old, the child weighs 45 pounds. Here, the age, 8, is measured in years, and the weight, 45, is measured in pounds.

Step 3: Determining the Units of the Derivative

The derivative \( f'(a) \) is the rate of change of the child's weight with respect to age. Therefore, its units are "pounds per year". It tells you how many pounds the child gains (or loses) per year of age.

Step 4: Interpreting the Derivative Value

The equation \( f'(8) = 4 \) means that at the age of 8, the child's weight is increasing at a rate of 4 pounds per year. This indicates a growth in weight as the child becomes older at this specific age.

Step 5: Predictions for the Derivative's Behavior

As a child grows older, one might expect the rate at which they gain weight (\( f'(a) \)) to either stay constant or decrease, especially after reaching a certain growth phase. Thus, \( f'(a) \) might not increase significantly and often decreases as age increases.

Key Concepts

Derivatives

In mathematics, derivatives represent the rate of change of a quantity with respect to another. When we talk about a function like weight with age, the derivative helps us determine how the weight changes as the child grows.

• The derivative, denoted as \( f'(a) \), shows how fast or slow the weight \( W \) changes when the age \( a \) changes.
• For children, their weight generally increases as they age, so we often see a positive derivative value.
• This means the slope of the function \( f(a) \) is upward as the child grows, indicating increasing weight.

Derivatives are fundamental in applied calculus, especially when predicting trends and understanding growth patterns.

Functions

Functions form a crucial part of calculus, acting as a rule that assigns each input to exactly one output. In the context of children's weight and age, the function \( W=f(a) \) gives us a way to relate a child's age \( a \), to their weight \( W \).

• Here, the function \( f \) translates the age into a weight value.
• Understanding the function values, like \( f(8) = 45 \), means at age 8, a child's weight is 45 pounds.
• The units give us clarity: age is in years and weight in pounds, ensuring we measure and interpret values correctly.

Functions provide a powerful way to model real-world phenomena, enabling predictions and insights through mathematical representations.

Rate of Change

Rate of change is a concept that describes how one quantity changes in relation to another. In calculus, it effectively measures how fast or slow a process, like weight gain, happens over time.

• The rate of change for weight with respect to age is given by the derivative \( f'(a) \).
• For example, if \( f'(8)=4 \), it indicates that at age 8, the child's weight is increasing by 4 pounds each year.
• The units, often pounds per year, help us quantify and understand how significant the growth rate is.
• This rate can vary based on the child's growth stage, with younger children typically gaining weight faster.

Understanding the rate of change equips us with insights into how dynamic processes unfold over time.

Children's Growth Patterns

Children's growth patterns represent the typical way children physically develop as they age, including changes in height, weight, and other body metrics. Recognizing these patterns helps predict and understand the changes.

• Young children generally experience rapid weight gain as part of their normal development.
• As they grow, this rate may slow, leading to a decrease in the derivative \( f'(a) \).
• Tracking the rate of weight change, particularly, helps assess whether a child's growth is within typical ranges.
• Predicting growth patterns involves understanding both the current and expected changes, using mathematical models and real-world data.

Exploring growth patterns assists caregivers and health professionals in ensuring healthy development through various growth stages.