Problem 16
Question
The average weight, \(W,\) in pounds, of an adult is a function, \(W=f(c),\) of the average number of Calories per day, \(c,\) consumed. (a) Interpret the statements \(f(1800)=155\) and \(f^{\prime}(2000)=0\) in terms of diet and weight. (b) What are the units of \(f^{\prime}(c)=d W / d c ?\)
Step-by-Step Solution
Verified Answer
(a) \(f(1800)=155\) means weight is 155 lbs at 1800 Cal; \(f'(2000)=0\) means weight is stable at 2000 Cal. (b) The units are pounds per Calorie.
1Step 1: Understand f(1800) = 155
The statement \( f(1800) = 155 \) signifies that when the average consumption is 1800 Calories per day, the average weight of an adult is 155 pounds. This means that \( f(c) \) maps the 1800 Calorie intake to an average weight of 155 pounds.
2Step 2: Interpret f'(2000) = 0
The statement \( f'(2000) = 0 \) means that when the average Caloric intake is 2000 Calories per day, the rate of change of weight with respect to calorie intake is zero. This suggests that at 2000 Calories, the weight is not increasing or decreasing with additional Calories consumed, indicating a stable point in the dietary function.
3Step 3: Determine units of the derivative
The derivative \( f'(c) = \frac{dW}{dc} \) represents the rate of change of the weight with respect to Calories. \( dW \) is in pounds and \( dc \) is in Calories/day. Therefore, the units of \( f'(c) \) are pounds per Calorie, indicating how much the weight changes with each additional Calorie consumed.
Key Concepts
Caloric IntakeDerivative InterpretationRate of Change
Caloric Intake
Understanding caloric intake is crucial in managing and interpreting how it affects average body weight. When we talk about caloric intake, we refer to the number of calories consumed by an individual each day. This daily intake directly influences an individual's weight over time.
In the function \( W = f(c) \), where \( W \) represents weight and \( c \) stands for caloric intake, we observe how different levels of caloric consumption can change the average weight. For instance, if \( f(1800) = 155 \), it reveals that a daily intake of 1800 calories is associated with an average weight of 155 pounds.
Managing caloric intake is crucial for maintaining, losing, or gaining weight. A diet that maintains a balanced caloric intake can help support a healthy weight. Monitoring caloric intake ensures that individuals can maintain energy balance which is vital for overall health. This balance is achieved when the number of calories consumed equals the number of calories burned by bodily functions and physical activity.
In the function \( W = f(c) \), where \( W \) represents weight and \( c \) stands for caloric intake, we observe how different levels of caloric consumption can change the average weight. For instance, if \( f(1800) = 155 \), it reveals that a daily intake of 1800 calories is associated with an average weight of 155 pounds.
Managing caloric intake is crucial for maintaining, losing, or gaining weight. A diet that maintains a balanced caloric intake can help support a healthy weight. Monitoring caloric intake ensures that individuals can maintain energy balance which is vital for overall health. This balance is achieved when the number of calories consumed equals the number of calories burned by bodily functions and physical activity.
Derivative Interpretation
Interpreting derivatives is essential in understanding how variables affect each other. In this context, the derivative of the weight function, \( f'(c) \), plays a key role in understanding how weight changes as caloric intake varies.
Consider the statement \( f'(2000) = 0 \). This derivative tells us that at an intake level of 2000 calories, there is no change in weight for small variations around this point. In practical terms, this signifies a stable point where changes in caloric intake do not lead to any changes in weight.
This concept is valuable in dietary planning. Knowing when your weight is stable can guide individuals and healthcare professionals in adjusting caloric intake for achieving desired weight goals. It also highlights the sensitivity of weight change to dietary habits and helps in identifying caloric levels that maintain current weight.
Consider the statement \( f'(2000) = 0 \). This derivative tells us that at an intake level of 2000 calories, there is no change in weight for small variations around this point. In practical terms, this signifies a stable point where changes in caloric intake do not lead to any changes in weight.
This concept is valuable in dietary planning. Knowing when your weight is stable can guide individuals and healthcare professionals in adjusting caloric intake for achieving desired weight goals. It also highlights the sensitivity of weight change to dietary habits and helps in identifying caloric levels that maintain current weight.
Rate of Change
The rate of change in mathematics gives us insights into how one quantity changes in response to changes in another quantity. Here, the rate of change of weight with respect to calorie intake, denoted as \( f'(c) \), is expressed in pounds per calorie.
This tells us how much the weight changes for each additional calorie consumed. Understanding this rate is crucial for making informed dietary choices. If \( f'(c) \) is positive, it indicates that increasing caloric intake will increase weight. Conversely, if \( f'(c) \) is negative, increasing caloric intake will decrease weight.
Understanding the rate of change helps in determining whether one needs to increase or decrease caloric intake to achieve a desired change in weight. By analyzing this, individuals can comprehend how sensitive their weight is to any variations in their calorie consumption, enabling targeted and efficient dietary management.
This tells us how much the weight changes for each additional calorie consumed. Understanding this rate is crucial for making informed dietary choices. If \( f'(c) \) is positive, it indicates that increasing caloric intake will increase weight. Conversely, if \( f'(c) \) is negative, increasing caloric intake will decrease weight.
Understanding the rate of change helps in determining whether one needs to increase or decrease caloric intake to achieve a desired change in weight. By analyzing this, individuals can comprehend how sensitive their weight is to any variations in their calorie consumption, enabling targeted and efficient dietary management.
Other exercises in this chapter
Problem 15
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