Problem 27 - Calculus

Question

A laboratory study investigating the relationship between diet and weight in adult humans found that the weight of a subject, \(W,\) in pounds, was a function, \(W=f(c),\) of the average number of Calories per day, \(c,\) consumed by the subject. (a) In terms of diet and weight, interpret the statements \(f(1800)=155, f^{\prime}(2000)=0.0003,\) and \(f^{-1}(162)=2200.\) (b) What are the units of \(f^{\prime}(c)=d W / d c ?\)

Step-by-Step Solution

Answer

(a) Weight at 1800 Calories: 155 lbs; rate at 2000 Calories: 0.0003 lbs/Cal; 162 lbs at 2200 Cal. (b) Units: pounds per Calorie.

Step 1: Interpret f(1800)=155

The statement \( f(1800) = 155 \) means that when a person consumes an average of 1800 Calories per day, their corresponding weight is expected to be 155 pounds. This indicates the function \( f \) maps the average calorie intake of 1800 to a weight of 155 pounds.

Step 2: Interpret f'(2000)=0.0003

The statement \( f'(2000) = 0.0003 \) signifies that at 2000 Calories per day, the rate of change of the subject's weight with respect to calorie intake is 0.0003 pounds per calorie. This implies that for each additional calorie consumed above 2000, the weight of the subject increases by 0.0003 pounds.

Step 3: Interpret f^{-1}(162)=2200

The statement \( f^{-1}(162) = 2200 \) means that to achieve and maintain a weight of 162 pounds, a subject needs to consume an average of 2200 Calories per day. This is the reverse relationship indicating the number of calories associated with a specific weight.

Step 4: Determine units of f'(c)=dW/dc

The expression \( f'(c) = \frac{dW}{dc} \) indicates the rate of change of weight with respect to calories. Since \( W \) has units of pounds and \( c \) has units of Calories, the units of \( f'(c) \) are pounds per Calorie. This describes how much a subject's weight changes for each unit change in the number of Calories consumed.

Key Concepts

Derivative

The derivative is a fundamental concept in calculus that describes how a function changes at any given point. In the context of the exercise, we have the statement \( f'(2000) = 0.0003 \). This tells us that at exactly 2000 Calories per day, the weight of the individual is increasing at a rate of 0.0003 pounds for each additional Calorie consumed.

To better understand this, recall that a derivative represents the slope of a function at a specific point. Here, the function \( f(c) \) denotes the subject's weight based on calorie intake. Thus, the derivative at a particular calorie intake, like 2000 Calories, provides insight into how the weight is expected to change with small adjustments in calorie consumption. This rate of change is particularly useful for making dietary adjustments while aiming for precise weight management.

By seeing derivatives in action—such as in this dietary model—we borrow from the power of calculus to predict outcomes, akin to seeing how fast something is moving at a specific instant rather than over a stretch of time.

Inverse Function

An inverse function essentially reverses the role of the input and output variables of a given function. In our exercise, \( f^{-1}(162) = 2200 \), implies that to maintain a weight of 162 pounds, an average of 2200 Calories should be consumed daily.

Understanding inverse functions helps in discovering the original input of a function that leads to a particular output. In simpler language, if the weight \( W \) can be considered as the output and the Caloric intake \( c \) as the input, the inverse function \( f^{-1}(W) \) will help you know how many calories \( c \) burn to achieve a desired weight \( W \).

This has practical applications, such as determining dietary needs for desired weight goals. Instead of guessing calorie intake required for weight control, inverse functions apply a mathematical certainty to diet planning—turning our dietary objectives from a guessing game into a science-based strategy.

Rate of Change

Rate of change is a key idea that helps us understand how one quantity changes in relation to another. This is central to interpreting derivatives. In our exercise, \( f'(c) = \frac{dW}{dc} \) is the mathematical representation of the rate of change of weight \( W \) with respect to calorie intake \( c \). Here, it is quantified as 0.0003 pounds per Calorie at 2000 Calories consumed per day.

The units, in this case, are pounds/Calorie, indicating how much weight change corresponds to an increment in Calorie intake.

This concept translates into real-world applications where understanding the rate of change enables researchers and dieticians to formulate dietary plans with measurable outcomes. For instance, if a person's weight increases slowly with calorie intake, as shown by a lower rate of change, they might require significantly more calories to see considerable weight gain.

Embracing the rate of change empowers individuals and professionals to tailor strategies that cater to specific health and fitness goals, ensuring more accurate and beneficial results.