Problem 84 - Calculus for Biology and Medicine

Question

Investigated how maximal heart rate depends on age. They found that if age \(x\) is given in years, then the maximum heart rate of a healthy adult can be predicted by the following formula: $$ H(x)=208-0.7 x $$ where \(H(x)\) is the maximum number of heart beats in one minute. The data from Tanaka et al. suggests that each additional year of age decreases \(H(x)\) by the same amount. (a) Explain in words what \(d H / d x\) represents. (b) Show that \(d H / d x\) is a constant.

Step-by-Step Solution

Answer

(a) It represents the rate of change of heart rate with age. (b) \( \frac{dH}{dx} = -0.7 \), a constant value.

Step 1: Understand the Formula

The given formula for maximum heart rate is \( H(x) = 208 - 0.7x \). Here, \( x \) is the age in years and \( H(x) \) is the maximum heart rate in beats per minute. Every additional year decreases the heart rate by 0.7 beats per minute.

Step 2: Define the Derivative

The derivative \( \frac{dH}{dx} \) represents the rate of change of the maximum heart rate with respect to age. It shows how much the heart rate changes per unit increase in age.

Step 3: Calculate the Derivative

To find \( \frac{dH}{dx} \), differentiate \( H(x) = 208 - 0.7x \) with respect to \( x \). The derivative of a constant is 0, and the derivative of \(-0.7x\) is \(-0.7\). Thus, \( \frac{dH}{dx} = -0.7 \).

Step 4: Interpret the Constant Derivative

Since \( \frac{dH}{dx} = -0.7 \) does not depend on \( x \), it is a constant value. This implies that the maximum heart rate decreases by a constant 0.7 beats per minute for each additional year of age.

Key Concepts

Derivatives

In calculus, derivatives help us understand how a function changes as its input changes. Specifically, a derivative gives us the rate at which one quantity changes with respect to another. Here, the function in question is the maximum heart rate, expressed as \(H(x) = 208 - 0.7x\). This function describes how the heart rate changes as age increases. The derivative \(\frac{dH}{dx}\) indicates the rate of change of the heart rate relative to age. In simpler terms, it tells us how much the heart rate drops for every additional year of age. Think of it as a helpful tool to measure the "slope" or steepness of the heart rate decrease as you get older.

Rate of Change

The concept of rate of change is essential in many fields, including biology. It describes how one variable changes in relation to another. In our context, it refers to how the maximum heart rate changes with age. The rate of change is essentially what the derivative tells us.For the given formula \(H(x) = 208 - 0.7x\), the rate of change is the derivative \(\frac{dH}{dx}\). After calculation, we find that \(\frac{dH}{dx} = -0.7\). This negative value means that as age increases, the heart rate decreases. With each passing year, there's a reduction of 0.7 beats per minute in maximum heart rate, reflecting a steady and predictable decline.

Maximum Heart Rate

Maximum heart rate refers to the highest number of beats per minute your heart can reach during intense physical activity. Understanding maximum heart rate is crucial, especially in designing workouts and monitoring cardiovascular health.In our study, the maximum heart rate is predicted using the formula \(H(x) = 208 - 0.7x\). Here, 208 represents a starting point or initial maximum heart rate when age is taken as zero. As one's age increases, the maximum heart rate decreases by a factor determined by the derivative, which is consistently 0.7.Knowing your maximum heart rate can help you tailor your exercise routines to be both safe and effective. It provides a benchmark that is useful, particularly in calculating target heart rate zones for various intensities of workouts.

Age Dependency

Age dependency in this context refers to how changes in our physiology, such as maximum heart rate, depend on age. Our bodies naturally undergo changes as we age, and this model shows a straightforward, linear relationship between age and heart rate.The formula \(H(x) = 208 - 0.7x\) provides a simple way to calculate how maximum heart rate decreases as one gets older. The constant term \(-0.7\) derived from the formula makes explicit the degree of dependence — each additional year of age causes a reduction of 0.7 beats per minute. This kind of age dependency informs health professionals and individuals about expected changes in heart health with aging. It highlights the importance of adjusting lifestyle and exercise routines as one grows older, ensuring they remain appropriate and effective.