Problem 37 - Applied Calculus

Question

Experiments suggest that the male maximum heart rate (the most times a male's heart can safely beat in a minute) decreases by 9 beats per minute during the first 21 years of his life, and by 26 beats per minute during the first 33 years. \(^{45}\) If you model the maximum heart rate as a function of age, should you use a function that is increasing or decreasing? Concave up or concave down?

Step-by-Step Solution

Answer

The function should be decreasing and concave up.

Step 1: Identifying the Type of Relationship

We know that the maximum heart rate decreases between the ages of 0-21 and 0-33. This implies that the heart rate is a decreasing function of age, since it reduces as age increases.

Step 2: Understanding Concavity from Given Data

The heart rate decreases by 9 beats over 21 years and by 26 beats over 33 years. To understand concavity, compare the rate of decrease relative to time intervals. For ages 0 to 21, rate of decrease = \( \frac{9}{21} \) beats per year. For ages 21 to 33, rate of decrease = \( \frac{26-9}{33-21} = \frac{17}{12} \) beats per year.Since \( \frac{17}{12} < \frac{9}{21} \), the rate per year decreases as age increases, suggesting the function is concave up.

Step 3: Summarizing Characteristics of the Function

Based on the analysis, the maximum heart rate as a function of age is a decreasing function because the heart rate decreases as age increases. It is also concave up because the rate of decrease slows down over time.

Key Concepts

Maximum Heart Rate

The concept of maximum heart rate is crucial when considering cardiovascular health. It refers to the most times a heart can safely beat in a minute during maximum physical exertion.
The formula often used to estimate this is "220 minus age"; however, individual factors might cause deviations.
Experiments suggest patterns for age-related changes in maximum heart rate, whereby increasingly fewer beats are possible per minute as one ages.

• From birth to 21 years, a male's maximum heart rate decreases by about 9 beats.
• From birth to 33 years, it decreases in total by about 26 beats.

The decrease signifies a natural decline, mainly due to aging and physiological changes, but staying informed about these changes helps maintain a healthier lifestyle.
Understanding how this measure changes can inform training and health strategies as people grow older.

Concavity

In mathematics, concavity describes how the slope or rate of change of a function behaves over time.
For the decreasing function of maximum heart rate in relation to age, we observe that it is concave up. Even though the heart rate decreases as age increases, the *rate* at which it decreases slows down.
This characteristic provides valuable insights:

• From age 0 to 21, the heart rate decreases by 9 beats, at a rate of 9/21 beats per year.
• From 21 to 33, it decreases by another 17 beats, but at a slower rate of 17/12 beats per year.

Since the decrease in heart rate slows as a person ages, the function depicting maximum heart rate shows a concave up pattern.
This means that even though the heart rate is reducing, it does so less dramatically as time goes on.

Age-Related Changes in Heart Rate

Age-related changes in heart rate encapsulate broader transformations that occur in the cardiovascular system across different life stages.
As people age, the heart's efficiency and capacity to pump declines gradually. This is reflected in the steady decrease in maximum heart rate as described above.

• During the younger years (0-21), a moderate decline is observed.
• The decline accelerates slightly during early adulthood (21-33).

These changes are mainly due to the natural aging process and other factors such as changes in cardiovascular fitness over time.
Understanding these changes is vital for designing appropriate exercise regimes and managing cardiac health effectively. As age progresses, individuals will need to adjust their physical activities to compensate for these natural changes, ensuring they do not overexert themselves.