Problem 2 - Calculus Early Transcendentals

Question

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute. $$\begin{array}{|c|c|c|c|c|c|}\hline t(\min ) & {36} & {38} & {40} & {42} & {44} \\ \hline \text { Heartbeats } & {2530} & {2661} & {2806} & {2948} & {3080} \\ \hline\end{array}$$ The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of \(t\) . $$\begin{array}{ll}{\text { (a) } t=36} & {\text { and } t=42 \quad \text { (b) } t=38 \text { and } t=42} \\ {\text { (c) } t=40 \text { and } t=42} & {\text { (d) } t=42 \text { and } t=44}\end{array}$$ What are your conclusions?

Step-by-Step Solution

Answer

The average estimated heart rate at 42 minutes is 69.605 beats per minute.

Step 1: Calculate Slope for t = 36 and t = 42

The formula for the slope of the secant line is \( m = \frac{\Delta y}{\Delta t} = \frac{y_2 - y_1}{t_2 - t_1} \). Here, \( y_1 = 2530 \) (heartbeats at \( t_1 = 36 \)) and \( y_2 = 2948 \) (heartbeats at \( t_2 = 42 \)). Substitute these values to find the slope:\[ m = \frac{2948 - 2530}{42 - 36} = \frac{418}{6} = 69.67 \] beats per minute.

Step 2: Calculate Slope for t = 38 and t = 42

Use the same formula for slope. Here, \( y_1 = 2661 \) at \( t_1 = 38 \) and \( y_2 = 2948 \) at \( t_2 = 42 \). Substitute these values:\[ m = \frac{2948 - 2661}{42 - 38} = \frac{287}{4} = 71.75 \] beats per minute.

Step 3: Calculate Slope for t = 40 and t = 42

Again, apply the slope formula. Here, \( y_1 = 2806 \) at \( t_1 = 40 \) and \( y_2 = 2948 \) at \( t_2 = 42 \). Substitute these values:\[ m = \frac{2948 - 2806}{42 - 40} = \frac{142}{2} = 71 \] beats per minute.

Step 4: Calculate Slope for t = 42 and t = 44

For this interval, \( y_1 = 2948 \) at \( t_1 = 42 \) and \( y_2 = 3080 \) at \( t_2 = 44 \). Substitute the values into the formula:\[ m = \frac{3080 - 2948}{44 - 42} = \frac{132}{2} = 66 \] beats per minute.

Step 5: Conclusion about the Heart Rate

The estimated heart rate at 42 minutes is obtained by averaging the secant slopes. The calculated slopes are approximately 69.67, 71.75, 71, and 66 beats per minute. The average slope is:\[ \frac{69.67 + 71.75 + 71 + 66}{4} = 69.605 \] beats per minute, which reflects the patient's heart rate at 42 minutes.

Key Concepts

Secant Line

The secant line is a fundamental concept in calculus. It represents a line that intersects two points on a curve. In real-life, we use the secant line to approximate the slope of a curve over a given interval. The slope of the secant line gives us an average rate of change between the two points. Imagine it like finding the average speed for a car trip between two cities on a map.

To calculate the slope of the secant line, we use the formula:

• \( m = \frac{y_2 - y_1}{t_2 - t_1} \)

Here, \( y_1 \) and \( y_2 \) are the heartbeats recorded at time \( t_1 \) and \( t_2 \), respectively. By calculating the secant line slopes between various intervals of time, we can estimate variations in heart rate over those intervals. This process simplifies complex curves into easier-to-manage straight lines.

Slope

The slope is a measure of how steep a line is. It tells us how much something changes versus another corresponding change. In our case, it's the change in heartbeats over the change in time, given in beats per minute.

When you see the slope formula \( m = \frac{y_2 - y_1}{t_2 - t_1} \), it describes the difference in heartbeats (\( y_2 - y_1 \)) over the difference in time (\( t_2 - t_1 \)). This helps us understand the average rate of heartbeat increase during an interval.

Different slopes give us different insights:

• Positive slope: Heart rate is increasing.
• Greater slope: Steeper, faster increase in heart rate.
• Smaller slope: Slower heart rate increase.

The slope provides a straightforward way to evaluate changes, like shifts in heartbeats over given times.

Tangent Line

The tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It has the same slope as the curve at that specific location. If you imagine your finger skimming across the surface of a ball, the tangent line is how your finger moves at that exact touchpoint.

In the context of the heart rate exercise, a tangent line represents the instant rate of change, or exactly how fast the heart rate is at a particular moment. While a secant line gives an average rate, the tangent line tells us exactly what's happening at an instant.

• If a tangent line at a point has a positive slope, the heart rate is increasing at that moment.
• If it's steep, the heart rate is rising quickly.

In practical applications, calculating the exact tangent line would require calculus and obtaining a limit, but secant lines help us approximate these changes effectively.

Heart Rate Estimation

Estimating heart rate using calculus involves analyzing changes over small and large time intervals. By calculating and understanding slopes of secant lines, we can deduce how fast the heart beats

• during specific intervals (using secant lines)
• at very specific moments (using tangent line approximations)

Using the heart rate data from the cardiac monitor, we calculated slopes of 69.67, 71.75, 71, and 66 beats per minute for different secant intervals. By averaging these rates, we get an estimate closer to reality. This average heart rate represents the overall heart activity at specific times post-surgery.

These calculations aren't just for math exercises. They can directly impact clinical decisions, offering insights into how a patient is recovering, helping doctors plan treatment and care accordingly. By mastering concepts like secant, slope, and tangent, we can better understand biological data and trends.