Question

Heart Rate A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

(a) Find the average heart rates (slopes of the secant lines) over the time intervals [40, 42] and [42, 44].

(b) Estimate the patient’s heart rate after 42 min by averaging the slopes of these two secant lines.

Short answer

a. The average rates of changes are 71 and 66 beats/minute respectively.

b. The patient’s heart rate after 42 min is 68.5 beats/minute.

Step-by-step solution

a.Step 1.  Given

Heart Rate A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

Step 2.  To determine

Find the average heart rates (slopes of the secant lines) over the time intervals [40, 42] and [42, 44].

Step 3.  Calculation

For the interval [40, 42] the points are (40, 2806) and (42, 2948).

So, the slope is:

$\frac{\left(2948-2806\right)}{42-40}=71$

For the interval [42, 44] the points are (42, 2948) and (44, 3080).

So, the slope is:

$\frac{\left(3080-2948\right)}{44-42}=66$

So, the average rates of changes are 71 and 66 beats/minute respectively.

b.Step 1.  Given

Heart Rate A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t min. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

Step 2.  To determine

Estimate the patient’s heart rate after 42 min by averaging the slopes of these two secant lines.

Step 3.  Calculation

From part (a), the average rates of changes are 71 and 66 beats/minute respectively.

So, the average of them is $\frac{\left(71+66\right)}{2}=68.5$.

Hence, the patient’s heart rate after 42 min is 68.5 beats/minute.