2The Concept of Periodic Functions

Transcription

1 The Concept of Periodic Functions EKG INTRODUCTION The heart is a fist-sized muscle that acts as an electrical generator lying in a conducting medium made of body tissue and fluids. The heart pumps oxygen-rich blood from the lungs throughout the body, then pumps deoxygenated blood from the tissues back to the lungs so that it can take on more oxygen. To be an effective pumping organ, the heart must be able to maintain this cyclic activity. During the alternate phases of the cardiac cycle, some portions of the heart become positively charged, while others become negatively charged. Within this cycle, depolarization (the elimination of electrical charge) and repolarization (the reignition of that charge) sweep over the heart muscles. This potential difference, or voltage gradient (as much as 100 millivolts), causes current to flow in the external medium between these regions of the heart. These currents are strongest close to the heart and weaker at greater distances from it. The largest voltage measured at the body surface is only around 1 millivolt. This voltage, however, is sufficient for an electrocardiograph to detect it at the body s surface. The electrocardiograph is a sensitive instrument that produces an electrocardiogram (EKG), which is a graphic tracing of the heart s electrical activity. A typical EKG shows a series of waveforms occurring in repetitive order. The waveforms arise from a flat baseline known as the isoelectric line.a deflection from this line signals electrical activity. Figure 2.1 shows a typical electrocardiogram. FIGURE 2.1 P Q R S T The principal features of the EKG are represented by the letters P, Q, R, S, and T. One heart cycle is represented by a group of waveforms beginning with the P-wave, followed by the QRSwave complex, and ending with the T-wave. The P-wave is caused by the depolarization of the atrial muscles and is related to their contraction. The QRS-wave complex, consisting of three waves, is caused by the depolarization of the ventricles and is related to their contraction. Atrial repolarization happens during the depolarization of the ventricles but its weak signal is undetectable on an EKG. The T-wave is caused by the repolarization of the ventricles. Figure 2.2 shows the P-wave to T-wave sequence that represents one heart cycle. We call the number of heart cycles in a minute the heart rate, which for a body at rest is typically 70 to 80 beats per minute. Factors that can contribute to serious cardiac conditions such as angina, cardiac arrest, and heart failure include a lack of exercise, bad eating habits, advanced age, and chronic disease. In situations Connecting Mathematics with Science: Experiments for Precalculus 2003 Key Curriculum Press 11

2 P R Q S T their shapes and consistency, doctors are able to learn more about the heart s conduction system as well as study any damaged areas and places that do not get enough oxygen. In this experiment you will investigate the cyclic pattern of the EKG and determine its periodic properties. The EKG sensor that you will use is for educational purposes only and cannot be used for medical diagnosis. FIGURE 2.3 EKG of a patient suffering third-degree heart block FIGURE 2.2 like these, an EKG would show waveforms significantly different from the normal patterns. (See Figures 2.3 and 2.4 for sample EKGs of patients with heart disease.) From monitoring factors such as time between the waveforms and FIGURE 2.4 Electrocardiogram of a patient suffering sinus arrhythmia OBJECTIVES Generate an EKG as a function of time. Analyze the periodic pattern of the function and determine its period. Give the general definition of periodic function in symbolic form. EQUIPMENT REQUIRED TI graphing calculator with unit-to-unit cable LabPro or CBL 2 interface Vernier EKG Probe CBL-DIN adaptor Electrode patches TI-Graph Link cable Stopwatch PROGRAMS DataMate TI-Graph Link 12 Connecting Mathematics with Science: Experiments for Precalculus 2003 Key Curriculum Press

3 EQUIPMENT SETUP PROCEDURE 1. Connect the LabPro or CBL 2 to the calculator with the unit-to-unit cable using the input/output ports located on the bottom edge of each unit. Press in the cable ends firmly. 2. Using CBL-DIN adaptor, connect the EKG sensor to the CH 1 port of the LabPro or CBL Turn on the calculator. 4. Run the DataMate program on the calculator. Press Clear and the program will automatically identify the EKG sensor connected to CH 1.* EXPERIMENTAL PROCEDURE 1. The student whose EKG is recorded should be calm and relaxed. 2. Use the stopwatch to measure the pulse of the student and record it in the Data section. 3. Peel three electrode patches from the backing paper. Seal the remaining electrodes in the airtight container. 4. Connect the electrodes to the student. a. Put the first electrode patch on the right wrist. Connect the black (reference) alligator clip to the right wrist electrode patch. This is the reference point for the isoelectric line (baseline). b. Put the second electrode patch on the inside of the right elbow. Connect the green (negative) alligator clip to the right elbow electrode patch. c. Put the third electrode patch on the inside of the left elbow. Connect the red (positive) alligator clip to the left elbow electrode patch. 5. Press 2: Start on the calculator to collect data. DATA Pulse beats per minute Using the graph of the data, record the times for the occurrences of the P-, Q-, R-, S-, and T-waves for three consecutive cycles in the table. (Use the trace feature. If you can t get three consecutive cycles, use as many data points as you can get from your graph.) *If you are using an old sensor, it will not be identified by the interface. In this case, set up the sensor manually by selecting the channel, pressing 1: Setup and then pressing Enter. Select the appropriate sensor from the menu provided. Connecting Mathematics with Science: Experiments for Precalculus 2003 Key Curriculum Press 13

4 Time (s) P Q R S T Cycle 1 (t 1 ) Cycle 2 (t 2 ) Cycle 3 (t 3 ) DATA ANALYSIS 1. Using TI-Graph Link, print the graph of the EKG and include it with the lab report. Indicate P-, Q-, R-, S-, and T-waves on the graph. 2. Calculate the time intervals between the consequent occurrences of the P-, Q-, R-, S-, and T-waves. Record the results of your calculations in the table. 3. Find an average time interval for each wave and record it in the table. Time interval (s) P Q R S T t 2 t 1 t 3 t 2 Average 4. Calculate the number of cardiac cycles per minute using your data. 5. Does this value agree with the heart rate (pulse) of the person whose EKG was recorded? Show your calculations. 6. Consider a generic function, f(x), that has a repetitive pattern. Let the interval between any two consecutive occurrences of a specific value in the same position within the repetitive pattern of the function be T (T is called the period of the function). Write the definition of this function in symbolic form. 14 Connecting Mathematics with Science: Experiments for Precalculus 2003 Key Curriculum Press

5 The Concept of Periodic Functions EKG BEFORE THE EXPERIMENT 1. If students have not used LabPro or CBL 2 before this experiment, ask them to bring their calculators to class in advance and download the DataMate program from LabPro or CBL 2 to the calculator. 2. You may want to ask your student groups to select in advance a student whose EKG will be taken. Then ask that student to avoid extensive physical exercise (running, stair climbing, fast walking, and so on) right before the experiment. Explain that this helps obtain better experimental data and makes the data analysis easier. TEACHING OBJECTIVES Students will be able to recognize the periodic pattern of an EKG as a function of time determine the period of the function by observing repetition of specific values of the function in this experiment, P-, Q-, R-, S-, and T-waves on an EKG verify the calculated period of the function by comparing results with the heart rate/pulse of the person whose EKG was measured formulate the property of an arbitrary function to be periodic in the following symbolic form: given f(x) with period T, then f(x + T) f(x), assuming that the domain of the function is all real numbers T E A C H E R N O T E S ACTIVITY NOTES This experiment allows you to introduce the concept of periodic function and the period of the function at a very conceptual level. Students are usually exposed to trigonometric functions as examples of periodic functions (and tend to think that only trigonometric functions are periodic), but this experiment shows them the idea of periodicity for an arbitrary function. The symbolic form of the definition of periodic function might be hard for students to come up with independently. You may need to lead students to make this generalization for an arbitrary periodic function f(x). You need to emphasize that the domain of a function must consist of all real numbers (except possibly a set of discrete points) for the function to be Connecting Mathematics with Science: Experiments for Precalculus 15

6 periodic. In the experiment, we can t record data for an infinite domain, so we have to generalize from a real-life situation to the ideal mathematical model. Experimental Procedure Notes T E A C H E R N O T E S 1. It is critical that the student undergoing the EKG remain calm. That means no talking, laughing, active breathing, or moving for the duration of the experiment the student s arm must be relaxed during the measurements. 2. The default time settings for this experiment are: 0.01 s between samples and 200 samples. This results in a total time of 2 seconds for the experiment. The default settings provide good experimental data for the analysis (see Sample Data, EKG 1), but they may create memory problems on the calculator, especially if the calculator already has other programs and other data; in this case, either a. clear the lists and remove all additional programs from the calculator before taking data or b. change the default setting to 0.02 s between samples and 100 samples. While that reduces the precision of the measurements, it allows students to complete the experiment (see Sample Data, EKG 2). Data Analysis Notes Students should be able to trace the EKG function either in the DataMate program or in StatPlot, then record time values for the points of the P-, Q-, R-, S-, and T-cycles (in this activity, they examine time coordinates of specific points, not their actual values; because the data are experimental, the actual values of potential at specific points vary slightly, but repetition of these points happens at regular time intervals) find differences between the times when the function has the same points and determine both that the function is periodic and that the time difference remains constant within measurement errors analyze the period, find the number of cardiac cycles per minute, and see if that is within reasonable range (compare it with the heart rate/pulse of the person whose EKG has been recorded) Note that due to its sharp feature, the R-wave gives the best approximation of the period of cardiac cycle and fewer errors due to the limitation on measurements. Sample Data All data were collected for a 13-year-old female student. The pulse before the experiment was 77 bpm. 16 Connecting Mathematics with Science: Experiments for Precalculus

7 EKG 1 Time (s) P Q R S T Cycle Cycle Cycle N/A EKG 2 Time (s) P Q R S T Cycle Cycle Cycle N/A Sample Graphs EKG over 2-second time interval, 200 samples (default settings); viewed in DataMate EKG over 2-second time interval, 100 samples (custom settings); viewed in DataMate T E A C H E R N O T E S Expected Calculations These are the calculations for the sample data EKG 1. Time interval (s) P Q R S T t 2 t t 3 t N/A Average N/A The period of the cardiac cycle is That means there are 60/0.77 or 78 cardiac cycles per minute, which is close to the measured pulse of 77 beats per minute. Connecting Mathematics with Science: Experiments for Precalculus 17

8 FURTHER EXPLORATIONS 1. Consider the transformations of the periodic function. How does the horizontal shift of the function change its appearance? What if the magnitude of the horizontal shift is equal to the period of the function? 2. How would the horizontal compressions and stretches of the function affect the period of the function? T E A C H E R N O T E S 18 Connecting Mathematics with Science: Experiments for Precalculus