Sunrise, Sunset and Mathematical Functions

Transcription

1 Teaching of Functions 13 Sunrise, Sunset and Mathematical Functions Activity #1: The table you are given reports the sunrise and sunset times for Manila, Philippines for each day of the year. Each day has been assigned a number using January 1 as Day 1 and then counting forwards for all other dates. The first few dates for next year are also included and that is why their values are greater than Convert each time to a 24 hour system using decimal parts of an hour rather than hours and minutes. Subtract the sunrise and sunset times to complete the last column in the table. Work in groups of 3 with each person doing one third of the dates and then sharing the information with the rest of the group members. 2. Use this data to draw a graph of daylight hours (y) versus day of the year (x). Daylight Hours by Day for Manila 13 Number of Hours of Daylight Day of the Year - Jan 1 is Day 1 3. Does this set of data form a function? If so why, if not explain. 4. Find December 21 on the graph. What do you notice? 5. Find another day with the same number of hours of daylight as March 21. What date would this be? 6. What would be true about the number of daylight hours for March 21, 2002 compared to March 21, 2003? 7. Graphs that have the property that they repeat over and over again in the same pattern are said to be periodic and the period refers to how often this repetition occurs. What would be the period of this relation?

2 14 Teaching of Functions Sunrise and Sunset Times for Manila Date Day No. Sunrise Sunset Hours of Daylight Jan 1 1 6:21 17:38 Jan :25 17:50 Feb :21 17:59 Mar :11 18:05 Mar :56 18:07 April :42 18:10 May :31 18:14 May :26 18:20 June :27 18:27 July :33 18:30 July :39 18:26 Aug :43 18:16 Sept :44 18:02 Oct :46 17:46 Oct :49 17:32 Nov :55 17:25 Dec :06 17:25 Dec :17 17:33 Jan :24 17:45 Feb :24 17:56 Feb :16 18:03

3 Teaching of Functions 15 Activity #2: There are other activities that can produce interesting mathematical graphs. You are provided with a diagram of the face of a clock with only the hour hand showing. Tape it to your desk anywhere you like as long as the top edge of the paper is between 1 and 10 cm from the top edge of the desk. Make sure that the edges of the paper are parallel to the edges of your desk. 1. Measure the distance (in cm) from the tip of the hour hand to the top edge of your desk for each hour, starting at 1 o'clock and ending at 12 o'clock. Remember that the distance must be the straight line distance to the top edge of the desk measured at right angles to this edge. 2. Record your data in the table below: Hour Dist Distance From Tip of Hour Hand to Desk Edge (cm) Position of Hour Hand 3. Draw a graph of hour (x-axis) versus distance to the top edge of the desk (y-axis). 4. Does this set of points form a function? Why/Why not? 5. Is it periodic? If so, what is the period? If not, why not?

4 16 Teaching of Functions 6. Assume that in our experiment, 1 o'clock represents 1 a.m. and that 12 o'clock is noon. If we call 1 p.m. 13:00 hours and continue making measurements all the way up to 24:00 hours explain why you could fill in another chart without making any measurements. 7. Make up another chart from 13:00 to 24:00 hours and add these points to the graph. What do you notice? 8. If you think of 00:00 (midnight) as an angle of 0, then what angle could you associate with (a) 1 a.m. (b) 7:00 a.m. (c) 11 a.m. (d) 3 p.m. 9. If you drew your graph with the angle on the x-axis and the distance along the y-axis, would the graph change? Establish what angle would be equivalent to each position of the hour hand recorded in your table. Re-label the x-axis below the original labels using the appropriate angles. Now if you graph angle on the x-axis and distance on the y-axis will the graph look the same or different? Explain. 10. Is there any reason you could not carry out this experiment to include the distances for the hour hand at 1:30, 2:30 and so on? Explain. 11. In what ways is this last graph similar to or different from the graph you drew for the number of daylight hours?

5 Teaching of Functions

6 18 Teaching of Functions Making Connections With Periodic Functions 1. In 1900, Willem Einthoven invented the electrocardiograph (EKG). This machine measures the electrical impulses that cause the heart to beat. Doctors use this record to help them in diagnosing heart problems; and, many medical shows on TV have video displays of the classic spiked pulse of the heartbeat. Einthoven was awarded the Nobel prize in 1924 for his invention. The display below shows a typical tracing from an EKG for a healthy adult. Typical EKG for Normal Adult Heartbeats Time (seconds) (a) Based on the portion of the graph that you can see, would you say that this is a periodic function? Explain. (b) Assume the graph illustrated is represented by y = f(x). Is f(0.5) = f(2)? Explain. (c) Find 2 values of a that make f(a) = f(3) a true statement. (d) How many complete cycles occur between t = 0.5 and t = 3.5 seconds? (e) What is the period of this graph? (f) Based on your answer to (e) what would you say is the heart rate in beats/minute for this person? Is this normal? (g) If the heart rate was 120 beats/minute, what would be the period of the EKG tracing in minutes? In seconds? (h) Animals such as cats and dogs have heart rates quite different to humans. The heart rate for an elephant is 46. What would be the period of the corresponding EKG tracing? (i) A canary s heart beats 1000 times per minute. What would be the period of the corresponding EKG tracing (if a canary would sit still for such a thing!)?

7 Teaching of Functions Some stock market analysts believe that stock prices vary in a predictable pattern and that if these patterns could be established it would make decisions about buying and selling stocks much easier. The average weekly stock price for the Acme Manufacturing Company for a five year period is illustrated in the display below: 32 Average Weekly Price of Acme Man ($) Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Month (a) Explain why this set of points forms a function. (b) What are the maximum and minimum values for this function? (c) When was the stock at its highest value over the past 5 years? (d) When was the stock at its lowest value over the past 5 years? (e) Based on the portion of the graph that you can see, would you say that this function is periodic? Explain. (f) What does the word cyclic mean? Would you say that these stock prices are cyclic? Explain.

8 20 Teaching of Functions Exploring Circular Functions - A 1. Use the function machine below and the circle to fill in the table of values for this function. For example, if you draw a 45 angle, the y-coordinate of the point where the angle cuts the circle will be This produces the ordered pair (45, 0.71) o INPUT PROCESSING INSTRUCTIONS Draw a line from O to make an angle with the x-axis equal to the input value. The output is the y-coordinate of the point where this line intersects the circle OUTPUT O Angle ( ) Function value 2. For every point on the unit circle will there be a corresponding angle? 3. For every angle will there be a corresponding point on the unit circle? 4. Are we justified in joining these points with a smooth continuous curve? Explain. 5. Plot these ordered pairs on the grid below. Comment on what you notice.

9 Teaching of Functions Compare your unit circle and function machine with that of your neighbour. In what ways are they the same or different? 7. Compare your graph with that of your neighbour. In what ways are they the same or different? 8. Where have you seen graphs similar to these before? 9. Would you say that the functions defined by these function machines represent periodic functions or not? Explain. The functions in these two exercises are called the sine (Exploration - A) and cosine y (Exploration - B) functions. The sine function is defined by sin θ = where r is the r radius of the circle used to generate the ordered pairs and is the angle. If the radius is 1 (as in this case) we can simply use the y-coordinate as we did here. The cosine function is x defined by cos θ = r 10. Sketch the graph of f(x) = sin θ and f(x) = cos θ in your notebook. Label the axes and state the domain and range for each function.

10 22 Teaching of Functions Exploring Circular Functions - B 1. Use the function machine below and the circle to fill in the table of values for this function. For example, if you draw a 60 angle, the x-coordinate of the point where the angle cuts the circle will be 0.5. This produces the ordered pair (60, 0.5). PROCESSING INSTRUCTIONS 60 o INPUT Draw a line from O to make an angle with the x-axis equal to the input value. The output is the x-coordinate of the point where this line intersects the circle. 0.5 OUTPUT O Angle ( ) Function value 2. For every point on the unit circle will there be a corresponding angle? 3. For every angle will there be a corresponding point on the unit circle? 4. Are we justified in joining these points with a smooth continuous curve? Explain. 5. Plot these ordered pairs on the grid below. Comment on what you notice.

11 Teaching of Functions Compare your unit circle and function machine with that of your neighbour. In what ways are they the same or different? 7. Compare your graph with that of your neighbour. In what ways are they the same or different? 8. Where have you seen graphs similar to these before? 9. Would you say that the functions defined by these function machines represent periodic functions or not? Explain. The functions in these two exercises are called the sine (Exploration - A) and cosine y (Exploration - B) functions. The sine function is defined by sin θ = where r is the r radius of the circle used to generate the ordered pairs and is the angle. If the radius is 1 (as in this case) we can simply use the y-coordinate as we did here. The cosine function is x defined by cos θ = r 10. Sketch the graph of f(x) = sin θ and f(x) = cos θ in your notebook. Label the axes and state the domain and range for each function.

12 24 Teaching of Functions Exploring Circular Functions - C As we saw in the previous exercises, the function machine can be programmed to produce different functions depending on the procedure that is part of the machine. This last set of processing instructions will produce yet another circular function. The function machine at the 5 30 o INPUT PROCESSING INSTRUCTIONS Draw a line from O to make an angle with the x-axis equal to the input value. The output is the y-coord of the point where this line intersects the tangent to the circle OUTPUT left is used in conjunction with the diagram on your handout. This diagram illustrates a circle along with a tangent line. Recall that a tangent is a line that intersects a circle in only one point. Once again, the radius of the circle is exactly one unit and the marks on the tangent line are 0.1 units. Use your own copy of the unit circle and the function machine above along with its defined rule to generate a table of values for this function for angles of -60, -45, -30, 0, 30, 45, , 390 and 420. (Hint: If the terminal ray of the angle doesn't hit the tangent line just extend the other end until it does). Make up a table similar to this one to help organize your work: O Angle Function value...

13 Teaching of Functions For every point on the tangent line will there be a corresponding angle? 2. For every angle will there be a corresponding point on the tangent line? Explain. The angles that do not produce values are 90 and 270 (and all coterminal equivalents). It is useful to draw dotted vertical lines at these angle values on the x-axis so that we know that the function is undefined at these values. 3. Now plot the ordered pairs in your table of values on a graph When a curve is not as well behaved as we would like, it is sometimes necessary to use more data points to get a good curve. This function is called the tangent function and you can get a few more points by using the button on your calculator. Get function values for 70, 75, 80 and 85 using the calculator and plot these ordered pairs on the graph to help you establish what the curve looks like. 5. Try to draw a smooth curve through the points. Comment on what you notice, particularly in what ways this curve is similar to or different from the graphs of the sine and cosine functions.

14 26 Teaching of Functions 6. Explain why the ordered pairs in the graph form a function. 7. Explain why the function appears to be periodic. 8. What is the period of this function? 9. What are the maximum and minimum values of this function? 10. For what values of the angle does the value of the function appear to be 0? 11. For what values of the angle does the value of the function appear to be undefined? 12. What are the domain and range of the tangent function? 13. Try to find tan 90 on your calculator. What happens? Explain.