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TF.03.2 - Periodic TF.03.2 - Periodic PhenomenonPhenomenon

MCR3U - Santowski

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(A) Review(A) Review

When we have data and information to analyze, we have seen a variety of mathematical models that we can use to analyze/interpret the information. To date, we have used linear models, quadratic models, rational functions, and modeling with triangles

Now we will present one other manner of modeling certain types of data – data that repeats itself after a given period of time hence the idea of periodic phenomenon

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(B) Modeling Periodic Data(B) Modeling Periodic Data

Prepare a scatter-plot of the following data representing the velocity of air of a

runner Time (s) 1 1.5 2 2.5 3 3.5 4 4.5 5

Velocity (L/s) 1.75 1.24 0 -1.24 -1.75 -1.24 0 1.24 1.75

Time (s) 5.5 6 6.5 7 7.5 8 8.5 9

Velocity (L/s) 1.24 0 -1.24 -1.75 -1.24 0 1.24 1.75

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(B) Modeling Periodic Data(B) Modeling Periodic Data

The scatter-plot of the data provided shows that the data is periodic (repeats itself)

Velocity of Breathing

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

Time (s)

Velo

city (L/s

)

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(B) Modeling Periodic Data(B) Modeling Periodic Data

Some questions to think about

from this graph:

(i) How long does it take for

one complete cycle of inhaling

and exhaling?

(ii) What is the maximum velocity of the air? The minimum velocity?

(iii) Describe the shape of the graph of the runner’s breathing

(iv) What is the runner’s breathing velocity after 16 s? What assumptions

did you make?

Velocity of Breathing

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 2 4 6 8 10

Time (s)

Vel

oci

ty (L

/s)

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(C) Graphs of Periodic (C) Graphs of Periodic PhenomenonPhenomenon

Graphs of periodic phenomenon have several important features that should be pointed out:

(i) the length of time it takes to complete one cycle of the data (one repetition, one pattern) is called the period

(ii) the graph has a maximum or high point occurring at regular intervals and likewise, it has a minimum point that occurs at regular intervals

(iii) a straight, horizontal line can be drawn through the graph that is halfway between the maximum and minimum points this line is called the equilibrium axis or axis of the curve

(iv) the distance from the equilibrium axis to the maximum (or minimum) is called the amplitude

(v) Notice that the cycle or the curve does not have to start at the equilibrium axis leads to a concept known as phase shift

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(D) Periodic Phenomenon(D) Periodic Phenomenon

Periodic phenomenon can also be described by equations, rather than simply just data points and their resultant scatter-plots

For example, the following equation describes how the time of sunset varies with the day of the year

Where n is the number of the days in the year and t is the time at which sunset occurs

We can graph the equation and determine the amplitude, period, equilibrium axis and maximum and minimum points and provide an interpretation of their meanings

t n n( ) . co s ( ) . 1 7 52

1 8 01 7 2 1 8 4 3

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(D) Periodic Phenomenon(D) Periodic PhenomenonHere is a graph of the equation and we can analyze the graph to find the following:

(i) The period is 365 days (as this is an annual cycle)

(ii) The axis of equilibrium is at y = 18:26 as this is the “average value” so sunset times on averageover the year is around 6:30 pm

(iii) The maximum sunset time is Around 20:10 on the 172nd day

(iv) The minimum sunset time isaround 16:45 on the 355th day

(v) The amplitude is around 1:45 as the time of the sunset seems to fluctuate about 1 hr and 45 min above and below the “average sunset time” of 18:27

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(E) Internet Links(E) Internet Links

Solve these on-line problems from the Introductory Exercises from U. of Sask EMR

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(E) Classwork/Homework(E) Classwork/Homework

Nelson textbook, p414-416, Q1-12