Lab2 trig models PDF

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Lab2 trig models...

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106L Labs: Modeling with Trigonometric Functions

Modeling with Trigonometric Functions Part I: Review of Transformations Before we begin constructing our model, it is important to understand some basic features of the sine function. Be sure to label the axes for each of your graphs below. 1. Sketch the graph of the function f (x) = sin x. 2. Sketch the graph of the function f (x) = 3 sin x. What is the effect of multiplying sin x by 3? 3. Sketch the graph of f (x) = sin(4x).   4. Sketch the graph of f (x) = sin x4 .  5. Sketch the graph of f (x) = sin x −  6. Sketch the graph of f (x) = sin x +

What is the effect of multiplying x by 4? What is the effect of multiplying x by 14 ?  π π from x? 2 . What is the effect of subtracting 2  π π to x? 2 . What is the effect of adding 2

Part II: Daylight Hours for Lisbon The following table gives the length of the day (that is, the number of hours of daylight) for Lisbon, Portugal. Date 1/2 1/18 2/19 3/7 3/23 4/24 5/10 5/26 6/11 6/27 7/13 7/29 8/14 9/15 10/1 10/17 11/18 12/4 12/20

Day 2 18 50 66 82 114 130 146 162 178 194 210 226 258 274 290 322 338 354

Hours of Daylight 9.1 9.5 10.7 11.4 12.2 13.7 14.3 14.8 15.1 15.1 14.9 14.4 13.8 12.4 11.7 10.9 9.5 9.2 9.0

This data is reproduced in the spreadsheet for this lab. 1

106L Labs: Modeling with Trigonometric Functions 1. Open the spreadsheet for this lab, make a copy, and rename it with your group names. 2. Plot the above data as a scatter plot. The shape of your scatter plot should look approximately like a sinusoidal function. Let’s try to model the daylight data by a function of the form f (x) = A sin[B(x − C)] + D. Our aim is to determine the values of A, B, C, and D. 3. Calculation of A The number A is called the amplitude and it is equal to half the difference between the highest point and the lowest point on the curve. Since we don’t have data for every day of the year, we don’t really know on which days the highest and lowest points occur and what their exact values are. By looking at the data and its plot, explain what might be reasonable values for the highest and lowest points. Then calculate A in cell G2. 4. Calculation of B The value of B affects the period of the function (how often the function repeats its standard pattern). If we increase B, the function shrinks. If we decrease B, the function stretches. The period of sin x is 2π . What is the period of the daylight function? How many times the period of sin x is the period of the daylight function? Calculate the value of B in cell G3. Explain your formula.

5. Calculation of D The number D determines the position of the daylight function along the y-axis with respect to the position of the function sin x. Decide which vertical direction and by how many units we need to shift sin x in order to match the daylight function. Calculate the value of D in cell G5. Explain your formula.

6. Calculation of C The number C is called the phase shift and it determines the position of the daylight function along the x-axis with respect to the position of the function A sin(Bx) + D. Decide which horizontal direction and how many units we need to shift sin x in order to match the daylight function. Calculate the value of C in cell G4.

7. Add data for your completed model in Column D, and insert a new chart plotting both the data and the model as lines1 . Make sure you use entries in column G so you can easily change your model if it doesn’t match. 1 Unfortunately, Google Sheets does not have the option of making a mixed scatter and line plot. Excel does, so if you know Excel well, feel free to use it. Best you can do in Google Sheets is add point markers to one of the lines.

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106L Labs: Modeling with Trigonometric Functions 8. There are actually other values of A, and C that will produce the same model function f (x) (even though the constants differ, the function f (x) will be the same at every value of x). Write down two of these different forms of f (x), one with the same value for A that you have above, and one with the negative of your value of A above. You can test out your answers by changing A and C in their cells!

Part III: Daylight Hours for Stockholm The table below gives the length of the day (that is, the number of hours of daylight) for Stockholm, Sweden. The data is reproduced in the second tab of your spreadsheet. Date 1/2 1/18 2/19 3/7 3/23 4/24 5/10 5/26 6/11 6/27 7/13 7/29 8/14 9/15 10/1 10/17 11/18 12/4 12/20

Day 2 18 50 66 82 114 130 146 162 178 194 210 226 258 274 290 322 338 354

Hours of Daylight 4.6 5.8 9.2 10.9 12.6 16.1 17.9 19.6 20.8 21.0 20.0 18.4 16.6 13.2 11.5 9.8 6.4 5.0 4.3

Carry out the same analysis as you did in Part II for this data, using the second tab of your spreadsheet.

Part IV: Tides in the Bay of Fundy On March 18, 1998, the high tide in the Bay of Fundy was at midnight and the water level then was 26 feet. The very next tide was 6 hours and 48 minutes later and the water level then was 2 feet. Assuming the water level y(t) varies sinusoidally, answer the following questions: 1. Find a formula involving a sin function for y(t). 3

106L Labs: Modeling with Trigonometric Functions 2. Find a formula involving a cos function for y(t). Note that we only need a successive maximum and minimum value for y(t) to create our sinusoidal model here.

Part V: Exercises 1. (a) Use Geogebra to graph the function y = sin(2x) − 2 sin(x) over the horizontal range [0, 2π]2 . Use the extrema tool in Geogebra to find decimal values of the points where the tangent line to the curve is horizontal. (b) Use the derivative to find the exact x-coordinates of all points on the interval [0, 2π ] where the tangent line to the graph of the function y = sin(2x) − 2 sin(x) is horizontal. (You will find the identities cos(2x) = cos2 (x)−sin2 (x) and cos2 (x) +sin2 (x) = 1 useful.) 2. Consider a particular point on a vibrating string as it moves vertically up and down. The position of this point (in mm) at time t (in seconds) is given by s(t) = 10 +

1 sin(10πt). 4

(a) What is the period of oscillation of this point on the vibrating string? (b) Find a formula for the velocity of the point on the string after t seconds. (c) Describe the position and the motion (up or down) of this point on the string at t = 0 and t = 0.3 seconds. (Your answers should have units.) 3. If a projectile is fired from ground level with initial velocity v0 and inclination angle α and if air resistance can be ignored, the horizontal distance (in feet) it travels is 1 2 v sin(α) cos(α). 16 0 (a) Assuming that a soccer player kicks the ball at a 40◦ angle of inclination, find the initial velocity needed for a kick of 120 feet. (No calculus needed.) (b) What value of α maximizes R? R=

4. A revolving beacon in a lighthouse makes one revolution every 15 seconds. The beacon is 200 feet from nearest point P on a straight shoreline. (a) Find the rate at which the beacon revolves in radians per second. (No calculus here.) (b) Find the rate at which a ray of light from the lighthouse moves along the shore at a point 400 feet from P . 2

If you don’t remember how to use Geogebra, look back to the Derivatives and Roots lab from Math 105L!

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106L Labs: Modeling with Trigonometric Functions

Periodic Data Handout Your assignment this week is to produce an attractive 2-sided, one page handout on periodic data you find yourselves. Instructions: • Brainstorm what data you’re going to look for. It should be: – Periodic, with relatively simple behavior3 ; – Relatively easy to find; – Interesting. In brainstorming, you may want to think about natural phenomena that are cyclical. The only datasets I will not accept is the length of day in some city, or the depth of tide, as we have covered examples of those in class. • Find the data. You may use any tools at your disposal, including online research, the library, etc. • Model it using trigonometric functions, using the methods we covered in lab. • Present your work in an attrative, concise fashion in your report. You may want to include the following: – A description of your data and why you expect it to be periodic, as well as why it is interesting. – Part or all of your data: If your data is hundreds of points, don’t include all of it, only a representative sample. See the data on the length of day in the lab for an example of this. – Graphs of your data, with your model superimposed. – How you calculated the various parameters. If your data has a natural period (such as 365 days in a year, like the length-of-day data), explain it. – Citations of your data source(s), as well as any other material you used. Of course, you are not limited to the above. Your only restriction is length. Grading: Each group will produce one handout. They will be graded for accuracy of data, attractiveness, correct use of language, well-labeled graphs, good citations (format is not relevant, as long as it is consistent), as well as precise, correct, and well-presented calculations.

3 If you are unsure whether your chosen data exhibits ‘simple’ behavior, plot it and show me the plot. If you’re planning to do this, make sure you give yourself plenty of time to find other data!

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