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Math 131 A candy bar quiz 2, Tom Linton, Feb 4, 2000
  1. Let f(x) = 3sin(3x - 3) + 2cos(2x - 2), then f(x) is periodic. The purpose of this quiz is to investigate properties of f(x) with the help of our calculators (so think twice before attempting any calculation by hand). A plot of f(x) is shown below. The plot indicates that f(x) is periodic and has hilltops and valley bottoms of two different heights and depths (high and low hills along with deep and shallow valleys). You should probably plot the graph below on your calculator at this point.
a plot of f(x) from x = 0 to 30
    1. The range of f(x) runs from the y-coordinate of the deeper valleys to the y-coordinate of the higher hills. One of the higher hills has coordinates (approximately equal to) x = 3.729 and y = 4.192. Find the y-coordinate of the deep valley near x = 2.57.

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    2. What is the approximate range of f(x)?

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    3. It looks like f(2) = 0 (from the graph above) calculate (accurately, as a decimal number) the value of f(2).





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    4. Your answer to the last problem should NOT have been exactly zero, find an accurate estimate of the value of x near x = 2, with f(x) = 0.

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    5. One fairly easy way to find the period, p, of f(x) is to calculate the difference between the x-coordinates of the small hilltop near x = 12 and the x-coordinate of the small hilltop near x = 6 (the period of f(x) is the x-distance between these two small hilltops). The small hilltop near x = 6 has (approximate) coordinates x = 5.7124 and y = 1. Find the x-coordinate of the small hilltop near x = 12.

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    6. What is the period of f(x)?

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    7. Find an accurate estimate of f(5).

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    8. Find a window, centered at (5, f(5) ), so that the graph of f(x) looks like a line on this window.