2.R: Periodic Functions (Review)
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2.1: Graphs of the Sine and Cosine Functions
For the exercises 1-8, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
1) f(x)=-3\cos x+3
- Answer
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amplitude: 3; period: 2\pi ; midline: y=3no asymptotes
2) f(x)=\dfrac{1}{4}\sin x
3) f(x)=3\cos\left ( x+\dfrac{\pi }{6} \right )
- Answer
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amplitude: 3; period: 2\pi ; midline: y=0; no asymptotes
4) f(x)=-2\sin\left ( x-\dfrac{2\pi }{3} \right )
5) f(x)=3\sin\left ( x-\dfrac{\pi }{4} \right )-4
- Answer
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amplitude: 3; period: 2\pi ; midline: y=-4; no asymptotes
6) f(x)=2\left (\cos\left ( x-\dfrac{4\pi }{3} \right )+1 \right )
7) f(x)=6\sin\left ( 3x-\dfrac{\pi }{6} \right )-1
- Answer
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amplitude: 6; period: dfrac{2\pi }{3}; midline: y=-1; no asymptotes
8) f(x)=-100\sin(50x-20)
2.2: Graphs of the Other Trigonometric Functions
For the exercises 1-4, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.
1) f(x)=\tan x-4
- Answer
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stretching factor: none; period: \pi midline: y=-4asymptotes: x=\dfrac{\pi }{2}+\pi k, where k is an integer
2) f(x)=2\tan \left ( x-\dfrac{\pi }{6} \right )
3) f(x)=-3\tan (4x)-2
- Answer
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stretching factor: 3; period: \dfrac{\pi }{4}; midline: y=-2; asymptotes: x=\dfrac{\pi }{8}+\dfrac{\pi }{4}k, where k is an integer
4) f(x)=0.2\cos(0.1x)+0.3
For the exercises 5-10, graph two full periods. Identify the period, the phase shift, the amplitude, and asymptotes.
5) f(x)=\dfrac{1}{3}\sec x
- Answer
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amplitude: none; period: 2\pi ; no phase shift; asymptotes: x=\dfrac{\pi }{2}k, where k is an integer
6) f(x)=3\cot x
7) f(x)=4\csc (5x)
- Answer
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amplitude: none; period: \dfrac{2\pi }{5}; no phase shift; asymptotes: x=\dfrac{\pi }{5}k, where k is an integer
8) f(x)=8\sec \left (\dfrac{1}{4}x \right )
9) f(x)=\dfrac{2}{3}\csc \left (\dfrac{1}{2}x \right )
- Answer
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amplitude: none; period: 4\pi ; no phase shift; asymptotes: x=2\pi k, where k is an integer
10) f(x)=-\csc (2x+\pi)
For the exercises 11-15, use this scenario: The population of a city has risen and fallen over a 20-year interval. Its population may be modeled by the following function: y=12,000+8,000\sin(0.628x), where the domain is the years since 1980 and the range is the population of the city.
11) What is the largest and smallest population the city may have?
- Answer
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largest: 20,000; smallest: 4,000
12) Graph the function on the domain of [0,40].
13) What are the amplitude, period, and phase shift for the function?
- Answer
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amplitude: 8,000; period: 10; phase shift: 0
14) Over this domain, when does the population reach 18,000? 13,000?
15) What is the predicted population in 2007? 2010?
- Answer
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In 2007, the predicted population is 4,413. In 2010, the population will be 11,924.
For the exercises 16a-16d, suppose a weight is attached to a spring and bobs up and down, exhibiting symmetry.
16) Suppose the graph of the displacement function is shown in the Figure below, where the values on the x-axis represent the time in seconds and the y-axis represents the displacement in inches.
- Give the equation that models the vertical displacement of the weight on the spring.
- At \text{time} = 0, what is the displacement of the weight?
- Answer
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5 in.
- At what time does the displacement from the equilibrium point equal zero?
- What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?
- Answer
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10 seconds
2.3: Inverse Trigonometric Functions
For the exercises 1-11, find the exact value without the aid of a calculator.
1) \sin ^{-1}(1)
2) \cos ^{-1}\left ( \dfrac{\sqrt{3}}{2} \right )
- Answer
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\dfrac{\pi }{6}
3) \tan ^{-1}(-1)
4) \cos ^{-1}\left ( \dfrac{1}{\sqrt{2}} \right )
- Answer
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\dfrac{\pi }{4}
5) \sin ^{-1}\left ( \dfrac{-\sqrt{3}}{2} \right )
6) \sin ^{-1}\left (\cos \left (\dfrac{\pi }{6} \right ) \right )
- Answer
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\dfrac{\pi }{3}
7) \cos ^{-1}\left (\tan \left (\dfrac{3\pi }{4} \right ) \right )
8) \sin \left (\sec^{-1} \left (\dfrac{3}{5} \right ) \right )
- Answer
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No solution
9) \cot \left (\sin^{-1} \left (\dfrac{3}{5} \right ) \right )
10) \tan \left (\cos^{-1} \left (\dfrac{5}{13} \right ) \right )
- Answer
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\dfrac{12}{5}
11) \sin \left (\cos^{-1} \left (\dfrac{x}{x+1} \right ) \right )
12) Graph f(x)=\cos x and f(x)=\sec x on the interval [0,2\pi ) and explain any observations.
- Answer
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The graphs are not symmetrical with respect to the line y=x.They are symmetrical with respect to the y-axis.
13) Graph f(x)=\sin x and f(x)=\csc x and explain any observations.
14) Graph the function f(x)=\dfrac{x}{1}-\dfrac{x^3}{3!}+\dfrac{x^5}{5!}-\dfrac{x^7}{7!} on the interval [-1,1] and compare the graph to the graph of f(x)=\sin x on the same interval. Describe any observations.
- Answer
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The graphs appear to be identical.
Practice Test
For the exercises 1-13, sketch the graph of each function for two full periods. Determine the amplitude, the period, and the equation for the midline.
1) f(x)=0.5\sin x
- Answer
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amplitude: 0.5; period: 2\pi midline y=0