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Ch. 1 - Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 1 - Functions문제 1.4.26
1장, 문제 1.4.26

Finding a Viewing Window


In Exercises 5–30, find an appropriate graphing software viewing window for the given function and use it to display that function’s graph. The window should give a picture of the overall behavior of the function. There is more than one choice, but incorrect choices can miss important aspects of the function.


y = 3 cos 60x

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Identify the function type: The given function is a trigonometric function, specifically a cosine function, y = 3 cos(60x). The amplitude is 3, and the frequency is 60.
Determine the period of the function: The period of a cosine function is given by the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( x \). Here, \( b = 60 \), so the period is \( \frac{2\pi}{60} = \frac{\pi}{30} \).
Set the x-axis viewing window: To capture at least one full period of the function, set the x-axis range to include at least \( 0 \) to \( \frac{\pi}{30} \). A good choice might be slightly wider, such as \( -\frac{\pi}{30} \) to \( \frac{2\pi}{30} \), to see the behavior before and after one period.
Set the y-axis viewing window: The amplitude of the function is 3, so the y-values will range from -3 to 3. Set the y-axis range slightly wider, such as -4 to 4, to ensure the peaks and troughs are clearly visible.
Use graphing software: Input the function y = 3 cos(60x) into graphing software with the determined viewing window. Adjust the window if necessary to ensure the graph displays the overall behavior of the function, including its periodic nature and amplitude.

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주요 개념

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Periodicity of Trigonometric Functions

Trigonometric functions like cosine are periodic, meaning they repeat their values in regular intervals. The period of a function y = cos(bx) is given by 2π/b. For y = 3 cos 60x, the period is 2π/60, which simplifies to π/30. Understanding the period helps in setting the viewing window to capture the function's repeating pattern.
추천 영상:
6:04
Introduction to Trigonometric Functions

Amplitude of Trigonometric Functions

The amplitude of a trigonometric function determines the height of its peaks and troughs. In the function y = 3 cos 60x, the amplitude is 3, indicating that the graph oscillates between -3 and 3. This information is crucial for setting the vertical limits of the viewing window to ensure the entire range of the function is visible.
추천 영상:
6:04
Introduction to Trigonometric Functions

Graphing Software Viewing Window

Choosing an appropriate viewing window in graphing software involves setting the x and y-axis limits to effectively display the function's behavior. For y = 3 cos 60x, the x-axis should cover multiple periods to show the function's periodic nature, while the y-axis should accommodate the amplitude. A well-chosen window reveals key features like symmetry and oscillation.
추천 영상:
13:49
Example 4: Norman Window
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Composition of Functions


A balloon’s volume V is given by V = s² + 2s + 3 cm³, where s is the ambient temperature in °C. The ambient temperature s at time t minutes is given by s = 2t − 3 °C. Write the balloon’s volume V as a function of time t.

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Use graphing software to graph the functions specified in Exercises 31–36.

Select a viewing window that reveals the key features of the function.


Graph four periods of the function f (x) = −tan 2x.

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What happens if you take B = 2π in the addition formulas? Do the results agree with something you already know?

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Can a function be both even and odd? Give reasons for your answer.

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Graphing


In Exercises 69–76, graph each function not by plotting points, but by starting with the graph of one of the standard functions presented in Figures 1.14–1.17 and applying an appropriate transformation.


y = (−2x)²/³

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Radians and Degrees


On a circle of radius 10 m, how long is an arc that subtends a central angle of (a) 4π/5 radians? (b) 110°?

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