교과서 질문
Graph the functions in Exercises 37–56.
y = |x − 2|
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Ch. 1 - Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
1장, 문제 1.4.35
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 the function f (x) = sin 2x + cos 3x.
검증된 단계별 안내1
Step 1: Understand the function f(x) = sin(2x) + cos(3x). This is a combination of two trigonometric functions, where sin(2x) oscillates with a frequency determined by the coefficient 2, and cos(3x) oscillates with a frequency determined by the coefficient 3. The sum of these functions creates a more complex waveform.
Step 2: Choose an appropriate graphing software or tool, such as Desmos, GeoGebra, or a graphing calculator. These tools allow you to input the function and adjust the viewing window to observe its behavior.
Step 3: Input the function f(x) = sin(2x) + cos(3x) into the graphing software. Ensure that the syntax matches the software's requirements (e.g., use 'sin(2x)' and 'cos(3x)' directly).
Step 4: Select a viewing window that reveals the key features of the function. For trigonometric functions, a good starting point is to set the x-axis range from -2π to 2π (or -6.28 to 6.28) to capture at least one full period of oscillation. Adjust the y-axis range to accommodate the amplitude of the function, which is determined by the sum of the maximum values of sin(2x) and cos(3x).
Step 5: Analyze the graph. Look for key features such as the amplitude, frequency, and points of intersection between sin(2x) and cos(3x). Observe how the combination of these functions creates a unique waveform and note any symmetry or periodicity in the graph.
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주요 개념
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Trigonometric Functions
Trigonometric functions, such as sine and cosine, are fundamental periodic functions that describe relationships between angles and sides in triangles. The sine function, sin(x), represents the y-coordinate of a point on the unit circle, while the cosine function, cos(x), represents the x-coordinate. Understanding their properties, including amplitude, period, and phase shift, is essential for analyzing and graphing their combinations.
추천 영상:
6:04
Introduction to Trigonometric Functions
Graphing Functions
Graphing functions involves plotting points on a coordinate system to visualize the relationship between the input (x-values) and output (f(x) values). For trigonometric functions, this includes identifying key features such as intercepts, maxima, minima, and periodicity. Using graphing software allows for precise representation and manipulation of these functions, making it easier to observe their behavior over specified intervals.
추천 영상:
5:53Graph of Sine and Cosine Function
Viewing Window
The viewing window in graphing software defines the range of x and y values displayed on the graph. Selecting an appropriate viewing window is crucial for revealing key features of the function, such as peaks, troughs, and periodicity. A well-chosen window allows for a comprehensive understanding of the function's behavior, ensuring that important characteristics are not overlooked.
추천 영상:
13:49Example 4: Norman Window
관련 실천
교과서 질문
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
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교과서 질문
Even and Odd Functions
In Exercises 47–62, say whether the function is even, odd, or neither. Give reasons for your answer.
f(x) = x² + x
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교과서 질문
In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x
341
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교과서 질문
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x - sin x
241
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교과서 질문
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 two periods of the function f (x) = 3 cot (x/2) + 1.
131
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