Textbook Question
Functions
In Exercises 1–6, find the domain and range of each function.
f(x) = 1 + x²
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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
Chapter 1, Problem 1.3.70
General Sine Curves
For
f(x) = A sin ((2π/B)(x – C) +D
identify A, B, C, and D for the sine functions in Exercises 67–70 and sketch their graphs.
y = L/2π sin (2πt/L), L > 0
Verified step by step guidance1
Identify the general form of the sine function: \( f(x) = A \sin\left(\frac{2\pi}{B}(x - C)\right) + D \). Here, \( A \) is the amplitude, \( B \) determines the period, \( C \) is the horizontal shift, and \( D \) is the vertical shift.
Compare the given function \( y = \frac{L}{2\pi} \sin\left(\frac{2\pi t}{L}\right) \) with the general form. Notice that \( A = \frac{L}{2\pi} \), \( B = L \), \( C = 0 \), and \( D = 0 \).
Determine the amplitude: The amplitude is the absolute value of \( A \), which is \( |A| = \left|\frac{L}{2\pi}\right| \). This represents the maximum vertical displacement of the sine wave from its midline.
Determine the period: The period of the sine function is given by \( B \), which is \( L \). This represents the length of one complete cycle of the sine wave.
Sketch the graph: Plot the sine wave with the identified parameters. The wave has an amplitude of \( \frac{L}{2\pi} \), a period of \( L \), no horizontal shift (\( C = 0 \)), and no vertical shift (\( D = 0 \)). The graph oscillates symmetrically about the x-axis.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Amplitude (A)
The amplitude of a sine function, represented by A, determines the height of the wave from its midline to its peak. It indicates how far the function's values stretch above and below the midline (y=0). In the context of the given function, a larger A results in a taller wave, while a smaller A compresses the wave vertically.
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Period (B)
The period of a sine function, denoted by B, is the distance along the x-axis required for the function to complete one full cycle. It is calculated using the formula Period = 2π/B. In the provided function, adjusting B alters the frequency of the wave, with a smaller B leading to more cycles within a given interval.
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Phase Shift (C) and Vertical Shift (D)
The phase shift, represented by C, indicates how much the graph of the sine function is shifted horizontally, while the vertical shift, denoted by D, moves the graph up or down. Specifically, C affects the starting point of the wave, and D adjusts the midline of the wave. Together, they help position the sine curve in the coordinate plane.
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Related Practice
Textbook Question
Functions and Graphs
Express the area and circumference of a circle as functions of the circle’s radius. Then express the area as a function of the circumference.
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Textbook Question
In Exercises 39–42, express the given quantity in terms of sin x and cos x.
sin (2π − x)
Textbook Question
In Exercises 9–16, determine whether the function is even, odd, or neither.
𝔂 = x⁴ + 1
x³ - 2x
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Textbook Question
Graph the functions in Exercises 37–56.
y = (x + 2)³/² + 1
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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
𝔂 = cos(x - 3) + 1
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