Textbook Question
Express the radius of a sphere as a function of the sphere’s surface area. Then express the surface area as a function of the volume.
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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 21
Graph the functions in Exercises 13–22. What is the period of each function?
sin (x − π/4) + 1
Verified step by step guidance1
Identify the function to be graphed: \( f(x) = \sin(x - \frac{\pi}{4}) + 1 \). This is a transformation of the basic sine function.
Determine the horizontal shift: The term \( x - \frac{\pi}{4} \) indicates a phase shift to the right by \( \frac{\pi}{4} \) units.
Determine the vertical shift: The '+1' outside the sine function indicates a vertical shift upwards by 1 unit.
Identify the period of the function: The period of the basic sine function \( \sin(x) \) is \( 2\pi \). Since there is no coefficient affecting the \( x \) inside the sine function, the period remains \( 2\pi \).
Graph the function: Start by plotting the basic sine curve, apply the horizontal shift to the right by \( \frac{\pi}{4} \), and then shift the entire graph upwards by 1 unit. The resulting graph will have the same shape as the sine curve but will be shifted accordingly.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Period of a Function
The period of a function is the length of the interval over which the function repeats itself. For trigonometric functions like sine and cosine, the period is a key characteristic that determines how often the function cycles through its values. For the sine function, the standard period is 2π, meaning it completes one full cycle over this interval.
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Graphs of Secant and Cosecant Functions
Transformation of Functions
Transformations involve shifting, stretching, or compressing the graph of a function. In the given function sin(x − π/4) + 1, the term (x − π/4) indicates a horizontal shift to the right by π/4 units, while the +1 indicates a vertical shift upward by 1 unit. Understanding these transformations is essential for accurately graphing the function.
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5:25Intro to Transformations
Graphing Trigonometric Functions
Graphing trigonometric functions requires knowledge of their basic shapes and how transformations affect these shapes. The sine function typically oscillates between -1 and 1, and when transformed, its amplitude and vertical position can change. For sin(x − π/4) + 1, the graph will oscillate between 0 and 2, reflecting the vertical shift.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Graph the functions in Exercises 23–26 in the ts-plane (t-axis horizontal, s-axis vertical). What is the period of each function? What symmetries do the graphs have?
s = −tan πt
Textbook Question
[Technology Exercise]
a. Graph y = cos x and y = sec x together for −3π/2 ≤ x ≤ 3π/2. Comment on the behavior of sec x in relation to the signs and values of cos x.
Textbook Question
Copy and complete the following table of function values. If the function is undefined at a given angle, enter “UND.” Do not use a calculator or tables.
Textbook Question
Graph the functions in Exercises 13–22. What is the period of each function?
sin (x/2)
Textbook Question
Graph y = sin x and y = ⌊sin x⌋ together. What are the domain and range of ⌊sin x⌋?