Textbook Question
Using the Addition Formulas
Use the addition formulas to derive the identities in Exercises 31–36.
sin (A − B) = sin A cos B − cos A sin B
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 1, Problem 1.8
In Exercises 5–8, determine whether the graph of the function is symmetric about the 𝔂-axis, the origin, or neither.
𝔂 = e⁻ˣ²
Verified step by step guidance1
To determine symmetry about the y-axis, check if the function y = f(x) satisfies f(x) = f(-x). Substitute -x into the function: y = e^(-(-x)^2). Simplify to see if it equals the original function.
To determine symmetry about the origin, check if the function y = f(x) satisfies f(-x) = -f(x). Substitute -x into the function: y = e^(-(-x)^2) and compare it to -e^(-x^2).
Simplify the expression e^(-(-x)^2) to e^(-x^2) and compare it to the original function e^(-x^2) to check for y-axis symmetry.
Compare e^(-x^2) with -e^(-x^2) to check for origin symmetry. If they are not equal, the function is not symmetric about the origin.
Conclude whether the function is symmetric about the y-axis, the origin, or neither based on the comparisons made in the previous steps.
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Symmetry about the y-axis
A function is symmetric about the y-axis if replacing x with -x in the function yields the same output. Mathematically, this means that f(-x) = f(x) for all x in the domain of the function. This type of symmetry indicates that the graph is a mirror image across the y-axis.
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06:21
Properties of Functions
Symmetry about the origin
A function is symmetric about the origin if replacing x with -x and y with -y results in the same equation. This is expressed as f(-x) = -f(x). Functions with this symmetry exhibit rotational symmetry of 180 degrees around the origin, meaning that if you rotate the graph, it looks the same.
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06:21Properties of Functions
Exponential functions
Exponential functions, such as y = e^(-x²), are characterized by a constant base raised to a variable exponent. These functions typically exhibit rapid growth or decay. Understanding their general shape and behavior is crucial for analyzing their symmetry properties, as they often do not possess symmetry about the y-axis or the origin.
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Related Practice
Textbook Question
Using the Half-Angle Formulas
Find the function values in Exercises 47–50.
sin² 3π/8
Textbook Question
Refer to the given figure. Write the radius r of the circle in terms of α and θ.
Textbook Question
Graph the functions in Exercises 37–56.
y = |x − 2|
Textbook Question
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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Textbook Question
In Exercises 19–32, find the (a) domain and (b) range.
_____
𝔂 = -1 + ∛ 2 - x
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