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2) y = sin(3x²ex)

2) y = sin(3x²ex)

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
**Equation Analysis:** The given equation is: \( y = \sin(3x^2 \cdot e^{-x}) \) ### Explanation: This mathematical equation represents a function \( y \) in terms of \( x \). The function utilizes: 1. **Sine Function** (\( \sin \)): A trigonometric function that oscillates between -1 and 1. 2. **Exponential Decay** (\( e^{-x} \)): An expression indicating the exponential base \( e \) raised to the power of \(-x\), which results in a decay as \( x \) increases. 3. **Quadratic Term** (\( 3x^2 \)): A polynomial term where \( x \) is squared and multiplied by 3. ### Components: - **Inside the sine function**, the expression \( 3x^2 \cdot e^{-x} \) is a product of a quadratic expression \( 3x^2 \) and an exponentially decaying function \( e^{-x} \). - **Behavior of the function**: As you vary \( x \), - The quadratic term \( 3x^2 \) will increase with increasing values of \( x \). - The exponential term \( e^{-x} \) will decrease, approaching zero for large \( x \). - The overall behavior of the function depends on the interplay between these terms, creating an oscillating pattern influenced by both polynomial growth and exponential decay. This function could be used to model phenomena where oscillatory behavior is modulated by growth and decay dynamics, such as in certain waveforms or signal processing applications.
Transcribed Image Text:**Equation Analysis:** The given equation is: \( y = \sin(3x^2 \cdot e^{-x}) \) ### Explanation: This mathematical equation represents a function \( y \) in terms of \( x \). The function utilizes: 1. **Sine Function** (\( \sin \)): A trigonometric function that oscillates between -1 and 1. 2. **Exponential Decay** (\( e^{-x} \)): An expression indicating the exponential base \( e \) raised to the power of \(-x\), which results in a decay as \( x \) increases. 3. **Quadratic Term** (\( 3x^2 \)): A polynomial term where \( x \) is squared and multiplied by 3. ### Components: - **Inside the sine function**, the expression \( 3x^2 \cdot e^{-x} \) is a product of a quadratic expression \( 3x^2 \) and an exponentially decaying function \( e^{-x} \). - **Behavior of the function**: As you vary \( x \), - The quadratic term \( 3x^2 \) will increase with increasing values of \( x \). - The exponential term \( e^{-x} \) will decrease, approaching zero for large \( x \). - The overall behavior of the function depends on the interplay between these terms, creating an oscillating pattern influenced by both polynomial growth and exponential decay. This function could be used to model phenomena where oscillatory behavior is modulated by growth and decay dynamics, such as in certain waveforms or signal processing applications.
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