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Please show the mathematical reasoning on why these are all equal curves

Please show the mathematical reasoning on why these are all equal curves

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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Please show the mathematical reasoning on why these are all equal curves

The image contains three graphs, each displaying a function and its shaded area under the curve between specified limits. 1. **Top Left Graph**: - **Function**: \( y = e^{\sqrt{x}} \) - **Axes**: The x-axis ranges from 0 to 1, and the y-axis scales accordingly. - **Description**: The graph shows the curve \( y = e^{\sqrt{x}} \) with the area under the curve from \( x = 0 \) to \( x = 1 \) shaded in orange. 2. **Top Right Graph**: - **Function**: \( y = 2xe^x \) - **Axes**: The x-axis ranges from 0 to 1, while the y-axis scales to accommodate the function’s rapid growth. - **Description**: The graph displays the curve \( y = 2xe^x \), with the area under the curve from \( x = 0 \) to \( x = 1 \) shaded in orange. 3. **Bottom Graph**: - **Function**: \( y = e^{\sin x} \sin 2x \) - **Axes**: The x-axis ranges from 0 to \(\pi/2\), with the y-axis set to capture oscillatory behavior. - **Description**: This graph illustrates the curve \( y = e^{\sin x} \sin 2x \). The area under the curve from \( x = 0 \) to \( x = \pi/2 \) is shaded in orange. The function shows oscillatory behavior due to the sine components. Each graph provides an example of how exponential and trigonometric functions can be visualized and analyzed to understand their behavior within given intervals.
Transcribed Image Text:The image contains three graphs, each displaying a function and its shaded area under the curve between specified limits. 1. **Top Left Graph**: - **Function**: \( y = e^{\sqrt{x}} \) - **Axes**: The x-axis ranges from 0 to 1, and the y-axis scales accordingly. - **Description**: The graph shows the curve \( y = e^{\sqrt{x}} \) with the area under the curve from \( x = 0 \) to \( x = 1 \) shaded in orange. 2. **Top Right Graph**: - **Function**: \( y = 2xe^x \) - **Axes**: The x-axis ranges from 0 to 1, while the y-axis scales to accommodate the function’s rapid growth. - **Description**: The graph displays the curve \( y = 2xe^x \), with the area under the curve from \( x = 0 \) to \( x = 1 \) shaded in orange. 3. **Bottom Graph**: - **Function**: \( y = e^{\sin x} \sin 2x \) - **Axes**: The x-axis ranges from 0 to \(\pi/2\), with the y-axis set to capture oscillatory behavior. - **Description**: This graph illustrates the curve \( y = e^{\sin x} \sin 2x \). The area under the curve from \( x = 0 \) to \( x = \pi/2 \) is shaded in orange. The function shows oscillatory behavior due to the sine components. Each graph provides an example of how exponential and trigonometric functions can be visualized and analyzed to understand their behavior within given intervals.
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