College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter3: Functions
Section3.4: Operations On Functions
Problem 123E
Related questions
Question
Find the area of the surface obtained by rotating the given curve about the x-axis.
(Cal2)
![The image illustrates a plot of the function \( y = \sqrt{1 + 2e^x} \). This function is graphed over a Cartesian coordinate system showing portions of the positive x-axis and y-axis.
### Graph Details:
- **Axes**: The x-axis and y-axis intersect at the origin (0,0). The graph extends to cover positive values of both x and y, with notable grid lines marked.
- **Curve**: The red curve represents the function \( y = \sqrt{1 + 2e^x} \). It starts relatively low and rises steeply as \( x \) increases, indicating an exponential growth moderated by the square root.
- **Visualization**: The graph shows half of the parabola due to the nature of the function. The function grows rapidly for positive values of x.
### Mathematical Function:
**Equation**:
\[ y = \sqrt{1 + 2e^x} \]
- **Exponential Component**: \( e^x \) indicates the exponential component, where \( e \) is the base of the natural logarithm.
- **Addition and Scaling**: The term \( 2e^x \) scales and shifts the exponential growth, and adding 1 ensures the argument of the square root is always positive.
- **Square Root**: The square root function modifies the rapid exponential growth to a more subdued increase.
### Interpretation:
The function \( y = \sqrt{1 + 2e^x} \) models a scenario where growth is exponential but is moderated by taking the square root, leading to a less rapid increase compared to purely exponential functions. This kind of function can be used in various fields like economics, biology, and engineering to model growth processes that escalate quickly but have some form of attenuation.
### Usage:
This graph and function are useful for teaching exponential growth and its transformations. It can also demonstrate how applying operations like square roots can alter the growth rate of functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f5b0ad-df1d-48c6-a9a3-5338b8167143%2F67abbacd-4ceb-42c3-988a-c8de88eda4b4%2F15npn6r_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image illustrates a plot of the function \( y = \sqrt{1 + 2e^x} \). This function is graphed over a Cartesian coordinate system showing portions of the positive x-axis and y-axis.
### Graph Details:
- **Axes**: The x-axis and y-axis intersect at the origin (0,0). The graph extends to cover positive values of both x and y, with notable grid lines marked.
- **Curve**: The red curve represents the function \( y = \sqrt{1 + 2e^x} \). It starts relatively low and rises steeply as \( x \) increases, indicating an exponential growth moderated by the square root.
- **Visualization**: The graph shows half of the parabola due to the nature of the function. The function grows rapidly for positive values of x.
### Mathematical Function:
**Equation**:
\[ y = \sqrt{1 + 2e^x} \]
- **Exponential Component**: \( e^x \) indicates the exponential component, where \( e \) is the base of the natural logarithm.
- **Addition and Scaling**: The term \( 2e^x \) scales and shifts the exponential growth, and adding 1 ensures the argument of the square root is always positive.
- **Square Root**: The square root function modifies the rapid exponential growth to a more subdued increase.
### Interpretation:
The function \( y = \sqrt{1 + 2e^x} \) models a scenario where growth is exponential but is moderated by taking the square root, leading to a less rapid increase compared to purely exponential functions. This kind of function can be used in various fields like economics, biology, and engineering to model growth processes that escalate quickly but have some form of attenuation.
### Usage:
This graph and function are useful for teaching exponential growth and its transformations. It can also demonstrate how applying operations like square roots can alter the growth rate of functions.
![In this educational content, we explore the mathematical expression given by:
\[ y = \sqrt{1 + 2e^x} \quad \text{for} \quad 0 \leq x \leq 3 \]
This equation represents a function \( y \) in terms of \( x \), defined over the interval from \( x = 0 \) to \( x = 3 \).
### Explanation of the Expression:
- \( y \): The dependent variable, whose value depends on \( x \).
- \( \sqrt{1 + 2e^x} \): This part of the expression indicates that \( y \) is the square root of the sum of 1 and twice the exponential function \( e \) raised to the power of \( x \).
- \( e \): Euler's number, which is approximately equal to 2.71828.
- \( e^x \): The exponential function of \( x \).
### Domain:
The function is defined for the interval \( 0 \leq x \leq 3 \).
### Notes:
- The square root function ensures that the value under the square root, \( 1 + 2e^x \), must be non-negative, which it is for all real values of \( x \).
- The exponential function \( e^x \) grows rapidly as \( x \) increases.
This function can be graphed to show its behavior over the interval \( 0 \leq x \leq 3 \). The graph would start at \( x = 0 \) and end at \( x = 3 \). It would depict how \( y \) changes as \( x \) ranges within its domain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f5b0ad-df1d-48c6-a9a3-5338b8167143%2F67abbacd-4ceb-42c3-988a-c8de88eda4b4%2F733monm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In this educational content, we explore the mathematical expression given by:
\[ y = \sqrt{1 + 2e^x} \quad \text{for} \quad 0 \leq x \leq 3 \]
This equation represents a function \( y \) in terms of \( x \), defined over the interval from \( x = 0 \) to \( x = 3 \).
### Explanation of the Expression:
- \( y \): The dependent variable, whose value depends on \( x \).
- \( \sqrt{1 + 2e^x} \): This part of the expression indicates that \( y \) is the square root of the sum of 1 and twice the exponential function \( e \) raised to the power of \( x \).
- \( e \): Euler's number, which is approximately equal to 2.71828.
- \( e^x \): The exponential function of \( x \).
### Domain:
The function is defined for the interval \( 0 \leq x \leq 3 \).
### Notes:
- The square root function ensures that the value under the square root, \( 1 + 2e^x \), must be non-negative, which it is for all real values of \( x \).
- The exponential function \( e^x \) grows rapidly as \( x \) increases.
This function can be graphed to show its behavior over the interval \( 0 \leq x \leq 3 \). The graph would start at \( x = 0 \) and end at \( x = 3 \). It would depict how \( y \) changes as \( x \) ranges within its domain.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 4 images
Recommended textbooks for you
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning
College Algebra (MindTap Course List)
Algebra
ISBN:
9781305652231
Author:
R. David Gustafson, Jeff Hughes
Publisher:
Cengage Learning