Compute the first-order partial derivatives of the function V = 15tr² (Use symbolic notation and fractions where needed.) av ər = 30лrh
Compute the first-order partial derivatives of the function V = 15tr² (Use symbolic notation and fractions where needed.) av ər = 30лrh
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![### Understanding First-Order Partial Derivatives
#### Problem Statement
Compute the first-order partial derivatives of the function \( V = 15\pi r^2 h \).
(Use symbolic notation and fractions where needed.)
#### Solution
To find the first-order partial derivatives of the given function with respect to \( r \) and \( h \), we differentiate the function \( V \) with respect to each variable separately, treating the other variable as a constant.
1. **Partial Derivative with respect to \( r \):**
\[
\frac{\partial V}{\partial r} = 30\pi rh
\]
2. **Partial Derivative with respect to \( h \):**
\[
\frac{\partial V}{\partial h} = 15\pi r^2
\]
##### Partial Derivative Explanation
- **\(\frac{\partial V}{\partial r}\)**: This represents how the volume \( V \) changes as the radius \( r \) changes, while keeping the height \( h \) constant.
- **\(\frac{\partial V}{\partial h}\)**: This represents how the volume \( V \) changes as the height \( h \) changes, while keeping the radius \( r \) constant.
In this problem, the calculated partial derivative with respect to \( h \) (i.e., \(\frac{\partial V}{\partial h}\)) is:
\[
\frac{\partial V}{\partial h} = 15 r^2
\]
but was incorrectly evaluated as \( \frac{\partial V}{\partial h} = 15hr^2 \). The correct form has been boxed.
### Conclusion
Understanding partial derivatives is crucial, especially in multivariable calculus, as it helps in understanding how a function changes with respect to one variable while keeping the other(s) constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F721da9e6-56ca-4fff-bad1-06b01dcc2531%2F4e4d1d0a-fdf6-45db-8160-a55f2f1d287f%2Felar9d4_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding First-Order Partial Derivatives
#### Problem Statement
Compute the first-order partial derivatives of the function \( V = 15\pi r^2 h \).
(Use symbolic notation and fractions where needed.)
#### Solution
To find the first-order partial derivatives of the given function with respect to \( r \) and \( h \), we differentiate the function \( V \) with respect to each variable separately, treating the other variable as a constant.
1. **Partial Derivative with respect to \( r \):**
\[
\frac{\partial V}{\partial r} = 30\pi rh
\]
2. **Partial Derivative with respect to \( h \):**
\[
\frac{\partial V}{\partial h} = 15\pi r^2
\]
##### Partial Derivative Explanation
- **\(\frac{\partial V}{\partial r}\)**: This represents how the volume \( V \) changes as the radius \( r \) changes, while keeping the height \( h \) constant.
- **\(\frac{\partial V}{\partial h}\)**: This represents how the volume \( V \) changes as the height \( h \) changes, while keeping the radius \( r \) constant.
In this problem, the calculated partial derivative with respect to \( h \) (i.e., \(\frac{\partial V}{\partial h}\)) is:
\[
\frac{\partial V}{\partial h} = 15 r^2
\]
but was incorrectly evaluated as \( \frac{\partial V}{\partial h} = 15hr^2 \). The correct form has been boxed.
### Conclusion
Understanding partial derivatives is crucial, especially in multivariable calculus, as it helps in understanding how a function changes with respect to one variable while keeping the other(s) constant.
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