Find second partial derivatives of the function f(x, y, z) = 3e*yz at the point xo = (-6, -5, 1). (Use symbolic notation and fractions where needed.)
Find second partial derivatives of the function f(x, y, z) = 3e*yz at the point xo = (-6, -5, 1). (Use symbolic notation and fractions where needed.)
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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![**Topic: Calculating Second Partial Derivatives**
In this lesson, we will find the second partial derivatives of a given function at a specified point.
**Function:**
\[ f(x, y, z) = 3e^{xyz} \]
**Point:**
\[ x_0 = (-6, -5, 1) \]
**Objective:**
Calculate the following second partial derivatives using symbolic notation and fractions if necessary:
1. \( f_{xx}(-6, -5, 1) = \)
2. \( f_{yy}(-6, -5, 1) = \)
3. \( f_{zz}(-6, -5, 1) = \)
4. \( f_{xy}(-6, -5, 1) = \)
5. \( f_{xz}(-6, -5, 1) = \)
6. \( f_{yz}(-6, -5, 1) = \)
These second partial derivatives help us understand the curvature of the function's surface at the given point in each respective direction. Understanding how to calculate these derivatives is crucial in fields like physics, engineering, and computer graphics, where multi-variable functions model real-world phenomena.
Let’s proceed to solve for each derivative using the derivative rules for partial differentiation.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b5ef7d0-c879-48bd-8064-864fb9c6c6dd%2Fad8ea43c-3c16-43ce-b98b-e0b7a758491a%2Fkrpz52w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Topic: Calculating Second Partial Derivatives**
In this lesson, we will find the second partial derivatives of a given function at a specified point.
**Function:**
\[ f(x, y, z) = 3e^{xyz} \]
**Point:**
\[ x_0 = (-6, -5, 1) \]
**Objective:**
Calculate the following second partial derivatives using symbolic notation and fractions if necessary:
1. \( f_{xx}(-6, -5, 1) = \)
2. \( f_{yy}(-6, -5, 1) = \)
3. \( f_{zz}(-6, -5, 1) = \)
4. \( f_{xy}(-6, -5, 1) = \)
5. \( f_{xz}(-6, -5, 1) = \)
6. \( f_{yz}(-6, -5, 1) = \)
These second partial derivatives help us understand the curvature of the function's surface at the given point in each respective direction. Understanding how to calculate these derivatives is crucial in fields like physics, engineering, and computer graphics, where multi-variable functions model real-world phenomena.
Let’s proceed to solve for each derivative using the derivative rules for partial differentiation.
![**Title: Exploring the Second Partial Derivatives of a Function**
**Objective**: Determine the second partial derivatives of the function \( f(x, y) = x^4 + x^5 + y^7 \) at the point \( x_0 = (3, 5) \).
### Second Partial Derivatives Formulas
1. **Second Partial Derivative with respect to \( x \)**:
\[
\frac{\partial^2 f}{\partial x^2} = \boxed{}
\]
2. **Second Partial Derivative with respect to \( y \)**:
\[
\frac{\partial^2 f}{\partial y^2} = \boxed{}
\]
3. **Mixed Partial Derivative \((x \rightarrow y)\)**:
\[
\frac{\partial^2 f}{\partial x \partial y} = \boxed{}
\]
4. **Mixed Partial Derivative \((y \rightarrow x)\)**:
\[
\frac{\partial^2 f}{\partial y \partial x} = \boxed{}
\]
**Instructions**: Calculate each derivative step by step at the specific point \( x_0 = (3, 5) \).
**Question Source**: Rogawski 4e Calculus Early Transcendentals
This exercise helps understand how to compute and interpret the second-order partial derivatives of a multivariable function, which play a crucial role in identifying the curvature and behavior of the function's graph around a given point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b5ef7d0-c879-48bd-8064-864fb9c6c6dd%2Fad8ea43c-3c16-43ce-b98b-e0b7a758491a%2Ffxi288f_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Exploring the Second Partial Derivatives of a Function**
**Objective**: Determine the second partial derivatives of the function \( f(x, y) = x^4 + x^5 + y^7 \) at the point \( x_0 = (3, 5) \).
### Second Partial Derivatives Formulas
1. **Second Partial Derivative with respect to \( x \)**:
\[
\frac{\partial^2 f}{\partial x^2} = \boxed{}
\]
2. **Second Partial Derivative with respect to \( y \)**:
\[
\frac{\partial^2 f}{\partial y^2} = \boxed{}
\]
3. **Mixed Partial Derivative \((x \rightarrow y)\)**:
\[
\frac{\partial^2 f}{\partial x \partial y} = \boxed{}
\]
4. **Mixed Partial Derivative \((y \rightarrow x)\)**:
\[
\frac{\partial^2 f}{\partial y \partial x} = \boxed{}
\]
**Instructions**: Calculate each derivative step by step at the specific point \( x_0 = (3, 5) \).
**Question Source**: Rogawski 4e Calculus Early Transcendentals
This exercise helps understand how to compute and interpret the second-order partial derivatives of a multivariable function, which play a crucial role in identifying the curvature and behavior of the function's graph around a given point.
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