3 Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) = 1-3x-2y near the origin. The quadratic approximation for f(x,y) is
3 Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f(x,y) = 1-3x-2y near the origin. The quadratic approximation for f(x,y) is
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Topic: Taylor Series Approximation in Multivariable Calculus**
### Taylor Series for Multivariable Functions
In this lesson, we will use Taylor's formula for \( f(x,y) \) at the origin to find the quadratic and cubic approximations of the function:
\[
f(x,y) = \frac{3}{1 - 3x - 2y}
\]
near the origin.
### Step-by-Step Solution
The process involves calculating partial derivatives of the function at the origin and using Taylor's series expansion to approximate the function up to the desired degree.
#### Quadratic Approximation
To find the quadratic approximation for \( f(x,y) \), we need to consider the first and second-order partial derivatives of the function.
**Computation Details:**
1. Compute the first-order partial derivatives:
- \( f_x = \frac{\partial}{\partial x} f(x,y) \)
- \( f_y = \frac{\partial}{\partial y} f(x,y) \)
2. Evaluate these derivatives at the origin \( (0,0) \).
3. Compute the second-order partial derivatives:
- \( f_{xx} = \frac{\partial^2}{\partial x^2} f(x,y) \)
- \( f_{xy} = \frac{\partial^2}{\partial x \partial y} f(x,y) \)
- \( f_{yy} = \frac{\partial^2}{\partial y^2} f(x,y) \)
4. Evaluate these derivatives at the origin \( (0,0) \).
5. Use the Taylor expansion formula for quadratic approximation:
\[
f(x,y) \approx f(0,0) + f_x(0,0) \cdot x + f_y(0,0) \cdot y + \frac{1}{2} f_{xx}(0,0) \cdot x^2 + f_{xy}(0,0) \cdot xy + \frac{1}{2} f_{yy}(0,0) \cdot y^2
\]
**Expression of Quadratic Approximation:**
\[
f(x,y) \approx [QUADRATIC APPROXIMATION HERE]
\]
*Note: Fill in the calculated values from the derivative evaluations.*
Next, follow similar steps for the cubic approximation, including third-order partial derivatives and evaluating them](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb21cc1b3-0a4b-474e-87d5-2a9f16683342%2F2c4df09e-d1fc-4e9e-8235-671eefdf0235%2Fqkomxoi_processed.png&w=3840&q=75)
Transcribed Image Text:**Topic: Taylor Series Approximation in Multivariable Calculus**
### Taylor Series for Multivariable Functions
In this lesson, we will use Taylor's formula for \( f(x,y) \) at the origin to find the quadratic and cubic approximations of the function:
\[
f(x,y) = \frac{3}{1 - 3x - 2y}
\]
near the origin.
### Step-by-Step Solution
The process involves calculating partial derivatives of the function at the origin and using Taylor's series expansion to approximate the function up to the desired degree.
#### Quadratic Approximation
To find the quadratic approximation for \( f(x,y) \), we need to consider the first and second-order partial derivatives of the function.
**Computation Details:**
1. Compute the first-order partial derivatives:
- \( f_x = \frac{\partial}{\partial x} f(x,y) \)
- \( f_y = \frac{\partial}{\partial y} f(x,y) \)
2. Evaluate these derivatives at the origin \( (0,0) \).
3. Compute the second-order partial derivatives:
- \( f_{xx} = \frac{\partial^2}{\partial x^2} f(x,y) \)
- \( f_{xy} = \frac{\partial^2}{\partial x \partial y} f(x,y) \)
- \( f_{yy} = \frac{\partial^2}{\partial y^2} f(x,y) \)
4. Evaluate these derivatives at the origin \( (0,0) \).
5. Use the Taylor expansion formula for quadratic approximation:
\[
f(x,y) \approx f(0,0) + f_x(0,0) \cdot x + f_y(0,0) \cdot y + \frac{1}{2} f_{xx}(0,0) \cdot x^2 + f_{xy}(0,0) \cdot xy + \frac{1}{2} f_{yy}(0,0) \cdot y^2
\]
**Expression of Quadratic Approximation:**
\[
f(x,y) \approx [QUADRATIC APPROXIMATION HERE]
\]
*Note: Fill in the calculated values from the derivative evaluations.*
Next, follow similar steps for the cubic approximation, including third-order partial derivatives and evaluating them
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