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Chapter 13: Problem 5

Explain how to approximate a function \(f\) at a point near \((a, b)\) where the values of \(f, f_{x},\) and \(f_{y}\) are known at \((a, b)\)

Short Answer

Expert verified
Answer: We can use the first-order Taylor series expansion to find a linear approximation of the function near the given point. The formula for the first-order Taylor series expansion is as follows: $$ L(x,y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) $$ By substituting the known values of \(f(a,b)\), \(f_x(a, b)\), and \(f_y(a, b)\) in the formula, we can approximate the function \(f\) near the point \((a, b)\).

Step by step solution

01

Notation

Before proceeding, let's define some notation: - \(f\): The original function - \(f_x\): Partial derivative of \(f\) with respect to \(x\) - \(f_y\): Partial derivative of \(f\) with respect to \(y\) - \((a, b)\): The point given where the values of \(f\), \(f_x\), and \(f_y\) are known
02

Understanding the Taylor series expansion

The Taylor series expansion is a technique to approximate a function around a given point. The first-order Taylor series expansion is a linear approximation that utilizes the values of the function and its partial derivatives at a specific point. The general form is given by: $$ f(x,y) \approx L(x,y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) $$
03

Replace known values in the Taylor series expansion

For this exercise, we know the values of \(f(a,b)\), \(f_x(a, b)\), and \(f_y(a, b)\). Replacing these values in the Taylor series expansion formula, we get the linear approximation of the function \(f\) near \(x=a\) and \(y=b\) as: $$ L(x,y) = f(a, b) + f_x(a, b)(x - a) + f_y(a, b)(y - b) $$
04

Example usage

Suppose we have a function \(f(x, y) = x^2y\), and we are given \(f(1,2)=2\), \(f_x(1, 2)=4\) and \(f_y(1, 2)=1\). Using the first-order Taylor series, we can approximate the value of the function near the point \((1, 2)\) as follows: $$ L(x,y) = 2 + 4(x - 1) + 1(y - 2) $$ So if we want to approximate the value of \(f\) at \(x=1.1\) and \(y=2.1\), the approximation will be: $$ L(1.1, 2.1) = 2 + 4(1.1 - 1) + 1(2.1 - 2) $$ Hence, our approximation of \(f(1.1, 2.1)\) as per the first-order Taylor series expansion is \(L(1.1, 2.1)\).

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