Math Problem Statement

Determine the linearization of the function F(x, y) = (y + sqrt(x), x + cuberoot(y)) at the point (9,8). Approximate the value of F at the point (8,9) using the linearization.

Solution

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Problem 7:
Determine the linearization of the function
F(x,y)=(y+x,x+y3)F(x, y) = (y + \sqrt{x}, x + \sqrt[3]{y})
at the point (9,8)(9, 8). Approximate the value of FF at the point (8,9)(8, 9) using the linearization.


To solve this, we need to follow these steps:

  1. Find the function value at the point (9,8)(9, 8).
  2. Compute the partial derivatives of the function F(x,y)F(x, y) with respect to both xx and yy.
  3. Set up the linearization formula: L(x,y)=F(9,8)+Fx(9,8)(x9)+Fy(9,8)(y8)L(x, y) = F(9, 8) + F_x(9, 8)(x - 9) + F_y(9, 8)(y - 8)
  4. Use the linearization to approximate F(8,9)F(8, 9).

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Math Problem Analysis

Mathematical Concepts

Multivariable Calculus
Linear Approximation
Partial Derivatives

Formulas

Linearization: L(x, y) = F(a, b) + F_x(a, b)(x - a) + F_y(a, b)(y - b)
Partial derivatives: F_x = ∂F/∂x, F_y = ∂F/∂y

Theorems

Differential Approximation
First-order Taylor Expansion

Suitable Grade Level

University Calculus (1st or 2nd Year)