Question

Let f(x, y) = x² / (y² + 1). Use the Linear Approximation at an appropriate point (a, b) to estimate f(5.05, 1.92). (Use decimal notation. Give your answer to two decimal places.)

Answer 1

To estimate f(5.05, 1.92) using linear approximation, we need to find the equation of the tangent line at an appropriate point (a, b). Let's choose (a, b) = (5, 2) as the point of approximation. The estimated value of f(5.05, 1.92) using linear approximation is 0.384.

Step-by-step explanation:

To estimate f(5.05, 1.92) using linear approximation, we need to find the equation of the tangent line at an appropriate point (a, b). Let's choose (a, b) = (5, 2) as the point of approximation.

First, find the partial derivatives:

• ∂f/∂x = 2x / (y² + 1)
• ∂f/∂y = -2xy / (y² + 1)²

Next, find the values of the partial derivatives at (a, b):

• ∂f/∂x = (2 * 5) / (2² + 1) = 10/5 = 2
• ∂f/∂y = (-2 * 5 * 2) / (2² + 1)² = -20/25 = -0.8

The equation of the tangent line at (5, 2) is:

y - b = (∂f/∂x)(x - a) + (∂f/∂y)(y - b)

y - 2 = 2(x - 5) - 0.8(y - 2)

Simplifying:

0.2y = 1.6x - 6

Now, plug in the values of x = 5.05 and y = 1.92 into the equation of the line:

0.2(1.92) = 1.6(5.05) - 6

0.384 = 8.08 - 6

0.384 = 2.08

Therefore, the estimated value of f(5.05, 1.92) using linear approximation is 0.384.