Final answer:
The linear approximation of the function f(x, y, z) = x² - y² - z² at the point (3, 2, 6) can be used to approximate the expression (3.02)² - (1.97)² - (5.99)² by substituting these values into the linear approximation equation which uses the partial derivatives calculated at (3, 2, 6).
Step-by-step explanation:
To find the linear approximation of the function f(x, y, z) = x² - y² - z² at the point (3, 2, 6), we need to calculate the partial derivatives of the function with respect to x, y, and z at that point. The partial derivatives are:
df/dx = 2x
df/dy = -2y
df/dz = -2z
At the point (3, 2, 6), these partial derivatives evaluate to:
df/dx (3, 2, 6) = 2 * 3 = 6
df/dy (3, 2, 6) = -2 * 2 = -4
df/dz (3, 2, 6) = -2 * 6 = -12
The linear approximation L(x, y, z) near the point (3, 2, 6) is given by:
L(x, y, z) = f(3, 2, 6) + (df/dx)(x - 3) + (df/dy)(y - 2) + (df/dz)(z - 6)
Plugging the values into this formula, we get:
L(x, y, z) = 3² - 2² - 6² + 6(x - 3) - 4(y - 2) - 12(z - 6)
To approximate (3.02)² - (1.97)² - (5.99)² using the linear approximation L(x, y, z), substitute x = 3.02, y = 1.97, and z = 5.99 into L(x, y, z).
L(3.02, 1.97, 5.99) = 3² - 2² - 6² + 6(3.02 - 3) - 4(1.97 - 2) - 12(5.99 - 6)
Computing this expression, we obtain the approximate value for the given expression.