To find the linear approximation of the function f(x, y, z) = x² + y² + z² at the point (3, 2, 6), we first need to calculate the value of the function and the partial derivatives at that point.
Step 1: Evaluate the function at (3, 2, 6)
Plugging in the values, we get:
f(3, 2, 6) = 3² + 2² + 6² = 9 + 4 + 36 = 49
Step 2: Calculate Partial Derivatives
Next, we need to find the partial derivatives of the function:
- Partial derivative with respect to x:
- Partial derivative with respect to y:
- Partial derivative with respect to z:
∂f/∂x = 2x
∂f/∂y = 2y
∂f/∂z = 2z
Now, we evaluate these derivatives at the point (3, 2, 6):
- ∂f/∂x (3, 2, 6) = 2(3) = 6
- ∂f/∂y (3, 2, 6) = 2(2) = 4
- ∂f/∂z (3, 2, 6) = 2(6) = 12
Step 3: Set Up the Linear Approximation
The linear approximation of a function of several variables can be expressed as:
L(x, y, z) = f(a, b, c) + ∂f/∂x(a, b, c)(x – a) + ∂f/∂y(a, b, c)(y – b) + ∂f/∂z(a, b, c)(z – c)
Substituting in our values (a = 3, b = 2, c = 6):
L(x, y, z) = 49 + 6(x – 3) + 4(y – 2) + 12(z – 6)
Step 4: Simplify the Linear Approximation
Expanding this yields:
L(x, y, z) = 49 + 6x – 18 + 4y – 8 + 12z – 72
L(x, y, z) = 6x + 4y + 12z – 49
Thus, the linear approximation of the function f(x, y, z) at the point (3, 2, 6) is:
L(x, y, z) = 6x + 4y + 12z – 49