Question

Find the derivative of the function at the given point in the direction of

f(x, y, z)=-2 x-7 y+8 z, (-9,-8,5), A=3 i-6 j-2 k

Answer 1

The derivative of the function at the given point in the direction of the given vector is 140.

Step-by-step explanation:

To find the derivative of the function f(x, y, z) = -2x - 7y + 8z at the point (-9,-8,5) in the direction of A = 3i - 6j - 2k, we first need to find the gradient of the function.

1. Take the partial derivatives of the function with respect to x, y, and z.
   • df/dx = -2
   • df/dy = -7
   • df/dz = 8
2. Create a gradient vector using the partial derivatives.
   • gradient vector = -2i - 7j + 8k
3. Find the dot product of the gradient vector and the direction vector A.
   • dot product = (-2)(3) + (-7)(-6) + (8)(-2) = -6 + 42 - 16 = 20
4. Multiply the dot product by the magnitude of the direction vector A.
   • derivative = 20 * |A| = 20 * sqrt((3)^2 + (-6)^2 + (-2)^2) = 20 * sqrt(9 + 36 + 4) = 20 * sqrt(49) = 20 * 7 = 140

Therefore, the derivative of the function at the point (-9,-8,5) in the direction of A is 140.