Mathematics: Consider the function f(x,y,z)equalse Superscript xyz minus 3, the point P(0 comma 3 comma negative 2 ), and the unit vector Bold uequalsleft angle seven ninths comma four ninths comma four ninths right angle . a. Compute the gradient of f and evaluate it at P. b. Find the unit vector in the direction of maximum increase of f at P. c. Find the rate of change of the function in the direction of maximum increase at P. d. Find the directional derivative at P in the direction of the given vector.

Consider the function f(x,y,z)equalse Superscript

Ответ:

karlyisaunicorn

19.01.2022

See the step by step solution.

Step-by-step explanation:

The main topics of this question are gradient and directional derivative of a multivariable function. Given a function f(x,y,z), its' gradient is defined by

$\nabla f = (\frac[\partial f][\partial x], \frac[\partial f][\partial y],\frac[\partial f][\partial z])$ . (i.e each of the components correspond to the partial derivative of the function f with respect each the its' variables). The gradient is a vector function, which means it could be evaluated at a given point P=(x_0,y_0,z_0) by replacing x,y,z in the formulas with respectively.

a. The given function is $e^{xyz}-3$ and the given point is P=(0,3,-2).

The partial derivatives of f are: $\frac[\partial f][\partial x]= yze^{xyz}, \frac[\partial f][\partial y]= xze^{xyz}, \frac[\partial f][\partial z]= xye^{xyz}$ .

Then, the gradient function of f is $\nabla f(x,y,z) =(yze^{xyz},xze^{xyz},xye^{xyz})$ . So, in this case, we want to know $\nabla f(0,3,-2)$ which is $\nabla f(0,3,-2)=(-6,0,0)$

b. The gradient function evaluated at a given point gives the vector in which there is maximum increase. On the other hand, an unit vector is a vector, whose norm is equal to 1. Then, we just need to divide the previous answer by its' norm. So, the unit vector $\vec{v}(-6,0,0)/6 = (-1,0,0)$ is the correct answer.

For questions c and d: The directional derivative (or rate of change) of a function f in the direction of vector v is given by the dot product of v and the gradient of f.

c. From b) we got that the vector of maximum increase at P is $\vec{v}=(-1,0,0)$ . So, the rate of change of f in that direction is (-6)*(-1)+0*0+0*0=6