$$ 1 \leftarrow $$ Compute the gradient of the following function and evaluate it at the given point $$ P $$. $$ g(x, y)=x^{2}-8 x^{2} y-6 x y^{2} ; P(-1,3) $$ The gradient is $$ \nabla g(x, y)=\square, \square $$

Question

UpStudy AI · Accepted Answer

The function is 
$$
g(x,y)=x^2-8x^2y-6xy^2.
$$

**Step 1. Compute the partial derivative with respect to $$x$$:**

Differentiate each term:
- $$\frac{\partial}{\partial x}(x^2) = 2x$$,
- $$\frac{\partial}{\partial x}(-8x^2y) = -8\cdot 2xy = -16xy$$ (since $$y$$ is treated as a constant),
- $$\frac{\partial}{\partial x}(-6xy^2)= -6y^2$$ (since $$y^2$$ is a constant with respect to $$x$$).

Thus, 
$$
\frac{\partial g}{\partial x}(x,y)=2x-16xy-6y^2.
$$

**Step 2. Compute the partial derivative with respect to $$y$$:**

Differentiate each term:
- $$\frac{\partial}{\partial y}(x^2)=0$$ (since $$x^2$$ is constant with respect to $$y$$),
- $$\frac{\partial}{\partial y}(-8x^2y)= -8x^2$$,
- $$\frac{\partial}{\partial y}(-6xy^2)= -6x\cdot 2y=-12xy$$.

Thus,
$$
\frac{\partial g}{\partial y}(x,y)=-8x^2-12xy.
$$

**Step 3. Evaluate the gradient at the point $$P(-1,3)$$:**

For the $$x$$-component:
$$
\frac{\partial g}{\partial x}(-1,3)=2(-1)-16(-1)(3)-6\cdot3^2 = -2+48-54=-8.
$$

For the $$y$$-component:
$$
\frac{\partial g}{\partial y}(-1,3)=-8(-1)^2-12(-1)(3)=-8+36=28.
$$

Thus, the gradient is:
$$
\nabla g(-1,3)=\left(-8, \,28\right).
$$

So, the final answer is:
$$
\nabla g(x,y)=2x-16xy-6y^2,\quad -8x^2-12xy,\quad \text{and} \quad \nabla g(-1,3)=\left(-8,28\right).
$$
The gradient of the function $$ g(x, y) = x^2 - 8x^2 y - 6xy^2 $$ is $$ \nabla g(x, y) = (2x - 16xy - 6y^2, -8x^2 - 12xy) $$. Evaluating the gradient at the point $$ P(-1, 3) $$ gives $$ \nabla g(-1, 3) = (-8, 28) $$.

Compute the gradient of the following function and evaluate it at the given point .

The gradient is

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Calculus Questions

Compute the gradient of the following function and evaluate it at the given point . The gradient is