Find all points (x, y) at which the tangent plane to the graph of $z=x^3$ is horizontal.

In given problem, find all first partial derivatives of each function.

$$f(x, y)=\left(4 x-y^2\right)^{3 / 2}$$

Respond with true or false to the following assertion. Be prepared to defend your answer.

If $f_x(0,0)=f_y(0,0)$, then $f(x, y)$ is continuous at the origin.

find the angle θ between the vectors. u = ⟨0, 2, 2⟩ v = ⟨3, 0, -4⟩

Find the measure of the smallest nonnegative angle between the vectors $\mathbf{v}=\langle 2,-3\rangle$ and $\mathbf{w}=\langle-3,4\rangle$. Round to the nearest tenth of a degree.

In given problem, find the gradient $\nabla f$.

$$f(x, y)=x^3 y-y^3$$

Find parametric equations and symmetric equations for the line.

The line of intersection of the planes $x+y+z=1$ and $x+z=0$

Find equations in standard form of the lines through point P that are (a) parallel to, and (b) perpendicular to, line L. P(-1, 2); L:x-3y=-2

Find the equation of the plane each of whose points is equidistant from (-2,1,4) and (6,1,-2).

Find an equation of the plane that satisfies the stated conditions. The plane through the points

$$P _ { 1 } ( - 2,1,4 ) , P _ { 2 } ( 1,0,3 )$$

that is perpendicular to the plane 4x-y+3z=2.

Based on your answer to the previous question, which values from the table do not make sense?

Let $\mathbf { u } , \mathbf { v } , \text { and } \mathbf { w }$ be vectors. Which of the following make sense, and which do not? Give reasons for your answers.

$$\mathbf { u } \times ( \mathbf { v } \cdot \mathbf { w } )$$

Which of the following do not make sense?

(a) $\mathbf{u} \cdot(\mathbf{v} \cdot \mathbf{w})$

(b) $(\mathbf{u} \cdot \mathbf{w})+\mathbf{w}$

(c) $\|\mathbf{u}\|(\mathbf{v} \cdot \mathbf{w})$

(d) $(\mathbf{u} \cdot \mathbf{v}) \mathbf{w}$

Let $\mathbf { u } , \mathbf { v } , \text { and } \mathbf { w }$ be vectors. Which of the following make sense, and which do not? Give reasons for your answers.

$$\mathbf { u } \times ( \mathbf { v } \times \mathbf { w } )$$