Higher Dimensions

Suppose that the temperature in a region of space is described by and that you are standing at the point $(1,2,-1)$.

1. Find the instantaneous rate of change of the temperature in the direction of $\vv=\langle 0, 1, 2\rangle$ at the point $(1,2,-1)$. Remember that you should first find a unit vector in the direction of $\vv$.

2. In what direction from the point $(1,2,-1)$ would you move to cause the temperature to decrease as quickly as possible?

3. How fast does t…

Consider the surface $x^2-y^2+z^2=4$.

1. View the surface as a level surface for a function $f(x,y,z)$ and find a normal vector to the surface at the point $(-1,1,2)$.

2. Now, find an equation for the tangent plane to the surface at $(-1,1.2)$.

Find the equation of the tangent plane to the surface given by $xz+2x^2y+y^2z^3=11$ at the point $(2,1,1)$.

Suppose that $\nabla f_P =\langle 2,-4,4\rangle$. Is $f$ increasing or decreasing at $P$ in the direction $\langle2,1,3\rangle$?