- Consider the elevation function $f(x,y) = ax^2+by$. A rover travels along the path $\vec r(t) =(ct+d,et+f)$.
- Compute $\frac{df}{dt}$.
- After obtaining $\frac{df}{dt}$, locate and label the quantities $f_x$, $f_y$, $\frac{dx}{dt}$, and $\frac{dy}{dt}$ in your answer.
- Look at the contour plot on the board. The region has a maximum value inside the shown circle. Draw vectors that point in the same direction as the gradient at the two points $A$ and $B$. Which gradient will be longer?
- How can we tell if the two vectors $(4,6)$ and $(6,9)$ are parallel? If we know $\vec u$ and $\vec v$ are parallel, what must be true?
- Consider the function $f(x,y,z) = 4x^2+4y^2+z^2$. We'll be analyzing the surface at the point $P=(1/2,0,\sqrt{3})$.
- Compute $f(1/2,0,\sqrt{3})$. Then draw the level surface that passes through this point. So draw the ellipsoid $4=4x^2+4y^2+z^2$.
- Compute the gradient $\vec\nabla f(x,y,z)$, and then give $\vec\nabla f(P)$.
- Compute the differential $df$, and then the differential at $P$.
- For a level surface, the output remains constant (so $df=0$). If we let $(x,y,z)$ be a point on the surface really close to $P$, then we have $dx=x-1$, $dy=y-0$ and $dz = z-?$. Plug this information into the differential to obtain an equation of the tangent plane to the surface.
- Give an equation of the tangent plane to the level surface of $f$ that passes through $(1,2,-3)$.
- Give an equation of the tangent plane to the level surface of $f$ that passes through $(a,b,c)$.
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