Please give a complete, justified solution to each question below. A single-term answer without explanation will receive no credit.
Please complete each question on its own sheet of paper (or more if necessary), and upload to Gradsescope.
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1. Let $f(x, y) = \frac{x+y}{1+x^2}$. Find the directional derivative of $f$ at the point $(1, -2)$ in the direction of:
2. a. Two surfaces are called \textbf{orthogonal} at a point of intersection if their normal lines are perpendicular at that point.
Show that surfaces with equations $F(x, y, z) = 0$ and $G(x, y, z) = 0$ are orthogonal at a point $P$ where $\nabla F \neq 0$ and $\nabla G \neq 0$ if and only if at the point $P$, $$ F_x G_x + F_y G_y + F_zG_z = 0. $$
b. Use part a. to show that the surfaces $z^2 = x^2 + y^2$ (a cone) and $x^2 + y^2 + z^2 = r^2$ (a sphere) are orthogonal at every point of intersection.
3. Consider the function $f(x,y)=x^2 - \frac{y^2}{9}$.
a. Sketch a contour plot with (at least) levels $k=0,\pm1,\pm2,\pm3$. Plot a point at $(2,6)$ and roughly sketch a ``path of greatest descent'' starting from this point. That is, a path along which the directional derivative is as low a number as possible.
b. Find a formula for your curve in part a. That is, find a curve $\vec r(t) = \langle x(t),y(t)\rangle$ such that $$ -\nabla f = \vec r'(t) $$ and $\vec r(0) = \langle 2,6 \rangle$. (This is technically an ODE, but one you can solve.)
c. In your curve from part b, solve for $y$ in terms of $x$ and plot the function. Is it close to your guess from part a?
d. Compare, qualitatively, what happnes to the limit $$\lim_{t\to\infty} \vec r(t) $$ in the three cases: