20241008

• I-Learn, Class Pictures, Learning Targets, Text Book Practice
• Prep Tasks: Unit 1 - Motion, Unit 2 - Derivatives, Unit 3 - Integration, Unit 4 - Vector Calculus

This is Day 3 in Unit 2.

Prep

We're in Unit 2 - Derivatives. Your homework assignment each day is to spend 1-2 hours working on the next 4 tasks from the current unit's prep.

Brain Gains (Rapid Recall, Jivin' Generation)

1. For the function $w=f(x,y,z) = y^2+z^2$, draw the level surface corresponding to $w=9$.
2. For the function $w=f(x,y,z) = y^2+z^2$, draw the level surface corresponding to $w=16$.
3. Compute the gradient of $f(x,y,z) = y^2+z^2$.
4. At the point $P = (0,3,0)$, draw the gradient $\vec \nabla f(P)$. Then repeat this at $P = (0,0,3)$ and $P = (2,-4,0)$.

f[x_, y_, z_] := y^2 + z^2
p1 = ContourPlot3D[f[x, y, z], {x, -5, 5}, {y, -5, 5}, {z, -5, 5}, Contours -> {9, 16}, ContourStyle -> Opacity[0.5]]
D[f[x, y, z], x]
gradf = D[f[x, y, z], {{x, y, z}}]
p2 = VectorPlot3D[gradf, {x, -5, 5}, {y, -5, 5}, {z, -5, 5}]
Show[p1, p2]

Group Problems

1. Compute $f_x$ and $\frac{\partial f}{\partial y}$ for each of the following (try to do it without computing $df$ first).
   • $f(x,y) = x^2y$
   • $f(x,y) = 3xy+4y^2$
   • $f(x,y) = \sin(xy^2)$
   • Check your work with Mathematica, using the D[] command.
2. A rover is located on a hill whose elevation is given by $z=f(x,y) = 3x^2+2xy+4y^2$.
   • Compute the differential $dz$, and state the gradient $\vec \nabla f$.
   • State $\dfrac{\partial f}{\partial x}$ and $f_y$. Remember you can check your answer with Mathematica. Using software will help increase confidence as you learn new things.
   • State the differential at the point $P=(x,y)=(1,1)$ (the spot where the rover currently resides) [Check: $dz = 8dx+10dy$].
   • What is the slope of the hill at $P=(1,1)$ in the direction $(dx,dy)=(1,0)$? [Check: 8.]
   • What is the slope of the hill at $P=(1,1)$ in the direction $(0,1)$?
   • What is the slope of the hill at $P=(1,1)$ in the direction $(3,4)$? [Check: $\frac{64}{5}$. The rise is $dz = 64$, with a run of $5$.]
3. Let $g(x,y) =x^2y$.
   • Give $g_x$ and $\dfrac{\partial g}{\partial y}$. Then state $\vec \nabla g$.
   • Find the slope of $g$ at $P=(3,1)$ in the direction $(-3,2)$.
   • Find the slope of $g$ at $P=(3,1)$ in the direction $(2,-5)$.
4. Consider the function $z=f(x,y) = x-y^2$.
   • Compute $\vec \nabla f$.
   • Construct a 2D contour plot by hand. So pick several values for $z$ and plot the resulting curves. If you end up with lots of horizontal lines in the $xy$-plane, you're doing this correctly. Write the height on each horizontal line you draw.
   • Construct a 3D surface plot by hand.
   • Remember that the gradient is a vector field. At a few points in your contour plot, add the gradient vector.
   • Check your work with Mathematica.

     f[x_, y_] := x^2 - y
     p1 = ContourPlot[f[x, y], {x, -2, 2}, {y, -2, 2}]
     Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}]
     Plot3D[f[x, y], {x, -2, 2}, {y, -2, 2}, MeshFunctions -> {#3 &}]
     gradf = D[f[x, y], {{x, y}}]
     p2 = VectorPlot[gradf, {x, -2, 2}, {y, -2, 2}]
     Show[p1, p2]
     

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