Rapid Recall
- Consider the function $z=\sin(x)+e^y$, where $x=3t$ and $y=t^2$. Compute $\frac{dz}{dt}$.
Solution
- Substition gives $z=\sin(3t)+e^{t^2}$. Differentiation gives $$\frac{dz}{dt}=\cos(3t)3+e^{t^2}2t.$$
- Suppose $dz = e^{x^2}dx+\cos(2y)dy$, $x=3t$, and $y=t^2$. Compute $\frac{dz}{dt}$.
Solution
- Note that $dx = 3dt$ and $dy = 2tdt$. Substitution then gives $$dz = e^{(3t)^2}3dt+\cos(2(t^2))2tdt.$$ Dividing by $dt$ completes the problem.
- For $f(x,y)=x^2-y^2$, draw the level curve that passes through the point $(0,1)$.
Solution
- We compute $f(0,1) = -1$. We then need to draw the curve $-1=x^2-y^2$ or $1=y^2-x^2$. It's a hyperbola opening up and down along the $y$-axis.
Group problems
- Compute $f_x$, $\frac{\partial f}{\partial y}$, $\vec \nabla f$, and $df$ for each of the following.
- $f(x,y) = x^2y$
- $f(x,y) = 3xy+4y^2$
- $f(x,y) = \sin(xy^2)$
- Consider the function $z=f(x,y)=x^2+y^2-4$.
- Construct a contour plot of $f$. So let $z=0$ and draw the resulting curve in the $xy$ plane. Then let $z=5$ and draw the resulting curve in the $xy$ plane. Then pick other values for $z$ and draw the resulting curve in the $xy$ plane. If you get a bunch of concentric circles, you're doing this right. On each circle you draw, write the height of that circle.
- Construct a 3D surface plot of the function.
- Consider the function $z=4-y^2$.
- Construct a 2D contour plot.
- Construct a 3D surface plot.
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