Math 215 - Multivariate Calculus - Schedule

Rapid Recall

Consider the function $z=\sin(x)+e^y$, where $x=3t$ and $y=t^2$. Compute $\frac{dz}{dt}$.

Solution

Substitution 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 $$dz = e^{(3t)^2}3\,dt+\cos(2(t^2))2t\,dt.$$ 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.

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.
Consider the following pairs of functions:
- $f(x,y)=xy^2$ and $\vec r(t) = (2t-1,t^2)$
- $f(x,y)=e^x \sin y$ and $\vec r(t) = (t^2,t^3)$
- $f(x,y)=ax+by$ and $\vec r(t) = (ct+d, et+f)$

For each pair, do the following:

Substitute $x(t)$ and $y(t)$ into $f(x,y)$ to obtain $f(\vec r(t))$, then compute $\frac{df}{dt}$.
Write $df$ in terms of $dx$ and $dy$ (focus only on $f(x,y)$).
Write $dx$ and $dy$ in terms of $t$ and $dt$ (focus only on $\vec r(t)$).
Substitute results of (3) into results of (2) to find $df$ in terms of $dt$. You should get the same answer as in part (1).
Label each piece of your answer to part (4) either $f_x$, $f_y$, $\frac{dx}{dt}$, or $\frac{dy}{dt}$.