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{\bf HW 4\\Math 261}

Please see the course syllabus for details on how to turn in your homework assignments.  %This one is due at the beginning of your class on \underbar{\bf Friday, October 4}.

\begin{enumerate}

\item (5 pts.) \textsc{True or False}
\begin{enumerate}
\item Let $x = 3, y = 4 , z = 5t$, and $f(x,y,z)$ be a function from $\bR^3$ to $\bR$. Then $\displaystyle \frac{\partial f}{\partial t} = \displaystyle \frac{\partial f}{\partial z}$.
\item Let$x = 3, y = 4 , z = 5t$, and $f(x,y,z)$ be a function from $\bR^3$ to $\bR$. Then $\displaystyle \frac{\partial f}{\partial t} = 5\displaystyle \frac{\partial f}{\partial z}$.
\item Let$x = 3, y = 4 , z = 5t$, and $f(x,y,z)$ be a function from $\bR^3$ to $\bR$. Then $\displaystyle \frac{\partial f}{\partial t} = 1/5 \displaystyle \frac{\partial f}{\partial z}$.
\item Let $h(x,y)$ be a function from $\bR^2$ to $\bR$ and $P$ a point in the domain of $h$. It is possible that the directional derivative of $h$ at $P$ is equal to $1$ for every direction vector.
\item Let $h(x,y)$ be a function from $\bR^2$ to $\bR$ and $P$ a point in the domain of $h$. It is possible that the directional derivative of $h$ at $P$ is equal to $0$ for every direction vector.
\end{enumerate}

\item (3 pts.) Suppose function $f(x,y)$ depends on variables $x$ and $y$, which are themselves functions of variables $\alpha$, $\beta$, and $\gamma$ (i.e., $x=x(\alpha,\beta,\gamma)$ and $y=y(\alpha,\beta,\gamma)$).  Fill in the blanks for the chain rule to compute 
$\displaystyle \frac{\partial f}{\partial \beta}$:

$$\displaystyle \frac{\partial f}{\partial\beta} = \frac{\partial\ \framebox(12,12){}}{\partial\ \framebox(12,12){}}\ \frac{\partial\ \framebox(12,12){}}{\partial\ \framebox(12,12){}} + \frac{\partial\ \framebox(12,12){}}{\partial\ \framebox(12,12){}}\ \frac{\partial\ \framebox(12,12){}}{\partial\ \framebox(12,12){}}$$

\vspace{0.2in}

\item  (3 pts.)  Let $$\displaystyle g(u,v) = u^2 + v^3,$$ 
$$\displaystyle u(t)= \cos(t),$$ 
$$\displaystyle v(t) = \ln(t).$$


Compute $\displaystyle  \frac{dg}{dt}$.  (Please use only the variable $t$ in your response, but do not bother multiplying everything out.)

\vspace{0.2in}

\item  (3 pts.)  Suppose $z$ is a function of $x$ and $y$ and that $\displaystyle x^2z^2 + y\sin(z) = 1.$  Find $\displaystyle \frac{\partial z}{\partial x}$.

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\item   (3 pts.)  Find the derivative of $f(x,y)= xy- y^2$ at point $(1,2)$ in the direction of $\mathbf{v}=\left<3,4\right>$.  Please simplify your answer to a number.  (Notice that $\mathbf v$ is not a unit vector!)
\vspace{0.2in}

\item   (3 pts.)  Find a direction vector $\mathbf{v}$ such that  the derivative of $f(x,y)= xy- y^2$ at point $(1,2)$  in the direction of $\mathbf{v}$ is equal to $0$.

\end{enumerate}

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