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\flushleft\underline{Homework 7 --- MATH 1586 Spring 2018} \\

Recall that the tangent plane to a surface $z=f(x,y)$ above the point $(a,b)$ is given by
$$z = f(a,b) + \dfrac{\partial f}{\partial x}(a,b) (x-a) + \dfrac{\partial f}{\partial y}(a,b) (y-b).$$
Recall that the gradient of a function $f(x,y)$ or a function $f(x,y,z)$ is given by
$$\nabla f(x,y) = \left\langle \dfrac{\partial f}{\partial x}, \dfrac{\partial f}{\partial y} \right\rangle$$
Recall the magnitude of a vector $\langle a,b \rangle$ is its length, i.e. $\lVert \langle a,b \rangle \rVert = \sqrt{a^2+b^2}$. We say that a vector is a unit vector if it is length $1$. You can make any vector $\vec{v}$ into a unit vector $\vec{u}$ pointing in the same direction as $\vec{v}$ by dividing by its magnitude: $\vec{u} = \dfrac{\vec{v}}{\lVert \vec{v} \rVert}.$ 

Let $\vec{u}$ be a unit vector and let $f(x,y)$ be a function. We say that the directional derivative of $f$ in the direction $\vec{u}=\langle a,b \rangle$ at $(x_0,y_0)$ is given by
$$D_{\vec{u}} f(x_0,y_0) = \vec{u} \cdot \nabla f(x_0,y_0) = a f_x(x_0,y_0) + b f_y(x_0,y_0).$$
Also recall the following theorem: \\
\textbf{Theorem}: Suppose that $f\left(\vec{x}\right)$ where $\vec{x}=\langle x_1,\ldots,x_n \rangle$ is a partial differentiable function of all variables $x_1,x_2,\ldots, x_n$. The maximum value of $D_u f\left(\vec{x}\right)$ is $\left\lVert \nabla f\left(\vec{x}\right) \right\rVert$ and it occurs in the same direction as the gradient vector $\nabla f\left(\vec{x} \right)$. $\blacksquare$ \\
Recall that to compute ``iterated integrals" like 
$$\displaystyle\int_a^b\int_c^d f(x,y) \mathrm{d}x\mathrm{d}y$$
is just to work ``inside-out". The region of integration of such an integral is the square whose top and bottom sides are above $[a,b]$ on the $x$-axis and whose left and right sides are next to $[c,d]$ on the $y$-axis.
\begin{enumerate}[1.]
\item Find an equation for the tangent plane to $f(x,y)=x^2y^3+xy^2$ above the point $(2,3)$. 

\item Calculate the gradient of $f(x,y)=x^2 e^{xy} + y\log(x)$. 

\item Find a unit vector that points in the same direction as the vector $\vec{v}=\langle 2,3 \rangle$. 

\item Find the directional derivative of $f(x,y)=xy^2+e^{xy}$ in the direction $\left\langle \dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{2}}{2} \right\rangle$ (note: this vector is already a unit vector) at $(x_0,y_0)=(1,2)$. 

\item Find the directional derivative of $f(x,y)=e^x\log(y) + x^2y$ in the direction of $\langle -1,2 \rangle$ at $(x_0,y_0)=(2,2)$. 

\item Calculate 
$$\displaystyle\int_0^1 \displaystyle\int_2^3 x^2y+xy^2 \mathrm{d}x \mathrm{d}y$$
and draw the region of integration.
\end{enumerate}

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