Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Cone $\mathbf{r}(u, v)=[u \cos v, \quad u \sin v, \quad c u]$.

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Elliptic paraboloid $\mathbf{r}(u, v)=\left[a u \cos v, \quad b u \sin v, u^2\right]$.

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. xy-plane r(u, v)=(u, v)(thus ui+vj)

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Hyperboloid $\mathbf{r}(u, v)=[a \sinh u \cos v, \quad b \sinh u \sin v, \quad c \cosh u]$.

Determine whether each statement makes sense or does not make sense; and explain your reasoning. The rational expressions

$$\frac { 7 } { 14 x } \text { and } \frac { 7 } { 14 + x }$$

can both be simplified by dividing each numerator and each denominator by 7.

Find a parametric representation for the surface. The part of the ellipsoid x^2+y^2+3z^2=1 that lies to the left of the xz-plane

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Ellipsoid r(u, v)=[a cos v cos u, b cos v sin u, c sin v]

Familiarize yourself with parametric representations of important surfaces by deriving a representation, by finding the parameter curves (curves u=const and v=const) of the surface and a normal vector N=ru*rv of the surface. Show the details of your work. Helicoid r(u, v)=[u cos v, u sin v, v]. Explain the name.

Familiarize yourself with parametric representations of practically important surfaces by deriving a representation $z=f(x, y) \text { or } g(x, y, z)=0 \text {. }$, by finding the parameter curves (curves $u=$ const and $v=$ const) on the surface and a normal vector $\mathbf{N}=\mathbf{r}_u \times \mathbf{r}_v$ of the surface. (Show the details of your work.) Hyperbolic paraboloid $\mathbf{r}(u, v)=\left[a u \cosh v, \quad b u \sinh v, u^2\right]$.

Using $\iint_R \nabla^2 w d x d y=\oint_C \frac{\partial w}{\partial n} d s$, evaluate $\oint_C \frac{\partial w}{\partial n} d s$ counterclockwise over the boundary curve $C$ of the region $R$. (Show the details of your work.) $w=\cosh x, \quad R$ the triangle with vertices $(0,0),(4,2),(0,2)$

Evaluate ∫_C F(r)*dr counterclockwise around the boundary C of the region R by Green’s theorem, where F=[6y², 2x-2y^4], R the square with vertices ±(2,2), ±(2,-2)

Evaluate ∫_C F(r)*dr counterclockwise around the boundary C of the region R by Green’s theorem, where F=[y,-x], C the circle x²+y²=1/4

Find the volume of the given region in space. The region beneath z=4x²+9y² and above the rectangle with vertices (0, 0), (3, 0), (3, 2), (0, 2) in the xy-plane.

Stocks A and B have the following historical returns:" $$ \begin{matrix} \text{Year} & \text{Stock A's Returns, }{r_A} & \text{Stock B's Returns, }{r_B}\\ \text{2011} & \text{(18.00\\%)} & \text{(14.50\\%)}\\ \text{2012} & \text{33.00} & \text{21.80}\\ \text{2013} & \text{15.00} & \text{30.50}\\ \text{2014} & \text{(0.50)} & \text{(7.60)}\\ \text{2015} & \text{27.00} & \text{26.30}\\ \end{matrix} $$ a. Calculate the average rate of return for each stock during the period 2011 through 2015. b. Assume that someone held a portfolio consisting of 50% of Stock A and 50% of Stock B. What would the realized rate of return on the portfolio have been each year? What would the average return on the portfolio have been during this period? c. Calculate the standard deviation of returns for each stock and for the portfolio. d. Calculate the coefficient of variation for each stock and for the portfolio. e. Assuming you are a risk-averse investor, would you prefer to hold Stock A, Stock B, or the portfolio? Why?