Find the first partial derivatives $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}$ and $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}$ for the following function:

$f(x,y)=\sqrt[\phantom{2}]{x^4+4y^2}$

Find the second partial derivatives $\frac{de{l}^{2}f}{del{x}^2}$ and $\frac{de{l}^{2}f}{del{y}^2}$ for the function in problem #$1.$

Find $\frac{?}{\text{b}\text{a}\text{r}}(grad)(f)$ for the following function: $f(x,y,z)=e^{(2x-3)}cos(y)sin(3-10z)$

Find $\frac{?}{\text{b}\text{a}\text{r}}(grad{)}^2(f)$ for the function in problem #$3.$

Given the expression: $e^{j2},$ Find the $($a$)$ real part, $($b$)$ magnitude, $($c$)$ phase, and $($d$)$ complex

conjugate. $($hint: $$ is real$)$

Show that the real part of hat$(A)hat(B)$ does not equal the real part of hat$(A)$ times the real part of hat$(B),$ for two

complex numbers hat$(A)=A_{pk}{e}^{j(t+)}$ and widehat$(B)=B_{pk}{e}^{j(t+)}$

Sketch the waveform of the following function: $p(x,t)=8cos(15t-7x).($For both parts, don't

forget to label the $x$ and $y-$axes$)$

a$.$ Sketch along a time axis from $0$ to $1$ seconds, holding space fixed at $x=0$

b$.$ Sketch along a spatial axis from $0$ to $1\mathrm{.}5$ meter, holding time fixed at $t=0$

Given a low frequency waveform $y(x,t)=14sin(500t-10x)$ which is propagating down a

string of unknown material, where $y$ and $x$ are in meters and $t$ in seconds. What are:

$($a$)$ the radian frequency $(),$

$($b$)$ the frequency $(f),$

$($c$)$ the wavenumber $(k),$

$($d$)$ the wavelength $(),$

$($e$)$ the speed of sound for this material $($c$)$

$($f$)$ If we only change the material to copper $(15\mathrm{.}75\frac{m}{s},$ how does the speed of sound

change?

Given that the speed of sound is $c=1500\frac{m}{s}$ in water, find the wavelengths for the follow

frequencies in this medium: $($a$)32$ Hz $,($b$)200$ Hz $,($c$)1\mathrm{.}3$ kHz $,($d$)21$ kHz $.($e$)$ Suppose the same

sound propagates in air, how do these wavelength change? $($i$.$e$.$ will they be longer or shorter?$)$

Justify your answer by stating a general algebraic formula.