Convert from cylindrical to cartesian coordinates and plot. $\left(3,45^{\circ}, 8\right)$

Describe the level curves $f(x, y)=c$ for each of the following functions. In particular, discuss any special values of $c$ at which the behavior of the level curves changes suddenly. Sketch the curves for $c=-1,0$, and 1 . (a) $f(x, y)=x+2 y$ (b) $f(x, y)=x^2-y^2$ (c ) $f(x, y)=y^2-x^2$ (d) $f(x, y)=x^2+y^2$ (e) $f(x, y)=x y$ (f) $f(x, y)=y-2 x^2$.

Sketch the curves. $(4 \cos t, 2 \sin t, \mathrm{t}) ; 0 \leqslant t \leqslant 2 \pi$.

Describe the domain of each of the given functions and evaluate the function at the indicated points. $f(x, y)=\frac{2 x-\sin y}{1+\cos x} ;(0, \pi / 4),(\pi / 4, \pi)$.

Sketch the graphs of the conics. $x^2-y=14$

Sketch the surfaces in three-dimensional space defined by each of the equations. $8 x^2+3 z^2=0$

Identify the equations as shifted conic sections and sketch their graphs. $2 x^2+4 y^2-6 y=8$

Sketch the graphs $x^2+4 y^2=4$ and $x^2-4 y^2=4$ on the same set of axes.

Convert from cylindrical to cartesian coordinates and plot. $(1, \pi / 2,0)$

A body of mass 42$kilograms moves in a circular path on a circle of radius$3$meters, making one revolution every$5$ seconds. Find the centripetal force acting on the body.

Describe the surface given in spherical coordinates by $\rho=\cos 2 \theta$.

A tank in the shape of a right circular cylinder of radius 10 feet and height 16 feet is half filled and lying on its side. Describe the air space inside the tank by suitably chosen cylindrical coordinates.

A circular membrane in space hes over the region $x^2+y^2 \leqslant a^2$. The maximum deflection $z$ of the membrane is $b$. Assume that $(x, y, z)$ is a point on the deflected membrane. Show that the corresponding point $(r, \theta, z)$ in cylindrical coordinates satisfies the conditions $0 \leqslant r \leqslant a, 0 \leqslant \theta \leqslant 2 \pi,|z| \leqslant b$.

A particle moving on the curve $\boldsymbol{\sigma}(t)=3 t^2 \mathbf{i}-$ $\sin t \mathbf{j}-e^t \mathbf{k}$ is released at time $t=\frac{1}{2}$ and flies off on a tangent. What are its coordinates at time $t=1 ?$