Unleash Your Parametric equations, polar coordinates, and vector-valued functions Potential with Expert Support and Comprehensive Practice

Find an equation of the plane.
The plane through the points (4, 1, 4), (5, -8, 6), and (-4, -5, 1)

Find the points on the ellipse $4x^2+y^2=4$ that are farthest away from the point (-1, 0).
(x, y) = ( ) (smaller y-value)
(x, y) = ( )(larger y-value)

Find the volume of the solid generated by revolving the standed region about the x-axis.

Te volume of the solid is ? cubic units.
The equation: $4x+3y=24$

Find a basis for the eigenspace corresponding to the eigenvalue of A given below.
$A=\left[\begin{array}{cccc}4& 0& 2& 0\\ 4& 2& 10& 0\\ 3& -4& 17& 0\\ 2& -2& 8& 3\end{array}\right],\lambda =3$

Find the area of the parallelogram with vertices A(-3,0) , B(-1,6) , C(8,5) and D(6,-1)

Evaluate the following integral in cylindrical coordinates triple integral ${\int }_{x=-1}^1{\int }_{y=0}^{\sqrt{1-x^2}}{\int }_{z=0}^2\left(\frac{1}{1+x^2+y^2}\right)dzdxdy$

Find the points on the cone $z^2=x^2+y^2$ that are closest tothe point (2, 2,0).

Find an area of shaded region

Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y:
$f(x,y)=\{\begin{array}{ll}x{e}^{-x(1+y)}& x\ge 0and\ge 0\\ 0& otherwise\end{array}$
a) What is the probability that the lifetime X of the first component exceeds 3?
b) What are the marginal pdfs

to what property does $(-6+3)+1=-6+(3+1)$ belong to

Given ${\displaystyle r\prime \left(t\right)=⟨\sec 2t,-\sin t⟩}$, find the arc length of the curve r(t) on the interval ${\displaystyle [-\frac{\pi }{3}]}$

Graph the plane curve defined by the parametric equations: $x=t^2+1,y=3t,-2\le t\le 2$.

1. A curve is given by the following parametric equations. x = 20 cost, y = 10 sint. The parametric equations are used to represent the location of a car going around the racetrack. a) What is the cartesian equation that represents the race track the car is traveling on? b) What parametric equations would we use to make the car go 3 times faster on the same track? c) What parametric equations would we use to make the car go half as fast on the same track? d) What parametric equations and restrictions on t would we use to make the car go clockwise (reverse direction) and only half-way around on an interval of [0, 2?]? e) Convert the cartesian equation you found in part “a” into a polar equation? Plug it into Desmos to check your work. You must solve for “r”, so “r = ?”

Polar coordinates of a point are given $(4,{90}^{\circ })$. Find the rectangular coordinates of each point.

Parametric equations and a value for the parameter t are given $x=(60cos{30}^{\circ })t,y=5+(60sin{30}^{\circ })t-16t2,t=2$. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t.

Solve. ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}(\frac{\sqrt[3]{x^3-2x^2+3}}{2x+1}}$

Represent the line segment from P to Q by a vector-valued function and by a set of parametric equations $P(-3,-6,-1),Q(-1,-9,-8)$.

Graph the plane curve defined by the parametric equations $x=125\cos tandy=125\sin t$

Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point. $x=3\ln (t),y=4t^{\frac{1}{2}},z=t^3,(0,4,1)$

Find parametric equations for the lines in The line through the origin parallel to the vector $2j+k$