Calculus 2 Equations: Expert Solutions and More

Find transient terms in this general solution to a differential equation, if there are any
${\displaystyle y=(x+C)\left(\frac{x+2}{x-2}\right)}$

Given the points (2,6) and (6,10), find at least three equations from exponential function that pass through these point

Sketch the vector field F by drawing a diagram. ${\displaystyle F(x,y)=\frac{yi+xj}{{(x^2+y^2)}^{\frac{1}{2}}}}$

Find sets of parametric equations and symmetric equations of the line that passes through the given point and is parallel to the given vector or line. (For each line, write the direction numbers as integers.)
Point: (-3, 4, 5)
Parallel to: ${\displaystyle \frac{x-1}{2}=\frac{y+1}{-3}=z-5}$
(a) Parametric equations
(b) Symmetric equations

Find the area of the region that lies inside both curves. ${\displaystyle r=\sqrt{3}\cos \theta ,r=\sin \theta }$

Find the length of the parametric curve defined over the given interval ${\displaystyle x=3t-6,y=6t+1,0\le t\le 1}$

A quadratic function f is given
(a)Express f in standard form
(b)Find the vertex and x- and y-intercepts of f.
(c)Sketch the graph.
(d)Find the domain and range of f.
${\displaystyle f\left(x\right)=x^2-2x+3}$

Find parametric equations for x, y, and z in terms of the polar coordinates r and ${\displaystyle \theta }$ to determine the points on the portion of the paraboloid $x+y+z=5$ that is on or above the plane $z=4$

Convert the point from rectangular coordinates to spherical coordinates. ${\displaystyle (-3,-3,\sqrt{2})}$

Replace the polar equations with equivalent
Cartesian equations. Then describe or identify the graph. ${\displaystyle r^2=-4r\cos \theta }$

${\displaystyle \left(\frac{dy}{dx}\right)=-\frac{y^2+x^2}{2xy}\text{and}y\left(1\right)=4}$
Please, solve the differential equation. Write the method you used and solve for the dependent variable it it is possible.

Find if the following first order differential equations seperable, linear, exact, almost exact, homogeneous, or Bernoulli. Rewrite the equation into standard form for the classification it fits.
${\displaystyle \left(\frac{dy}{dx}\right)=x^2[\left(x^3\right)\left(y\right)-\left(\frac{1}{x}\right)]}$

A vector-valued function $r(t)$ with its defining parametric equations is given by the following.
$x=f(t)$
$y=g(t)$
$r(t)=ft(i)+g(t)j$
Now find the vector-valued function $r(t)$ with the given parametric equations.

Determine the x-y coordinates of the points where the following parametric equations will have horizontal or vertical tangents ${\displaystyle x=t^3-3t}$
${\displaystyle y=3t^2-9}$

Polar coordinates of point P are given. Find all of its polar coordinates.
${\displaystyle P=(1,-\frac{\pi }{4})}$

Represent the line segment from P(−2, −3, 8), Q(5, 1, −2) by a vector-valued function and by a set of parametric equations.

Surface s is a part of the paraboloid ${\displaystyle z=4-x^2-y^2}$ that lies above the plane $z=0$.$(6+7+7=20pt)$

a) Find the parametric equation ${\displaystyle \overrightarrow{r}(u,v)}$ of the surface with polar coordinates ${\displaystyle x=u\cos \left(v\right),y=u\sin \left(v\right)}$ and find the domain D for u and v.

b) Find ${\displaystyle {\overrightarrow{r}}_u,{\overrightarrow{r}}_v,}$ and ${\displaystyle {\overrightarrow{r}}_u\cdot {\overrightarrow{r}}_v}$.

c) Find the area of the surface

Express the point with cartesian coordinates Q(3,-1) in polar form

$x=t-\sin t$
$y=1-\cos t-$ curve given by the parametric equation ${\displaystyle =\frac{\pi }{3}}$ in point
Find the equation of the tangent line

Write an equations in indicated coordinate system. Also write a parametric equation and a vector equation for each of the following equations.

1) ${\displaystyle \frac{x-1}{2}=\frac{y-5}{3}=\frac{z-2}{4}}$ cylindrical coordinates, sherical coordinates

2) ${\displaystyle 3x+4y+5z=6}$ cylindrical coordinates, sherical coordinates

3) ${\displaystyle x^2+y^2+z^2=100}$ cylindrical coordinates, sherical coordinates

4) ${\displaystyle x^2+y^2=100}$ cylindrical coordinates, sherical coordinates