91影视

For the following exercises, determine the equation of the ellipse using the information given. $$ \text { Foci located at }(0,-3),(0,3) \text { and eccentricity of } \frac{3}{4} $$

Find the equation of the tangent line to \(x=\sin (t), y=\cos (t)\) at \(t=\frac{\pi}{4}\).

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{3}, \quad y=3 \ln t, t \geq 1 $$

Find rectangular coordinates for the given point in polar coordinates. $$ (-4.5,6.5) $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{n}, y=n \ln t, t \geq 1, \text { where } n \text { is a natural number } $$

For the curve \(x=4 t, y=3 t-2\), find the slope and concavity of the curve at \(t=3\).

For the following exercises, determine the equation of the hyperbola using the information given. $$ \text { Vertices located at }(5,0),(-5,0) \text { and foci located at }(6,0),(-6,0) $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ \begin{aligned} &x=\ln (5 t) \\ &y=\ln \left(t^{2}\right) \end{aligned} \text { where } 1 \leq t \leq e $$

Find a definite integral that represents the arc length.\(r=e^{\theta}\) on the interval \(0 \leq \theta \leq 1\)

Determine whether the graphs of the polar equation are symmetric with respect to the \(x\) -axis, the \(y\) -axis, or the origin. $$ r=3 \sin (2 \theta) $$