91影视

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=2 t+4, y=t-1 $$

Finding a Second Derivative Calculate the second derivative \(d^{2} y / d x^{2}\) for the plane curve defined by the parametric equations \(x(t)=t^{2}-3, y(t)=2 t-1,-3 \leq t \leq 4\)

Find the symmetry of the rose defined by the equation \(r=3 \sin (2 \theta)\) and create a graph.

For the following exercises, determine the equation of the parabola using the information given. $$ \text { Focus }(4,0) \text { and directrix } x=-4 $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ \boldsymbol{x}=\cosh t, \quad y=\sinh t $$

Convert the parametric equations of a curve into rectangular form. No sketch is necessary. State the domain of the rectangular form. $$ x=t^{2}, y=2 \ln t, t \geq 1 $$

Describe the graph of each polar equation. Confirm each description by converting into a rectangular equation. $$ \theta=\frac{\pi}{4} $$

The pairs of parametric equations represent lines, parabolas, circles, ellipses, or hyperbolas. Name the type of basic curve that each pair of equations represents. $$ \begin{aligned} &x=3 \cos t \\ &y=4 \sin t \end{aligned} $$

Convert the rectangular equation to polar form and sketch its graph. $$ x=8 $$

Use the familiar formula from geometry to find the length of the curve and then confirm using the definite integral.\(r=6 \sin \theta+8 \cos \theta\) on the interval \(0 \leq \theta \leq \pi\)