91影视

Finding an Area of a Polar Region Find the area of one petal of the rose defined by the equation \(r=3 \sin (2 \theta)\).

Finding the Derivative of a Parametric Curve Calculate the derivative \(\frac{d y}{d x}\) for each of the following parametrically defined plane curves, and locate any critical points on their respective graphs. $$ \begin{array}{ll} \text { a. } x(t)=t^{2}-3, & y(t)=2 t-1, \quad-3 \leq t \leq 4 \\ \text { b. } x(t)=2 t+1, & y(t)=t^{3}-3 t+4, \quad-2 \leq t \leq 5 \\ \text { c. } x(t)=5 \cos t, & y(t)=5 \sin t, \quad 0 \leq t \leq 2 \pi \end{array} $$

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

Convert each of the following points into polar coordinates. a. \((1,1)\) b. \((-3,4)\) c. \((0,3)\) d. \((5 \sqrt{3},-5)\)

Convert \((-8,-8)\) into polar coordinates and \(\left(4, \frac{2 \pi}{3}\right)\) into rectangular coordinates.

Find the area inside the cardioid defined by the equation \(r=1-\cos \theta\).

Given a parabola opening upward with vertex located at \((h, k)\) and focus located at \((h, k+p)\), where \(\rho\) is a constant, the equation for the parabola is given by $$ y=\frac{1}{4 p}(x-h)^{2}+k $$ This is the standard form of a parabola.

Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=\cos (t), y=\sin (t),(0,2 \pi] $$

Calculate the derivative \(d y / d x\) for the plane curve defined by the equations $$ x(t)=t^{2}-4 t, \quad y(t)=2 t^{3}-6 t, \quad-2 \leq t \leq 3 $$ and locate any critical points on its graph.

Finding a Tangent Line Find the equation of the tangent line to the curve defined by the equations $$ x(t)=t^{2}-3, \quad y(t)=2 t-1, \quad-3 \leq t \leq 4 \text { when } t=2 $$