Problem 1
Finding an Area of a Polar Region Find the area of one petal of the rose defined by the equation \(r=3 \sin (2 \theta)\).
Problem 1
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. $$ \begin{array}{ll} \text { a. } x(t)=t^{2}-3, & y(t)=2 t-1, \quad-3 \leq t \leq 4 \\ \text { b. } x(t)=2 t+1, & y(t)=t^{3}-3 t+4, \quad-2 \leq t \leq 5 \\ \text { c. } x(t)=5 \cos t, & y(t)=5 \sin t, \quad 0 \leq t \leq 2 \pi \end{array} $$
Problem 1
Convert each of the following points into polar coordinates. a. \((1,1)\) b. \((-3,4)\) c. \((0,3)\) d. \((5 \sqrt{3},-5)\)
Problem 1
Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=t^{2}+2 t, y=t+1 $$
Problem 2
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.
Problem 2
Find the area inside the cardioid defined by the equation \(r=1-\cos \theta\).
Problem 2
Convert \((-8,-8)\) into polar coordinates and \(\left(4, \frac{2 \pi}{3}\right)\) into rectangular coordinates.
Problem 2
Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=\cos (t), y=\sin (t),(0,2 \pi] $$
Problem 2
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.
Problem 3
Sketch the curves below by eliminating the parameter \(t\). Give the orientation of the curve. $$ x=2 t+4, y=t-1 $$
