Problem 6
Calculate the second derivative \(d^{2} y / d x^{2}\) 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 6
Use technology (CAS or calculator) to sketch the parametric equations. $$ x=t^{2}+t, \quad y=t^{2}-1 $$
Problem 6
Create a graph of the curve defined by the function \(r=4+4 \cos \theta\).
Problem 6
Find the total arc length of \(r=3 \sin \theta\).
Problem 6
Put the equation \(9 x^{2}+4 y^{2}-36 x+24 y+36=0\) into standard form and graph the resulting ellipse.
Problem 7
Rewrite each of the following equations in rectangular coordinates and identify the graph. a. \(\theta=\frac{\pi}{3}\) b. \(r=3\) c. \(r=6 \cos \theta-8 \sin \theta\)
Problem 7
Use technology (CAS or calculator) to sketch the parametric equations. $$ x=e^{-t}, \quad y=e^{2 t}-1 $$
Problem 7
Determine a definite integral that represents the area.Region enclosed by \(r=4\)
Problem 7
Finding the Area under a Parametric Curve Find the area under the curve of the cycloid defined by the equations $$ x(t)=t-\sin t, \quad y(t)=1-\cos t, \quad 0 \leq t \leq 2 \pi $$
Problem 7
Put the equation \(9 x^{2}+16 y^{2}+18 x-64 y-71=0\) into standard form and graph the resulting ellipse.
