Eliminating the Parameter from the Function
Find an equation for the line tangent to the curve at the point defined by the given value of $t$. Also, find the value of $\frac{d^{2} y}{d x^{2}}$ at this point.
\[
x=8 \cos t, y=4 \sin t, t=\frac{\pi}{4}
\]
Converting to Polar Coordinates
Find the exact length of the polar curve $r=\cos ^{4}(\theta / 4)$.
Length $=$
Identifying and Graphing Circles
A circle is described by the parametric equations: \(x = 5cos(t)\) and \(y = 5sin(t)\). What is the radius of the circle, and graph the circle.
Identifying and Graphing Limacons
Find the slope of the tangent to the curve $r=4-4 \cos \theta$ at the value $\theta=\pi / 2$
Identifying and Graphing Roses
Identify and graph the rose given by the polar equation \(r = 5\cos(3\theta)\).
Identifying and Graphing Cardioids
Consider the polar equation \(r = 1 + \cos(\theta)\). Identify this equation as a cardioid and sketch its graph.