91Ó°ÊÓ

(a) Find the equation of the tangent line to the curve $$ x=e^{t}, \quad y=e^{-t} $$ at \(t=1\) without eliminating the parameter. (b) Find the equation of the tangent line in part (a) by eliminating the parameter.

Find the area inside the curve \(r^{2}=\sin 2 \theta\).

(a) Find the equation of the tangent line to the curve $$ x=2 t+4, \quad y=8 t^{2}-2 t+4 $$ at \(t=1\) without eliminating the parameter. (b) Find the equation of the tangent line in part (a) by eliminating the parameter.

Prove: The line tangent to the ellipse $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $$ at the point \(\left(x_{0}, y_{0}\right)\) has the equation $$ \frac{x x_{0}}{a^{2}}+\frac{y y_{0}}{b^{2}}=1 $$

Determine a shortest parameter interval on which a complete graph of the polar equation can be generated, and then use a graphing utility to generate the polar graph. \(r=\sin \frac{\theta}{2}\)

Find all values of \(t\) at which the parametric curve has (a) a horizontal tangent line and (b) a vertical tangent line. $$ x=2 \sin t, y=4 \cos t \quad(0 \leq t \leq 2 \pi) $$

Determine a shortest parameter interval on which a complete graph of the polar equation can be generated, and then use a graphing utility to generate the polar graph. \(r=1-2 \sin \frac{\theta}{4}\)

Prove: The line tangent to the hyperbola $$ \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 $$ at the point \(\left(x_{0}, y_{0}\right)\) has the equation $$ \frac{x x_{0}}{a^{2}}-\frac{y y_{0}}{b^{2}}=1 $$

A radial line is drawn from the origin to the spiral \(r=a \theta\) \((a>0\) and \(\theta \geq 0\) ). Find the area swept out during the second revolution of the radial line that was not swept out during the first revolution.

Determine a shortest parameter interval on which a complete graph of the polar equation can be generated, and then use a graphing utility to generate the polar graph. \(r=0.5+\cos \frac{\theta}{3}\)