Problem 97
For the following exercises, find the polar equation for the curve given as a Cartesian equation. $$ y^{2}=4+x^{2} $$
Problem 98
For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. $$ x=\ln (t), y=t^{2}-1, t=1 $$
Problem 99
For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. $$ r=3+\cos (2 \theta), \theta=\frac{3 \pi}{4} $$
Problem 100
For the following exercises, find the equation of the tangent line to the given curve. Graph both the function and its tangent line. Find \(\frac{d y}{d x}, \frac{d x}{d y}\), and \(\frac{d^{2} x}{d y^{2}}\) of \(y=\left(2+e^{-t}\right), x=1-\sin (t)\)
Problem 101
For the following exercises, find the area of the region. $$ x=t^{2}, y=\ln (t), 0 \leq t \leq e $$
Problem 102
For the following exercises, find the area of the region. $$ r=1-\sin \theta \text { in the first quadrant } $$
Problem 103
For the following exercises, find the arc length of the curve over the given interval. $$ x=3 t+4, y=9 t-2,0 \leq t \leq 3 $$
Problem 104
For the following exercises, find the arc length of the curve over the given interval. \(r=6 \cos \theta, 0 \leq \theta \leq 2 \pi .\) Check your answer by geometry.
Problem 105
For the following exercises, find the Cartesian equation describing the given shapes. A parabola with focus \((2,-5)\) and directrix \(x=6\)
Problem 106
For the following exercises, find the Cartesian equation describing the given shapes. An ellipse with a major axis length of 10 and foci at \((-7,2)\) and \((1,2)\)
