Tangent Lines to Parametrized Curves


In Exercises 1−14, 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 d²y/dx² at this point.


x = t + eᵗ, y = 1 − eᵗ, t = 0

Eccentricities and Directrices

Exercises 29–36 give the eccentricities of conic sections with one focus at the origin along with the directrix corresponding to that focus. Find a polar equation for each conic section.

e = 2, x = 4

Lengths of Curves

Find the lengths of the curves in Exercises 25–30.

x = cos t, y = t + sin t, 0 ≤ t ≤ π

Examples of Polar Equations

[Technology Exercise] Graph the lines and conic sections in Exercises 65–74.

r = −2 cos θ

Cartesian to Polar Equations

Replace the Cartesian equations in Exercises 53–66 with equivalent polar equations.

x - y = 3

Graphing Sets of Polar Coordinate Points

Graph the sets of points whose polar coordinates satisfy the equations and inequalities in Exercises 11–26.

θ = π/2, r ≥ 0