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 = sec² t − 1, y = tan t, t = −π/4

Centroids

Find the coordinates of the centroid of the curve x = cos t, y = t + sin t, 0 ≤ t ≤ π.

Finding Cartesian from Parametric Equations

In Exercises 19–24, match the parametric equations with the parametric curves labeled A through F.

x = cos t, y = sin 3t

Finding Lengths of Polar Curves

Find the lengths of the curves in Exercises 21–28.

The curve r = cos³(θ/3), 0 ≤ θ ≤ π/4

Shifting Conic Sections

You may wish to review Section 1.2 before solving Exercises 39-56.

Exercises 53-56 give equations for hyperbolas and tell how many units up or down and to the right or left each hyperbola is to be shifted. Find an equation for the new hyperbola, and find the new center, foci, vertices, and asymptotes.

x²/4 − y²/5 = 1, right 2, up 2

Hyperbolas

Exercises 27-34 give equations for hyperbolas. Put each equation in standard form and find the hyperbola's asymptotes. Then sketch the hyperbola. Include the asymptotes and foci in your sketch.

8x² − 2y² = 16

Polar to Cartesian Equations

Replace the polar equations in Exercises 27–52 with equivalent Cartesian equations. Then describe or identify the graph.

r = 3 cos θ