16. Parametric Equations & Polar Coordinates

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

81–88. Arc length Find the arc length of the following curves on the given interval.

x = 3 cos t, y = 3 sin t + 1; 0 ≤ t ≤ 2π

67–72. Derivatives Consider the following parametric curves.

b. Make a sketch of the curve showing the tangent line at the point corresponding to the given value of t.

x = 2 + 4t, y = 4 − 8t; t = 2

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

e. There are two points on the curve x=−4 cos t, y=sin t, for 0≤t≤2π, at which there is a vertical tangent line.

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

Find $\frac{d^{3}y}{d{x}^3}$ for the parametric curve at the given point.

$x=2t^3$, $y=t^4$, $t=1$

Lengths of Curves

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

x = cos t, y = t + sin t, 0 ≤ t ≤ π

77–80. Slopes of tangent lines Find all points at which the following curves have the given slope.

x = 2 + √t, y = 2 - 4t; slope = -8

Find the slope of the parametric curve x=−2t ³ +1, y=3t ², for −∞<t<∞, at the point corresponding to t=2. 

Use calculus to find the arc length of the line segment x=3t+1, y=4t, for 0≤t≤1. Check your work by finding the distance between the endpoints of the line segment. 

67–72. Derivatives Consider the following parametric curves.

a. Determine dy/dx in terms of t and evaluate it at the given value of t.

x = 2 + 4t, y = 4 − 8t; t = 2

22–23. Arc length Find the length of the following curves.

x = cos 2t, y = 2t - sin 2t; 0 ≤ t ≤ π/4

81–88. Arc length Find the arc length of the following curves on the given interval.

x = eᵗ sin t, y = eᵗ cos t; 0 ≤ t ≤ 2π

Find the length of the curve below on the interval $[0,4]$.

$x=\frac{1}{3}{(2t+3)}^{\frac{3}{2}}$, $y=\frac{1}{2}{t}^2+t$

Cycloid

a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).