16. Parametric Equations & Polar Coordinates

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=cos t+t sin t,y=sin t−t cos t; t=π/4

Write the equation of the tangent line in cartesian coordinates for the given parameter $t$.

$x=8\cos t$, $y=6\sin t$, $t=\frac{\pi }{4}$

Find $\frac{d^{2}y}{d{x}^2}$ for the parametric curve at the given point.

$x=8\cos t$, $y=6\sin t$, $t=\frac{\pi }{4}$

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

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

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

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

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. 

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

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

x = 2 cos t, y = 8 sin t; slope = -1

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

x = 4 cos t, y = 4 sin t; slope = 1/2

73–76. Tangent lines Find an equation of the line tangent to the curve at the point corresponding to the given value of t.

x=t ²−1, y=t ³ +t; t=2

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 = t + 1/t, y = t − 1/t; t = 1

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

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

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

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