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.