Active Calculus

Which is a parametric equation for the curve $16={(x-9)}^2+{(y-2)}^2\text{?}$

Each set of parametric equations below describes the path of a particle that moves along the circle $x^2+(y-1{)}^2=4$ in some manner. Match each set of parametric equations to the path that it describes.

Consider the curve given by the parametric equations

a.) Determine the point on the curve where the tangent is horizontal.

$t=$

b.) Determine the points $t_1\text{,}$ $t_2$ where the tangent is vertical and $t_1<t_2$ .

$t_1=$

$t_2=$

The functions $f(t)$ and $g(t)$ are shown below.

If the motion of a particle whose position at time $t$ is given by $x=f(t)\text{,}$ $y=g(t)\text{,}$ sketch a graph of the resulting motion and use your graph to answer the following questions:

(a) The slope of the graph at $(0.25,0.5)$ is

(c) The slope of the graph at $(1.75,0.5)$ is

(a) Find $dy/dx$ and $d^{2}y/d{x}^2$ in terms of $t\text{.}$

$dy/dx$ =

$d^{2}y/d{x}^2$ =

(b) Using "less than" and "greater than" notation, list the $t$-interval where the curve is concave upward.

Use upper-case "INF" for positive infinity and upper-case "NINF" for negative infinity. If the curve is never concave upward, type an upper-case "N" in the answer field.

$t$-interval: $<t<$

Find the length of the curve $x=1+3t^2,y=4+2t^3,0\le t\le 1.$

Length =

Area =

Find the area of the region enclosed by the parametric equation