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Active Calculus 2nd Ed

Section 9.5 Parametric Curves

Exercises Exercises

1.

Which is a parametric equation for the curve 81=(x+7)2+(y5)2?
  • c(t)=(81cos(t)7,5+81sin(t))
  • c(t)=(5+9cos(t),9sin(t)7)
  • c(t)=(9cos(t)7,5+9sin(t))
  • c(t)=(5+81cos(t),81sin(t)7)

2.

Each set of parametric equations below describes the path of a particle that moves along the circle x2+(y1)2=4 in some manner. Match each set of parametric equations to the path that it describes.

3.

Consider the curve given by the parametric equations
x=t(t248),y=2(t248)
a.) Determine the point on the curve where the tangent is horizontal.
t=
b.) Determine the points t1, t2 where the tangent is vertical and t1<t2 .
t1=
t2=

4.

Find an equation for each line that passes through the point (4, 3) and is tangent to the parametric curve
x=3t2+1,y=2t3+1.
If there are multiple answers then separate distinct answers with commas.

5.

The functions f(t) and g(t) are shown below.
f(t) g(t)
If the motion of a particle whose position at time t is given by x=f(t), y=g(t), sketch a graph of the resulting motion and use your graph to answer the following questions:
(a) The slope of the graph at (1.75,0.5) is
(enter undef if the slope is not defined)
(b) At this point the particle is moving
and
.
(c) The slope of the graph at (0.25,0.5) is
(enter undef if the slope is not defined)
(d) At this point the particle is moving
and
.

6.

Consider the parametric curve given by
x=t312t,y=8t28
(a) Find dy/dx and d2y/dx2 in terms of t.
dy/dx =
d2y/dx2 =
(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<

7.

Calculate the length of the path over the given interval.
(sin8t,cos8t),0tπ

8.

Find the length of the curve x=1+3t2,y=4+2t3,0t1.
Length =

9.

Use the parametric equations of an ellipse
x=acos(θ),y=bsin(θ),0θ2π,
to find the area that it encloses.
Area =

10.

Find the area of the region enclosed by the parametric equation
x=t33t
y=6t2
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