Evaluate the indefinite integral as an
infinite series:
arctan(
x
2
)
dx
.
203.
Use series to approximate the definite
integral correct to three decimal places:
1
0
x
cos(
x
5
)
dx
.
204.
Use series to evaluate the limit:
lim
x
→
0
5
x

arctan(5
x
)
x
3
.
Curves Defined by Parametric
Equations
Directions:
Eliminate the parameter to find a
Cartesian equation of the curve.
205.
x
=
e
t

6,
y
=
e
2
t
206.
x
= ln
t
,
y
=
√
t
,
t
≥
36
Directions:
Determine what curve is represented
by the parametric equations. Be sure to indicate
direction as well as any starting or ending points.
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207.
x
= 2 cos(3
t
),
y
= 2 sin(3
t
), 0
≤
t
≤
2
π
/
3
208.
x
= sin(4
t
),
y
= sin
2
(4
t
)
Calculus with Parametric Curves
Directions:
Answer the questions.
209.
Find an equation of the tangent to
x
=
t
4
+ 2,
y
=
t
3
+
t
at the point
corresponding to
t
= 1.
210.
Find an equation of the tangent to
x
= 2 + ln
t
,
y
=
t
2
+ 2 at the point (2
,
3)
by both eliminating the parameter and
without eliminating the parameter.
211.
Consider
x
= 3 +
t
2
,
y
=
t
2
+
t
3
.
(a)
Find
dy
dx
and
d
2
y
dx
2
.
(b)
For which values of
t
is the curve
concave upward?
212.
Consider
x
= 10

t
2
,
y
=
t
3

27
t
.
(a)
Find the points on the curve where
the tangent is horizontal.
(b)
Find the points on the curve where
the tangent is vertical.
213.
Consider
x
= 3 cos
θ
,
y
= sin 2
θ
.
(a)
Find the points on the curve where
the tangent is horizontal.
(a)
Find polar coordinates (
r,
θ
) of the
point (2
,

2), where
r >
0 and
0
≤
θ
≤
2
π
.
(b)
Find polar coordinates (
r,
θ
) of the
point (2
,

2), where
r <
0 and
0
≤
θ
≤
2
π
.
(c)
Find polar coordinates (
r,
θ
) of the
point (1
,
√
3), where
r >
0 and
0
≤
θ
≤
2
π
.
(d)
Find polar coordinates (
r,
θ
) of the
point (1
,
√
3), where
r <
0 and
0
≤
θ
≤
2
π
.
219.
Find a Cartesian equation for the polar
curve
r
= 6 sin
θ
and identify the curve.
Directions:
Sketch the graph of the given polar
equation.
220.
r
= sin
θ
221.
r
=

7 cos
θ
222.
r
= 5(1

sin
θ
),
θ
≥
0
Polar Coordinates
Directions:
Answer the questions.
218.
The Cartesian coordinates of a point are
given.
(b)
Find the points on the curve where
the tangent is vertical.
214.
Consider
x
= 5 cos
t
,
y
= 2 sin
t
cos
t
. Show
that this curve has two tangents at (0
,
0)
and find their equations.
215.
Consider
x
=
t

t
2
,
y
= 4
t
3
/
2
/
3,
3
≤
t
≤
5. Setup an integral that
represents the length of the curve and
then evaluate that integral.
216.
Find the length of the curve:
x
= 1 + 3
t
2
,
y
= 9 + 2
t
3
, 0
≤
t
≤
3.
217.
Find the length of the curve:
x
=
e
t
+
e

t
,
y
= 5

2
t
, 0
≤
t
≤
3.