1432
www.math.uh.edu/~pamb
THINK:
On Math each row place two letters that can be attached to the end of
the word to the left and to the beginning of the word to the right to
form a longer word in each case.
When completed the six added letters will give a word reading
downwards.
THAT
__
ORAL
PRIM
__
COVE
__
HER
OWL
What is the word?
POPPER
1. For what values of t does the curve given by the parametric
equations x = t 3 t 2 1 and y = t 4 + 2t 2 8t have a vertical
tangent?
a) 0 only
b) 1 only
2
c) 0 and only
3
2
d) 0 , , and 1
3
2. Give the area of the region that is enclosed by the polar curve
r = 1 + 2sin() and lies below the x-axis.
a. 3 3 + 2
4
4
b.
3 + 2
2
c.
3 +2
d.
33
+ 4
2
2
3. Give a parameterization for the line segment from
(2, 1) to ( 4, 3).
a. x(t) = 2 3t,
y(t) = 1 + t,
0t1
b. x(t) = 4 + 6t, y(t) = 3 2t,
0t1
c. x(t) = 2 6t,
0t1
y(t) = 1 + 2t,
d. x(t) = 4 + 3t, y(t) = 3 t,
0t1
4. Give the slope of the tangent line to the curve given
parametrically by (cos(t), sin(2t)) at the point where t = /6.
a. 2
b. 2
c.
3
d. 3
e. None of these.
5. Let L be the tangent line to the curve given parametrically by
(cos(t), sin(2t)) at the point where t = /6. The line L strikes the
y-axis at the point (0, a). a = ?
33
a. a =
4
b. a = something other than
33
4
Suppose a particle is traveling on a path represented by a portion of
the xy plane. In this case, assume that the y-axis represents the
up/down direction. Given this information, suppose the position of
the particle is given at time t in seconds by
r ( t ) = t , 16 t 2 + 4 0 t + 1 0 for t > 0 until the particle strikes the
(
)
ground. Give the time when the particle strikes the ground, and
determine the velocity and acceleration of the particle at this time.
Arc Length and Speed
Section 9.8
Velocity Speed and (magnitude of velocity)
If the position of a particle at time t is given by
r(t) = (x(t), y(t))
then the velocity is given by
r(t) = v(t) = (x(t), y(t))
and the speed is given by
speed = v ( t ) =
x' ( t ) + y' ( t )
2
2
A particle is traveling on an elliptic path in the xy-plane so that its
position at time t is given by r ( t ) = ( 2 cos ( t ) ,3 sin ( t ) ) .
3
Give the position, velocity and
speed of the particle at time t = /4.
2
1
3
3
2
1
1
1
2
3
4
2
3
4
If a curve C is given parametrically by
(x(t), y(t)) for a t b
and x(t) and y(t) are continuous functions,
then the length of the curve is given by
L (c) =
b
a
x ' ( t ) + y ' ( t )
2
2
dt
Give an integral which represents the length of the curve given
parametrically by r ( t ) = ( 2 cos ( t ) ,3 sin ( t ) ) for 0 t 2.
3
2
1
3
3
2
1
1
1
2
3
4
2
3
4
Give an integral which represents the length of the curve given
parametrically by r ( t ) = ( 2 cos ( 3t ) ,3 sin ( 4t ) ) for 0 t 2.
3
2
1
3
3
2
1
1
1
2
3
4
2
3
4
Give a formula for the length of the curve given by the graph of
f (x) = 2x +1 for 1 x 3.
How can we find the length of the curve given by the graph of f (x)
for a x b?
L (c) =
b
a
1 + f ' ( x )
2
dx
Verify that the formula works for the previous example.
Give a formula for the length of the curve given by the graph of
f (x) = x2 for 1 x 1.
How can we find the length of a polar curve?
L (c) =
( ) + ' ( )
2
2
d
Summary:
L (c) =
b
1 + f ' ( x )
2
dx
Arc length (rectangular)
x [a, b]
a
L (c) =
b
x ' ( t ) + y ' ( t ) dt Arc length (parametric) t [a, b]
2
2
a
v( t ) =
L (c) =
x' ( t ) + y' ( t )
2
( ) + ' ( )
2
2
2
d
Speed (parametric)
Arc length (polar)
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