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Test 1 Practice … this is in addition to the week 3 supplemental problems and the
problems at the end of the projectile motion handout.
- Multiple Choice - For each question write into the provided blank the letter that corresponds to
the one correct answer.
A. r t ( )
G
is some unknown function for which ( )
T = −. Which of the
following couldpossibly be B ˆ ( 5 )?
a. 4,3,0 b.
c. 1,0,
d. 0,0, − 1 e. None of (a) - (d) f. All of (a) - (d)
B. r t ( )
G
is some unknown function for which ( )
T = −. Which of the
following couldpossibly be N ˆ^ ( 5 )?
a. 4,3,0 b.
c. 1,0,
d. 0,0, − 1 e. None of (a) - (d) f. All of (a) - (d)
C. r t ( )
G
is some unknown function for which the osculating plane at the point
r ( 12 ) =( 3,3,7)
G
has equation x + y = 6. Which of the following couldpossibly
be B ˆ ( 12 )?
a.
− b.
c.
d. 0,1,0 e. None of (a) - (d) f. All of (a) - (d)
D. r t ( )
G
is some unknown function for which the osculating plane at the point
r ( 12 ) =( 3,3,7)
G
has equation x + y = 6. Which of the following couldpossibly
be N ˆ ( 12 )?
a.
− b.
c.
d. 0,1,0 e. None of (a) - (d) f. All of (a) - (d)
E. r t ( )
G
is some unknown function for which r ′(^8 ) = 2,0,
G
. Which of the
following couldpossibly be r ′′( 8 )
G
?
a. −7,0,2 b. 0,0,0 c.
d. 0,1,0 e. None of (a) - (d) f. All of (a) - (d)
F. r t ( )
G
is some unknown function for which r t ( )
G
has the same non-zero
constant value for all values oft. Which of the following must be true about the
motion described by r t ( )
G
?
a. The motion must be circular b. r t ( ) ⋅ r ′( ) t = 0 ∀ t
G G
c. r ′( ) t
G
is a constant function d. r ′^ ( t ) ⋅ r ′′( ) t = 0 ∀ t
G G
e. (a) and (b) only f. (c) and (d) only
G. Which of the following functions describes elliptical motion along a plane?
a. r t ( ) = sin 3( t ) ,cos 3( t ) ,sin 3( t )+ 2
G
b. r t ( ) = sin ( ) t ,cos ( ) t , t
G
c. r t ( ) = sin ( ) t ,cos ( ) t ,cos 2( t )
G
d. ( )
2 2 r t = 1 − t , − 1 − t , t
G
e. None of (a) - (d) f. All of (a) - (d)
H. Which of the following functions describes spiraling motion up a cone?
a. r t ( ) = t sin 3( t ) , t cos 3( t ) , sin 3 t ( t )+ 2
G
b. r t ( ) = t sin ( ) t , cos t ( ) t , t
G
c. r t ( ) = sin ( ) t ,cos ( ) t ,cos 2( t )
G
d. r t ( ) = sin ( ) t ,cos ( ) t , t
G
e. None of (a) - (d) f. All of (a) - (d)
- Find an equation of the osculating circle of the ellipse 2 2
9 x + 4 y = 36 at the point ( 0, 3).
Include a sketch of the circle on Figure 2. Organize your algebra in a way such that it is clear what you are doing, why you are doing it, and what your “final answer” is!!
Recall:
( )
2
2
2 3 / 2
,
x y
d y
dx x y dy
dx
⎢⎣ ⎝^ ⎠⎥⎦
8. Name three surfaces upon which the curve r ( t ) = cos ( ) t , cos ( ) t −2, sin( ) t
G
must lie. No
work need be, nor should be, shown.
9. On of the curves in figures 3-6 is the function r ( t ) = t sin ( ) t + 4, t cos ( ) t − 2, sin t ( ) t + 1
G
.
Which is it?
Figure 3 Figure 4
Figure 5 Figure 6
Figure 2: 2 2 9 x + 4 y = 36
- For each statement write T into the provided blank if the statement is always true and write F
into the provided blank if the statement is sometimes (or always) false. In all cases you should
assume that r ( ) t
G
is differentiable (first and second derivatives) at all points and that all
referenced vectors are non-zero vectors.
a. If r ′′^ ( 4 )= 6, 0, 0
G
, then N ˆ^ ( 4 ) = 1, 0, 0.
b. The Binormal vector at a given point is perpendicular to the normal line at the same point.
c. N ˆ ( ) t ⊥ r ′( ) t
G
at every value of t.
d. T ˆ ( ) t ⊥ r ′′( ) t
G
at every value of t.
e. If r ′^ ( 4 )= 6, 0, 0
G
, then T ˆ^ ( 4 )= 1, 0, 0.
f. For motion along a circle, the velocity vector and the acceleration vector are perpendicular at every value of t.
g. The binormal vector is a unit vector.
h. The distance traveled along r ( t )
G
between times t = a and t = b is given by
b
a
r t dt.
i. The velocity is constant for a function that has the property that
v t ( ) ⋅ v ′( ) t = 0 ∀ t
G G
.
