math54exam4review
TRANSCRIPT
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 1/16
Schedule Coverage Sample Questions
Math 54 Fourth Exam Review Mathematics 54 - Elementary Analysis 2
Institute of Mathematics
University of the Philippines-Diliman
7 November 2014
1/16
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 2/16
Schedule Coverage Sample Questions
Reminder
FOURTH LONG EXAMINATION
11 November 2014, Tuesday
Discussion Class Room and Time
2/16
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 3/16
Schedule Coverage Sample Questions
Coverage
Vector-Valued Functions
Domain and GraphCurves as Intersection of Surfaces
Operations on Vector-Valued Functions
Calculus of Vector-Valued Functions
Limits and Continuity, Derivatives, Integrals
Geometric Consequences (Tangent Vector/Tangent Line, Arc Length)
Moving Trihedral
Unit Tangent, Normal, and Binormal Vectors
Osculating, Rectifying, Normal Planes
Arc Length Parametrization and Curvature
Reparametrization of Vector-Valued FunctionsCurvature and the Osculating Circle
Curvilinear Motion
Position, Velocity, Acceleration Vectors
Components of Velocity and Acceleration
Projectile Motion
3/16
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 4/16
Schedule Coverage Sample Questions
Sample Questions
Write TRUE if the statement is always true, and FALSE if otherwise.
1. The unit tangent vector to the graph of R at t is perpendicular to the
normal plane to the graph of R at t . TRUE
2. R (t ) ·R ′(t )= 0, for any t . FALSE
3. If F and G have differentiable component functions, then
(F ×G )′(t )=F ′(t )×G (t )+G ′(t )×F (t ). FALSE
4. If s is a parameter representing the length of arc of the graph of R froma fixed point following the orientation of R , thenR ′(s )=T (s ) for any s .
TRUE
5. Given a vector functionR , s the length of arc of the graph of R from a
fixed point following the orientation of R , and κ(t ) the curvature at t ,
the following equation holds for any t : R ′′(t )2 =d 2s
dt 2
2+(κ(t ))2
ds
dt
4
.
TRUE
4/16
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 5/16
Schedule Coverage Sample Questions
Sample Questions
Given: R (t )=
2t +4, tan−1 t −2
4 , 16e t
2−4t −12 − 16
t 2
−4
1. Find domR .
5/16
S h d l C S l Q ti
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 6/16
Schedule Coverage Sample Questions
Sample Questions
Given: R (t )=
2t +4, tan−1 t −2
4 , 16e t
2−4t −12 − 16
t 2
−4
2. Evaluate limt →−2+
R (t ).
6/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 7/16
Schedule Coverage Sample Questions
Sample Questions
Given: R (t )=
2t +4, tan−1 t −2
4 , 16e t
2−4t −12 − 16
t 2
−4
3. Find a vector equation to the line tangent to the graph of R (t ) at t = 6.
7/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 8/16
Schedule Coverage Sample Questions
Sample Questions
Find the vector equation of the curve of intersection of the surfaces
2x −3 y +z = 6 and x 2
9 + y 2
4 = 1.
8/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 9/16
Schedule Coverage Sample Questions
Sample Questions
Given: Vector functionR such thatR (0)
= ⟨−6,3,2
⟩, R ′(0)
=
5,R ′′
(0)= ⟨
1,−
1,4⟩
1. Findh ′(0) where h (t )= (R ·R ′)(t ).
9/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 10/16
Schedule Coverage Sample Questions
Sample Questions
Given: Vector functionR such thatR (0)
= ⟨−6,3,2
⟩, R ′(0)
=
5,R ′′(0)= ⟨
1,−
1,4⟩
2. Find (R ′ ◦ f )′(1) where f (t )= ln(8t −7).
10/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 11/16
g p Q
Sample Questions
Given: Position vectorR and unit tangent vector T T ′(t )
=3cos5t ı̂
−4cos5t ˆ
−5sin5t k̂ ,
R (0)= 2 ˆ −5 k̂ , T (0)= k̂ , and curvature κ(0)= 1
2.
1. Find the unit normal and the unit binormal vector at t = 0.
11/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 12/16
Sample Questions
Given: Position vectorR and unit tangent vector T T ′(t )
=3cos5t ı̂
−4cos5t ˆ
−5sin5t k̂ ,
R (0)= 2 ˆ −5 k̂ , T (0)= k̂ , and curvature κ(0)= 1
2.
2. Find the center of the osculating circle at the point where t = 0.
12/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 13/16
Sample Questions
Let C be the curve having vector equationR (t )= 3
16t 2 ı̂ + 1
2t
32 ˆ + 3
4t k̂ .
Find the arc length of the graph of R from the origin to the point (3, 4,3).
13/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 14/16
Sample Questions
A particle moves such that its velocity vector at any time t is
V (t )=
1
t +1+3, −cosht , 8−2t
.
At t = 0, the particle is at the point (4,0,−1). Find:
1. the position vectorR (t ) of the particle at any t ;
14/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 15/16
Sample Questions
A particle moves such that its velocity vector at any time t is
V (t )=
1
t +1+3, −cosht , 8−2t
.
At t = 0, the particle is at the point (4,0,−1). Find:
2. the tangential and normal components of acceleration of the particle
at t =
0.
15/16
Schedule Coverage Sample Questions
7/23/2019 Math54Exam4Review
http://slidepdf.com/reader/full/math54exam4review 16/16
Sample Questions
A projectile is fired from the ground at an angle of elevation of 60◦, hitting
a target that is 100
3 m away and 60 m above the ground. Find the initial
speed of the projectile. (Assume that the acceleration due to gravity g is
10 m/s2.)
16/16