EXAM #1 MATH 241 CALCULUS 3- SANTILLI
Chapters 12 and 13 - ALL WORK IN YOUR BLUE BOOK! Problems equally weighted
1.) An object is moving through space with an acceleration vector of
a 3t2ˆ
j 10 ˆ
k m/s2
. Initially, the object’s position and
velocity are
r 1,1,1
meters and
v 1ˆ
i 2ˆ
k m/s
respectively.
a.) Find the object’s position vector function for any time, t.
b.) Find the time of object’s impact (ground is located on the x-y plane).
c.) Find the location of impact, i.e., the impact position vector.
d.) Find the impact velocity vector and the impact speed.
e.) Find the angles of impact (
, ,
) by the use of directional cosines.
f.) Find the range of the particle in space. (straight-line distance between initial and impact positions)
g.) Find the trajectory path-length of the particle. (total arc length of the object’s flight)
h.) Find the maximum altitude experienced by the object during its flight.
2.) Let
r (t) 2 t ˆ
i et1t ˆ
j ln t1 ˆ
k
a.) Find the domain of
r (t)
b.) Find
lim
t0r (t)
c.) Find
T (1)
3.) Reparametrize the curve
r (t)et ˆ
i etsint ˆ
j etcost ˆ
k
with respect to the arc length measured from the point
1,0,1
in
the direction of increasing t and then show that
r (s)ˆ
T (s)
.
4.) The helix
r (t) cost ˆ
i sint ˆ
j t ˆ
k
intersects the curve
r (t) (1 t)ˆ
i t2ˆ
j t3ˆ
k
at the point
1,0,0
. Find the angle
of intersection of these curves.
5.) A particle’s path is
r (t) 4t ˆ
i 3 cos t ˆ
j 3 sin t ˆ
k
. Find the following when the particle is located at
( , 3
2,3
2)
.
a.) Velocity and Speed
b.) Acceleration in terms of its base vectors then in terms of its tangential and normal components.
c.) Show that the two forms of acceleration in part b are equivalent.
d.) TNB frame
e.) Curvature and torsion of path
f.) Equations of the osculating, normal, and rectifying planes.
g.) Radius of the osculating circle.
h.) Graph the particle’s path for
0t2
6.) Find the equation of the plane that passes through the line of intersection of the planes
x z 1
and
y2z3 0
and is
perpendicular to the plane
x y 2z1
.
7.) Reduce the following quadric surfaces into standard form, classify the surface, and sketch it. Label your graph.
a.)
x24y2z22x0
b.)
4x29y2z236 0
c.)
x2y24y z 4 0
d.)
rcos
8.) Show that the line that passes through the points (1,2,7) and (-2,3,-4) and the line that passes through the points (2,-1,4) and
(5,7,-3) are skew, then find the distance between them.
9.) Find the vector function that represents the curve of intersection of
x2y216
and
xyz
.
10.) We find the total curvature K of the portion of a smooth curve that runs from
0
ss
to
1
ss
by integrating the curvature
along its arc length from
0
ss
to
1
ss
. Find the total curvature of the parabola
2
xy
,
x
.
11.) A baseball is hit 3 feet above the ground at 100 ft/sec and at an angle of π/4 with respect to the ground.
a.) Find the maximum height reached by the baseball.
b.) Will the ball clear a 10-foot-high fence located 300 feet from home plate?
12.) A surface consists of all points P such that the distance from P to the plane y=1 is twice the distance from P to the point
(0,-1,0). Find the equation for this surface and identify it.
13.) Sketch the curve
kjtittr ˆ
cost 2
ˆ
sin
ˆ
sin)(
between
20 t
. Indicate with arrows the direction in which t
increases. Also identify and sketch the surfaces that the curve lies within.
14.) Find the length of the curve
r (t) 2t3 2,cos2t,sin2t
,
0t1
.
15.) Find the point at which the line
21-
4
,5 z
y
x
intersects the plane
2x y z 5
.