Yue Huang Ruedemann MAT 267 ONLINE B Spring 2020
Assignment Section 10.7 due 03/26/2020 at 11:59pm MST
1. (1 point) Find the domain of the vector functions, r(t),
listed below.
You may use ”-INF” for −∞and use ”INF” for ∞as necessary,
and use ”U” for a union symbol if a union of intervals is needed.
a) r(t) = Dln(9t),√t+11,1
√19−tE
b) r(t) = √t−4,sin(8t),t2
c) r(t) = e−4t,t
√t2−64 ,t1/3
Answer(s) submitted:
•
•
•
(incorrect)
2. (1 point) Let r(t) = (√t+3)i+t2−4
t+2j+
sin(−5πt)k.
Then
lim
t→1r(t)=i+j+k.
Answer(s) submitted:
•
•
•
(incorrect)
3. (1 point) Find the limit:
lim
t→0e−6t−1
t,t9
t10 −t9,−4
19 +t
h, , i
Answer(s) submitted:
•
•
•
(incorrect)
4. (1 point) The curve c(t) = hcost,sint,tilies on which of
the following surfaces.
Enter Tor Fdepending on whether the statement is true or
false.
(You must enter Tor F– True and False will not work.)
1. a plane
2. a sphere
3. an ellipsoid
4. a circular cylinder
Answer(s) submitted:
•
•
•
•
(incorrect)
5. (1 point) Match the parametric equations with the graphs
labeled A - F. As always, you may click on the thumbnail image
to produce a larger image in a new window (sometimes exactly
on top of the old one).
1. x=sin3tcost,y=sin3tsint,z=t
2. x=t,y=1/(1+t2),z=t2
3. x=cos4t,y=t,z=sin4t
4. x=t2−2,y=t3,z=t4+1
5. x=cost,y=sint,z=sin5t
6. x=cost,y=sint,z=lnt
A B C D E F
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
6. (1 point) Find a vector function that represents the curve
of intersection of the paraboloid z=7x2+5y2and the cylinder
y=6x2. Use the variable t for the parameter.
r(t) = ht,,i
Answer(s) submitted:
•
•
(incorrect)
1
7. (1 point) Consider the paraboloid z=x2+y2. The plane
7x−6y+z−2=0 cuts the paraboloid, its intersection being a
curve.
Find ”the natural” parametrization of this curve.
Hint: The curve which is cut lies above a circle in the xy-
plane which you should parametrize as a function of the variable
t so that the circle is traversed counterclockwise exactly once as
t goes from 0 to 2*pi, and the paramterization starts at the point
on the circle with largest x coordinate. Using that as your start-
ing point, give the parametrization of the curve on the surface.
c(t) = (x(t),y(t),z(t)), where
x(t) =
y(t) =
z(t) =
Answer(s) submitted:
•
•
•
(incorrect)
8. (1 point) Find the derivative of the vector function
r(t) = ln(20 −t2)i+√16 +tj−9e2tk
r0(t) = h, , i
Answer(s) submitted:
•
•
•
(incorrect)
9. (1 point) For the given position vectors r(t)compute the
unit tangent vector T(t)for the given value of t.
A) Let r(t) = hcos5t,sin5ti.
Then T(π
4)h,i
B) Let r(t) = ht2,t3i.
Then T(3) = h,i
C) Let r(t) = e5ti+e−3tj+tk.
Then T(4)=i+j+
k.
Answer(s) submitted:
•
•
•
•
•
•
•
(incorrect)
10. (1 point) Find parametric equations for the tangent line
at the point
cos5
6π,sin5
6π,5
6πon the curve x=cost,y=sint,z=t
x(t)=
y(t)=
z(t)=
(Your line should be parametrized so that it passes through
the given point at t=0).
Answer(s) submitted:
•
•
•
(incorrect)
11. (1 point) Find the parametric equations for the tangent
line to the curve
x=t4−1,y=t4+1,z=t2
at the point (15,17,4). Use the variable tfor your parameter.
x=,
y=,
z=
Answer(s) submitted:
•
•
•
(incorrect)
12. (1 point) Evaluate
Z8
0ti+t2j+t3kdt =i+j+k.
Answer(s) submitted:
•
•
•
(incorrect)
13. (1 point) If r(t) = cos(−2t)i+sin(−2t)j+2tk
compute r0(t)=i+j+k
and Rr(t)dt=i+j+k+C
with Ca constant vector.
Answer(s) submitted:
•
•
•
•
•
•
(incorrect)
2
14. (1 point) Find a vector parametrization of the curve
x=−3z2in the xz-plane. Use tas the parameter in your answer.
~r(t) =
Answer(s) submitted:
•
(incorrect)
15. (1 point)
Are the following statements true or false?
? 1. A parametrization of the graph of y=ln(x)for x>0 is
given by x=et,y=tfor −∞<t<∞.
? 2. The line parametrized by x=7,y=5t,z=6+tis par-
allel to the x-axis.
? 3. The parametric curve x= (3t+4)2,y=5(3t+4)2−9,
for 0 ≤t≤3 is a line segment.
Answer(s) submitted:
•
•
•
(incorrect)
16. (1 point) Find a vector parametric equation ~r(t)for the
line through the points P= (4,3,1)and Q= (9,−1,−3)for
each of the given conditions on the parameter t.
(a) If~r(0) = h4,3,1iand~r(6) = h9,−1,−3i, then
~r(t) =
(b) If~r(6) = Pand~r(10) = Q, then
~r(t) =
(c) If the points Pand Qcorrespond to the parameter values
t=0 and t=−4, respectively, then
~r(t) =
Answer(s) submitted:
•
•
•
(incorrect)
17. (1 point)
The function r(t)traces a circle. Determine the radius, cen-
ter, and plane containing the circle
r(t) = 7i+ (9cos(t))j+ (9sin(t))k
Plane : x=
Circle’s Center : ( , , )
Radius :
Answer(s) submitted:
•
•
•
•
•
(incorrect)
18. (1 point)
Use cos(t)and sin(t), with positive coefficients, to parame-
trize the intersection of the surfaces x2+y2=9 and z=4x3.
r(t) = h, , i
Answer(s) submitted:
•
(incorrect)
19. (1 point)
Find a parametrization, using cos(t)and sin(t), of the follow-
ing curve:
The intersection of the plane y=7 with the sphere x2+y2+z2=
74
r(t) = h, , i
Answer(s) submitted:
•
(incorrect)
20. (1 point)
Find the solution r(t)of the differential equation with the
given initial condition:
r0(t) = hsin4t,sin5t,9ti,r(0) = h8,8,9i
r(t) = h,,i
Answer(s) submitted:
•
•
•
(incorrect)
Generated by c
WeBWorK, http://webwork.maa.org, Mathematical Association of America
3
