Question

1.Let *y*=6*x*^2. Find a parametrization of the
osculating circle at the point *x*=4.

2. Find the vector *O**Q*−→− to the center of the
osculating circle, and its radius *R* at the point
indicated. *r*⃗
(*t*)=<2*t*−sin(*t*),
1−cos(*t*)>,*t*=*π*

*3.* Find the unit normal vector *N*⃗ (*t*)
of *r*⃗ (*t*)=<10*t*^2, 2*t^*3>
at *t*=1.

4. Find the normal vector to *r*⃗
(*t*)=<3⋅*t*,3⋅cos(*t*)> at
*t*=*π*4.

5. Evaluate the curvature of *r*⃗
(*t*)=<3−12*t*, *e^(*2*t*−24),
24*t*−*t*2> at the point *t*=12.

6. Calculate the curvature function for *r*⃗
(*t*)=<7, *e^*1*t*, 10*t*>.

7. Let *r*⃗ (*t*)=<11cos*t*,
11sin*t*> Find a parametrization of the osculating circle
at the point *t*=*π*2.

Answer #1

A space curve C is parametrically parametrically defined by
x(t)=e^t^(2) −10,
y(t)=2t^(3/2) +10,
z(t)=−π,
t∈[0,+∞).
(a) What is the vector representation r⃗(t) for C ?
(b) Is C a smooth curve? Justify your answer.
(c) Find a unit tangent vector to C .
(d) Let the vector-valued function v⃗ be defined by
v⃗(t)=dr⃗(t)/dt
Evaluate the following indefinite integral
∫(v⃗(t)×i^)dt. (cross product)

At what point do the curves r1 =〈 t
, 1 − t , 3 + t2 〉 and r2 =
〈 3 − s , s − 2 , s2 〉 intersect? Find the angle of
intersection.
Determine whether the lines L1 :
r1 = 〈 5 − 12t , 3 + 9t ,1 − 3t 〉 and
L2 : r2 = 〈 3 + 8s , −6s , 7
+ 2s 〉are parallel, skew, or intersecting. Explain. If...

1.
(1 point)
Find the distance the point P(1, -6, 7), is to the plane through
the three points
Q(-1, -1, 5), R(-5, 2, 6), and S(3, -4, 8).
2.
(1 point) For the curve given by
r(t)=〈−7t,−4t,1+7t2〉r(t)=〈−7t,−4t,1+7t2〉,
Find the derivative
r′(t)=〈r′(t)=〈 , , 〉〉
Find the second derivative
r″(t)=〈r″(t)=〈 , , 〉〉
Find the curvature at t=1t=1
κ(1)=κ(1)=

show that L: x=1-2t .y=t .z=-t
and the plane P: 6x-3y+3z=1
are parpeudicular then find the point of intersect

1.) Let f ( x , y , z ) = x ^3 + y + z + sin ( x + z ) + e^( x
− y). Determine the line integral of f ( x , y , z ) with respect
to arc length over the line segment from (1, 0, 1) to (2, -1,
0)
2.) Letf ( x , y , z ) = x ^3 * y ^2 + y ^3 * z^...

Find a parametrization for the line perpendicular to
(4, −1, 1),
parallel to the plane
4x + y −
8z = 1,
and passing through the point
(1, 0, −7).
(Use the parameter t. Enter your answers as a
comma-separated list of equations.)

Consider the following vector function.
r(t) = <9t,1/2(t)2,t2>
(a) Find the unit tangent and unit normal vectors
T(t) and
N(t).
(b) Use this formula to find the curvature.
κ(t) =

1. Let f(x, y) = 2x + xy^2 , x, y ∈ R.
(a) Find the directional derivative Duf of f at the point (1, 2)
in the direction of the vector →v = 3→i + 4→j .
(b) Find the maximum directional derivative of f and a unit
vector corresponding to the maximum directional derivative at the
point (1, 2).
(c) Find the minimum directional derivative and a unit vector in
the direction of maximal decrease at the point...

Find a parametrization for the line perpendicular to (2, −1, 1),
parallel to the plane 2x + y − 6z = 1, and passing through the
point (1, 0, −3). (Use the parameter t. Enter your answers as a
comma-separated list of equations.)

(1 point) Match the following nonhomogeneous linear equations
with the form of the particular solution yp for the method of
undetermined coefficients.
? A B C D
1. y′′+y=t(1+sint)
? A B C D
2. y′′+4y=t2sin(2t)+(5t−7)cos(2t)
? A B C D
3.
y′′+2y′+2y=3e−t+2e−tcost+4e−tt2sint
? A B C D
4.
y′′−4y′+4y=2t2+4te2t+tsin(2t)
A.
yp=t(A0t2+A1t+A2)sin(2t)+t(B0t2+B1t+B2)cos(2t)
B.
yp=A0t2+A1t+A2+t2(B0t+B1)e2t+(C0t+C1)sin(2t)+(D0t+D1)cos(2t)
C.
yp=Ae−t+t(B0t2+B1t+B2)e−tcost+t(C0t2+C1t+C2)e−tsint
D. yp=A0t+A1+t(B0t+B1)sint+t(C0t+C1)cost

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