Question

- r(t)=[cos(t),sin(t),cos(3t)]
- r(t)=[tcos(t),tsin(t),t)
- r(t)=[cos(t),sin(t),t2]
- r(t)=[t2cos(t),t2sin(t),t]
- r(t)=[cos(t),t,sin(t)]

Sketch the graphs.

Answer #1

**1.)**

**2.)**

**3.)**

**4.)**

**5.)**

If equation 3 is like

r(t)=[cos(t),sin(t),t^{2}]

then the graph will be like this

If equation 4 is like

r(t)=[t^{2}cos(t),t^{2}sin(t),t]

then the graph will be like this

** This take a long time to solve. So, please comment below if you have any doubt regarding this answer before rating this answer.**

Sketch the graph of each curve by finding surfaces on which they
lie.
(A) <t,tcos(t),tsin(t)>
(B) <cos(t),sin(t),sin(2t)>
(C) <t,cos(t),sin(t)>
Please Sketch in a 3D graph.

Find the derivative r '(t) of the
vector function r(t).
<t cos 3t , t2, t sin 3t>

With the parametric equation x=cos(t)+tsin(t), y=sin(t)-tcos(t)
, 0 ≤ t ≤ 2π)
Find the length of the given curve. (10 point)
2) In the circle of r = 6, the area
above the r = 3 cos (θ) line
Write the integral or integrals expressing the area of this
region by drawing. (10 point)

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

Question 4.Let C be the curve in three dimensions with the
following parameterization for 0≤t≤2:
(sin(πt71),t2−3t+ 2,t3−3t).
Find the value of
∫C (cos(x+y) +ze^xz)dx+ cos(x+y)dy+ (xe^xz+ 2)dz.

Given that the acceleration vector is a(t)=(-9 cos(3t))i+(-9
sin(3t))j+(-5t)k, the initial velocity is v(0)=i+k, and the initial
position vector is r(0)=i+j+k, compute:
A. The velocity vector v(t)
B. The position vector r(t)

Given that the acceleration vector is a ( t ) = (−9 cos( 3t ) )
i + ( −9 sin( 3t ) ) j + ( −5 t ) k, the initial velocity is v ( 0
) = i + k, and the initial position vector is r ( 0 ) = i +j + k,
compute: the velocity vector and position vector.

6.) Let ~r(t) =< 3 cos t, -2 sin t > for 0 < t < pi.
a) Sketch the curve. Make sure to pay attention to the parameter
domain, and indicate the orientation of the curve on your graph. b)
Compute vector tangent to the curve for t = pi/4, and sketch this
vector on the graph.

17.)Find the curvature of r(t) at the
point (1, 0, 0).
r(t) =
et
cos(t),
et
sin(t), 3t
κ =

Find a unit tangent vector to the curve r = 3 cos 3t
i + 3 sin 2t j at t =
π/6 .

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