Math 241
Quiz #3 version: 1
Instructor: Wenqiang Feng
Name:
(1) (5 points) Find parametric equation for the tangent line to the curve r(t) = ht, 2t2 , t3 i
at the point (1,2,1).
Solution.
Step 1. Find out the specific t


1

=t
2 = 2t2



1 = t3
⇒ t = 1.
Step 2. Find out the direction vector of the tangent line
• direction vector for all t:
E
D
E
d D
t, 2t2 , t3 = 1, 4t, 3t2
r0 (t) =
dt
• direction vector for specific t=1:
r0 (1) = h1, 4, 3i .
Step 3. Write out the parametric equation for the tangent line
T (t) = r(1) + tr0 (1)
= h1, 2, 1i + t h1, 4, 3i
Hence, the parametric equation for the tangent line is as follows



x
= 1 + t,
y = 2 + 4t,



z = 1 + 3t.
J
(2) (5 points) Find the arc-length of the curve r = h− cos(2t), sin(2t), ti over the interval
0 ≤ t ≤ π.
Solution.
Step 1. Find out r0 (t)
d
h− cos(2t), sin(2t), ti = h2 sin(2t), 2 cos(2t), 1i
r0 (t) =
dt
Step 2. Find out the length of r0 (t)
kr (t)k =
q
4 sin2 (2t) + 4 cos2 (2t) + 12
=
q
4 (sin2 (2t) + cos2 (2t)) + 1
0
√
=
4+1
√
=
5.
Step 3. Compute the integral for 0 ≤ t ≤ π
Z π
Z π√
√ Z
kr0 (t)k dt =
5dt = 5
0
0
π
1dt =
√
√
5(π − 0) = 5π.
0
J
Math 241
Quiz #3 version: 2
Instructor: Wenqiang Feng
Name:
(1) (5 points) Find parametric equation for the tangent line to the curve r(t) = h2t2 , t3 , 4ti
at the point where t = 1.
Solution.
Step 1. Find out the specific point at t=1
D
E
r(1) = 2 · 12 , 13 , 4 · 1 = h2, 1, 4i .
Step 2. Find out the direction vector of the tangent line
• direction vector for all t:
E
d D 2 3 E D
r0 (t) =
2t , t , 4t = 4t, 3t2 , 4
dt
• direction vector for specific t=1:
r0 (1) = h4, 3, 4i .
Step 3. Write out the parametric equation for the tangent line
T (t) = r(1) + tr0 (1)
= h2, 1, 4i + t h4, 3, 4i
Hence, the parametric equation for the tangent line is as follows


x

= 2 + 4t,
y = 1 + 3t,



z = 4 + 4t.
J
(2) (5 points) Find the arc-length of the curve r = h4t, − cos(t), sin(t)i from the point
(0, −1, 0) to the point (2π, 0, 1).
Solution.
Step 1. Find out r0 (t)
d
h4t, − cos(2t), sin(t)i = h4, sin(t), cos(t)i .
dt
Step 2. Find out the length of r0 (t)
r0 (t) =
0
kr (t)k =
q
42 + sin2 (t) + cos2 (t)
q
16 + (sin2 (t) + cos2 (t))
√
=
16 + 1
√
=
17.
=
Step 3. compute the domain of the integral



0
= 4t,
−1 = − cos(t),



0 = sin(t).



2π
= 4t,
⇒ t = 0, 0 = − cos(t),



1 = sin(t).
⇒t=
π
.
2
3
Step 4. Compute the integral for 0 ≤ t ≤
Z π/2
0
0
kr (t)k dt =
Z π/2 √
0
17dt =
√
π
2
17
Z π/2
0
1dt =
√
π
17( − 0) =
2
√
17
π.
2
J