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MATH 2433 (Section 20708)
TA: Jeric Alcala
Office: PGH 612
Email: [email protected]
On Section 12.1 - 12.3:
1. Find the derivative of the following:
(a) r(t) = 2t2i +j
2t− 3+ 3 tan2(2t)k
2
(b) r(t) =√
2ti + arctan tj + 3cos tk
3
(c) r(t) = ln(5t)i +t
t− 1k
4
LabPop04a # 1:
Find r′(t) if r(t) = (et − 3)i + jt− 1
+ 2 cos(3t)k.
a.
(et,
1
(t− 1)2,−6 sin(3t)
)b.
(et − 3, 1
(t− 1)2,−6 sin(3t)
)c.
(et,− 1
(t− 1)2,−6 sin(3t)
)d.
(et,− 1
(t− 1)2,−2 sin(3t)
)e. None of these.
5
2. Calculate the required limit.
(a) limt→0
r(t) given r(t) =sin(4t)
3ti +
(t+ 1
et
)j− arctan(t+ 1)k
6
(b) limt→1
r(t) given r(t) = 4 cos(πt)i + 3 sin(πt)j +t− 1|t− 1|
k
7
(c) limt→0
r(t) given r(t) =1− cos t
2ti +√
2t2 − 1j + (ln t)k
8
LabPop04a # 2:
Find limt→0
r(t) if r(t) = (−2 sin(2t))i−(
sin(3t)
t
)j +
(2
et
)k.
a. (−2,−3, 0)
b. (0, 3, 2)
c. (0,−3, 2)
d. (0,−3, 0)
e. None of these.
9
3. Let f(t) = t2 + 1, r(t) = (1 − t2)i + 2 cos(2t)j + e−2tk, and s(t) = sin ti +
(ln t)j + t3k. Find:
(a)d
dt[f(t)r(t)]
10
(b)d
dt[r(t) · s(t)]
11
(c)d
dt[r(t)× s(t)]
12
LabPop04a # 3:
Given differentiable real-valued function f(t) and differentiable vector-valued
function r(t), which of the following is always TRUE?
I.d
dt[r (f(t))] = r′(f(t))f(t) + f ′(t)r(f(t))
II.d
dt[f(t)r(t)] = f ′(t)r(t) + f(t)r′(t)
a. I only
b. II only
c. Both I and II
d. Neither I nor II
13
4. Find
∫ π0
r(t) dt if r(t) = (2t)i + 2 sec2(t)j + cos(t/2)k.
14
5. Find the tangent vector r(t) at the given point (or that corresponding to given
t) and the equation of the tangent line at that point.
(a) r(t) = (2t2 + t)i +j
et+ 3 tan(2t)k at t = 0
15
(b) r(t) = cos(πt)i + arctan tj + sin(πt)k at t = −1
16
(c) r(t) = (t2 − 1)i + ln(t+ 1)j + (t+ 3)k at point (−1, 0, 3)
17
LabPop04a # 4:
Find the tangent vector to r(t) = 2t2i + 2 sin(πt)j + 2k at t = 1.
a. (4,−2, 0)
b. (2,−2π, 0)
c. (4,−2π, 0)
d. (4,−2π, 2)
e. None of these.
18
6. Find the unit tangent vector, principal normal vector and osculating plane at
the given point (or that corresponding to given t) to the given curve.
(a) r(t) = t2i + 3tj + 2k at t = 1
19
(b) r(t) = cos(2t)i + sin(t)j + tan(t+ π)k at t = 0
20
(c) r(t) = et−1i + t2j + (t+ 2)k at the point (1, 1, 3)
21
7. Given the curve r(t) = (t2− 1)i+ 3tj, find the points where r(t) and r′(t): (a)
are perpendicular, (b) have the same direction, (c) have opposite directions.
22
LabPop04a # 5:
Find the principal normal vector to r(t) = (2 sin t)i + (2 cos t)j.
a. (− sin t,− cos t, 0)
b. (− sin t, cos t, 0)
c. (−2 sin t,−2 cos t, 0)
d. (− cos t, sin t, 0)
e. None of these.
23