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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.
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2. Calculate the required limit.
(a) limt→0
r(t) given r(t) =sin(4t)
3ti +
(t+ 1
et
)j− arctan(t+ 1)k
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(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
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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.
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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)]
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(b)d
dt[r(t) · s(t)]
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(c)d
dt[r(t)× s(t)]
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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
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4. Find
∫ π0
r(t) dt if r(t) = (2t)i + 2 sec2(t)j + cos(t/2)k.
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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
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(b) r(t) = cos(πt)i + arctan tj + sin(πt)k at t = −1
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(c) r(t) = (t2 − 1)i + ln(t+ 1)j + (t+ 3)k at point (−1, 0, 3)
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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.
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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
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(b) r(t) = cos(2t)i + sin(t)j + tan(t+ π)k at t = 0
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(c) r(t) = et−1i + t2j + (t+ 2)k at the point (1, 1, 3)
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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.
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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.
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