Velocity and Acceleration

CALCULUS
Trancedental Functions. Velocity and Acceleration
1. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = 3 log(−t)
2. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −3e2t
3. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −3 ln(−2t)
4. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −5 sin(2t)
5. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −4 cos(t)
6. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −3 ln(5t)
7. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = − ln(−2t)
8. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −2 sin(t)
9. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = −(10)−4t
10. Find the velocity, acceleration, and jerk functions for the following position function: s(t) = 4 cos(−t)
10. v(t) = 4 sin(−t)
9. v(t) = 4(ln 10)1 10−4t
a(t) = −4 cos(−t)
j(t) = −4 sin(−t)
a(t) = −16(ln 10)2 10−4t
5. v(t) = 4 sin(t)
a(t) = 4 cos(t)
3
−3
a(t) = 2
j(t) =
6. v(t) =
t
t
1
−1
a(t) = 2
7. v(t) =
j(t) =
t
t
8. v(t) = −2 cos(t)
a(t) = 2 sin(t)
j(t) = 64(ln 10)3 10−4t
j(t) = 2 cos(t)
j(t) = −4 sin(t)
−6
t3
−2
t3
2. v(t) = −6e2t
a(t) = −12e2t
j(t) = −24e2t
3
−6
−3
a(t) =
j(t) =
3. v(t) =
t
t2
t3
4. v(t) = −10 cos(2t)
a(t) = 20 sin(2t)
j(t) = 40 cos(2t)
1. v(t) =
3
(ln 10)t
a(t) =
−3
(ln 10)t2
1 of 3
Answers:
j(t) =
6
(ln 10)t3
c 2009 La Citadelle
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CALCULUS
Trancedental Functions. Velocity and Acceleration
Solutions:
1.
Apply the formula(s):
d f (t)
d
b
= (ln b)bf (t) f (t)
dt
dt
d n
t = ntn−1
dt
s(t) = 3 log(−t)
d
d
3
s(t) =
[3 log(−t)] =
dt
dt
(ln 10)t
d
d
3
−3
a(t) = v(t) =
=
dt
dt (ln 10)t
(ln 10)t2
d
6
d
−3
j(t) = a(t) =
=
2
dt
dt (ln 10)t
(ln 10)t3
v(t) =
2.
Apply the formula(s):
d f (t)
d
e
= ef (t) f (t)
dt
dt
s(t) = −3e2t
d
s(t) =
dt
d
a(t) = v(t) =
dt
d
j(t) = a(t) =
dt
v(t) =
3.
d −3e2t = −6e2t
dt
d −6e2t = −12e2t
dt
d −12e2t = −24e2t
dt
1 d
d
ln f (t) =
f (t)
Apply the formula(s):
dt
f (t) dt
d n
t = ntn−1
dt
s(t) = −3 ln(−2t)
d
−3
[−3 ln(−2t)] =
dt
t
d −3
3
= 2
dt t
t
d
d 3
−6
j(t) = a(t) =
= 3
dt
dt t2
t
d
s(t) =
dt
d
a(t) = v(t) =
dt
v(t) =
4.
Apply the formula(s):
d
d
sin f (t) = (cos f (t)) f (t)
dt
dt
d
d
cos f (t) = −(sin f (t)) f (t)
dt
dt
s(t) = −5 sin(2t)
d
s(t) =
dt
d
a(t) = v(t) =
dt
d
j(t) = a(t) =
dt
d
[−5 sin(2t)] = −10 cos(2t)
dt
d
[−10 cos(2t)] = 20 sin(2t)
dt
d
[20 sin(2t)] = 40 cos(2t)
dt
d
d
5.
Apply the formula(s):
sin f (t) = (cos f (t)) f (t)
dt
dt
s(t) = −4 cos(t)
v(t) =
d
s(t) =
dt
d
a(t) = v(t) =
dt
d
j(t) = a(t) =
dt
v(t) =
d
d
cos f (t) = −(sin f (t)) f (t)
dt
dt
d
[−4 cos(t)] = 4 sin(t)
dt
d
[4 sin(t)] = 4 cos(t)
dt
d
[4 cos(t)] = −4 sin(t)
dt
c 2009 La Citadelle
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CALCULUS
6.
Trancedental Functions. Velocity and Acceleration
Apply the formula(s):
d
1 d
ln f (t) =
f (t)
dt
f (t) dt
d n
t = ntn−1
dt
s(t) = −3 ln(5t)
d
−3
[−3 ln(5t)] =
dt
t
d −3
3
= 2
dt t
t
d
−6
d 3
j(t) = a(t) =
= 3
2
dt
dt t
t
d
s(t) =
dt
d
a(t) = v(t) =
dt
v(t) =
7.
Apply the formula(s):
d
1 d
ln f (t) =
f (t)
dt
f (t) dt
d n
t = ntn−1
dt
s(t) = − ln(−2t)
d
−1
[− ln(−2t)] =
dt
t
d −1
1
= 2
dt t
t
d
d 1
−2
j(t) = a(t) =
= 3
2
dt
dt t
t
d
s(t) =
dt
d
a(t) = v(t) =
dt
v(t) =
8.
Apply the formula(s):
d
d
sin f (t) = (cos f (t)) f (t)
dt
dt
d
d
cos f (t) = −(sin f (t)) f (t)
dt
dt
s(t) = −2 sin(t)
d
s(t) =
dt
d
a(t) = v(t) =
dt
d
j(t) = a(t) =
dt
d
[−2 sin(t)] = −2 cos(t)
dt
d
[−2 cos(t)] = 2 sin(t)
dt
d
[2 sin(t)] = 2 cos(t)
dt
d
d f (t)
10
= (ln 10)10f (t) f (t)
9.
Apply the formula(s):
dt
dt
s(t) = −(10)−4t
d
d v(t) = s(t) =
−(10)−4t = 4(ln 10)1 10−4t
dt
dt
d d
4(ln 10)1 10−4t = −16(ln 10)2 10−4t
a(t) = v(t) =
dt
dt
d
d j(t) = a(t) =
−16(ln 10)2 10−4t = 64(ln 10)3 10−4t
dt
dt
d
d
10.
Apply the formula(s):
sin f (t) = (cos f (t)) f (t)
dt
dt
s(t) = 4 cos(−t)
v(t) =
d
s(t) =
dt
d
a(t) = v(t) =
dt
d
j(t) = a(t) =
dt
v(t) =
d
d
cos f (t) = −(sin f (t)) f (t)
dt
dt
d
[4 cos(−t)] = 4 sin(−t)
dt
d
[4 sin(−t)] = −4 cos(−t)
dt
d
[−4 cos(−t)] = −4 sin(−t)
dt
c 2009 La Citadelle
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