§ 2.6 – Higher-Order Derivatives
Fall 2011, Math 1210-007
Notation
Derivative
f
y
D
1st f ′ (x)
y′
Dx y
2nd f ′′ (x)
y ′′
Dx2 y
3rd f ′′′ (x)
y ′′′
Dx3 y
4th f (4) (x)
y (4)
Dx4 y
5th f (5) (x)
y (5)
Dx5 y
nth f (n) (x) y (n) Dxn y
Example: 1
Leibniz
dy
dx
d2 y
dx2
d3 y
dx3
d4 y
dx4
d5 y
dx5
dn y
dxn
1 x5 .
Find the 6th derivative of y = 120
Velocity and Acceleration
• position/height/directed distance: s
• (instantaneous) velocity: v =
ds
dt
• speed: |v|
• acceleration: a =
Example: 2
dv
d2 s
= 2
dt
dt
An object moves along a horizontal coordinate line with a directed distance from
the origin, in feet, of s = t3 − 9t2 + 24t, where t is in seconds.
(a)
(b)
(c)
(d)
(e)
What are the velocity and acceleration at time t?
When is the object moving to the left?
When is its acceleration negative?
When is it not moving?
Draw a schematic diagram that shows the motion of the object.
Falling Body Problems
Close to sea level and without air resistance, an object thrown straight upward/downward has a
height above ground (in feet) of
s=
1
· |{z}
−32 t2 + v0 t + s0 = −16t2 + v0 t + s0
|{z}
|{z}
2
accel. due
to gravity
Example: 3
initial
velocity
initial
height
Some random guy decides to throw a brick directly into the air with an initial velocity
of 14 feet per second (ft/sec). The height of the brick, in feet, after t seconds is
s = −16t2 + 14t + 8.
(a) At what height is the brick released from his hand?
(b) Roughly how high does the brick go before it starts to return to the ground,
i.e. what is its maximum height?
(c) How long does the guy have to move out of the way before he gets hit in the
head by the brick, assuming he is 6 feet tall?