Higher order derivatives
Objective:
To be able to find higher order derivatives
and use them to find velocity and
acceleration of objects.
TS: Explicitly assess information and draw
conclusions.
Do you remember your different notations
for derivatives?
f '( x)
y'
dy
dx
Well these are the same notations for higher
power derivatives! Any guesses on what each
means?
f ''( x)
y '''
the sec ond derivative of f
the third derivative
2
d y
2
dx
the sec ond derivative
And to find them you just take the
derivative again...and again…if necessary!
For example to get from f’’(x) to f’’’(x) you
just take the derivative of f’’(x).
And to get from f’(x) to f(4)(x) you would
just take the derivative of f’(x) three times.
Example A
Find the second derivative of f(x) = x4 – 2x3
f '( x)  4x3  6x2
f ''( x)  12x 2 12x
Example B
Given f '''( x)  2 x 1 find f (4) ( x)
f '''( x )  2( x  1)
1
2
f (4) ( x)  2(1/ 2)( x  1)1/2 (1)
f (4)  ( x 1)1/2
Example C
Given g ( x)  3x3  9x  1, solve the following equation g ''(x)  0
g '( x)  9x 2  9
g ''( x)  18x
We want g ''( x)  0 so 18x  0
x0
Position, Velocity & Acceleration
Velocity is the rate of change of position
with respect to time.
D
Velocity 
T
Acceleration is the rate of change of velocity
with respect to time.
V
Acceleration 
T
Position, Velocity & Acceleration
Warning: Professional driver, do not attempt!
When you’re driving your car…
Position, Velocity & Acceleration
squeeeeek!
…and you jam on the brakes…
Position, Velocity & Acceleration
…and you feel the car slowing down…
Position, Velocity & Acceleration
…what you are really feeling…
Position, Velocity & Acceleration
…is actually acceleration.
Position, Velocity & Acceleration
I felt that
acceleration.
Position, Velocity & Acceleration
Example D: A crab is crawling along the edge of your desk.
Its location (in feet) at time t (in seconds) is given by
P (t ) = t 2 + t.
A) Where is the crab after 2 seconds?
B) How fast is it moving at that instant (2 seconds)?
Position, Velocity & Acceleration
A crab is crawling along the edge of your desk.
Its location (in feet) at time t (in seconds) is given by
P (t ) = t 2 + t.
A) Where is the crab after 2 seconds?
P  2   2   2
2
P  2   6 feet
Position, Velocity & Acceleration
A crab is crawling along the edge of your desk.
Its location (in feet) at time t (in seconds) is given by
P (t ) = t 2 + t.
B) How fast is it moving at that instant (2 seconds)?
Velocity is the rate of change of position.
P t   t 2  t
V  t   P '  t   2t  1
Velocity function
P ' 2  2  2  1
P ' 2  5
feet per second
Position, Velocity & Acceleration
Example E:
A disgruntled calculus student
hurls his calculus book in the air.
Position, Velocity & Acceleration
The position of the
calculus book:
p  t   16t 2  96t
t is in seconds and
p(t) is in feet
A) What is the maximum height attained by the book?
B) At what time does the book hit the ground?
C) How fast is the book moving when it hits the ground?
Position, Velocity & Acceleration
A) What is the maximum height attained by the book?
The book attains its maximum height when its velocity is 0.
p  t   16t 2  96t
v  t   p  t   32t  96
0  32t  96
32t  96
t  3 seconds
Velocity function
p  3  16  3  96  3
2
p  3  144  288
p  3  144
feet
Position, Velocity & Acceleration
B) At what time does the book hit the ground?
The book hits the ground when its position is 0.
p  t   16t 2  96t
0  16t 2  96t
0  16t (t  6)
16t  0
t 6  0
t  0 sec.
t  6 sec.
Position, Velocity & Acceleration
C) How fast is the book moving when it hits the ground?
Good guess: 0 ft/sec
This is incorrect.
v  t   32t  96
v  6   32  6   96
v  6   192  96
v  6   192  96
v  6   96 ft/sec
Downward direction
Position, Velocity & Acceleration
Acceleration: the rate of change of velocity
with respect to time.
Velocity function
Acceleration function
v  t   32t  96
a  t   v  t   32
ft/sec2
How is the acceleration function related to the position function?
Acceleration is the second derivative of position.
a  t   p  t 
Position, Velocity & Acceleration
Example F:
A red sports car is traveling, and its position P (in miles) at time t (in hours)
is given by P (t ) = t 2 – 7t.
A) When is the car 30 miles from where it started?
B) What is the velocity at the very moment the car is 30 miles away?
C) What is the acceleration at the very moment the car is 30 miles away?
D) When does the car stop?
Position, Velocity & Acceleration
A red sports car is traveling, and its position P (in miles) at time t (in hours)
is given by P (t ) = t 2 – 7t.
A) When is the car 30 miles from where it started?
30  t 2  7t
0  t 2  7t  30
0   t  10  t  3
t  10  0
t 3 0
t  10 hours t  3
Position, Velocity & Acceleration
A red sports car is traveling, and its position P (in miles) at time t (in hours)
is given by P (t ) = t 2 – 7t.
B) What is the velocity at the very moment
the car is 30 miles away?
V  t   P '  t   2t  7
V  t   P '  t   2t  7
P ' 10   2 10   7
P ' 10   13 Miles per hour
Position, Velocity & Acceleration
A red sports car is traveling, and its position P (in miles) at time t (in hours)
is given by P (t ) = t 2 – 7t.
C) What is the acceleration at the very moment
the car is 30 miles away?
V  t   P '  t   2t  7
A  t   P ''  t   2
Miles per hour2
Position, Velocity & Acceleration
A red sports car is traveling, and its position P (in miles) at time t (in hours)
is given by P (t ) = t 2 – 7t.
D) When does the car stop?
V  t   P '  t   2t  7
0  2t  7
7  2t
t  3.5 hours
Conclusion
The height/distance of an object can be
given by a position function.
Velocity measures the rate of change of
position with respect to time.
The velocity function is found by taking the
derivative of the position function.
Conclusion
In order for an object traveling upward to obtain
maximum position, its instantaneous velocity must equal
0.
As an object hits the ground, its velocity is not 0, its
height is 0.
Acceleration measures the rate of change of velocity
with respect to time.
The acceleration function is found by taking the
derivative of the velocity function.