Higher order derivativesHigher order derivatives

Objective:Objective:

To be able to find higher order derivatives To be able to find higher order derivatives and use them to find velocity and and use them to find velocity and acceleration of objects.acceleration of objects.

TS: Explicitly assess information and draw TS: Explicitly assess information and draw conclusions.conclusions.

Do you remember your different notations Do you remember your different notations for derivatives?for derivatives?

'( )f x 'y dy

dx

Well these are the same notations for higher Well these are the same notations for higher power derivatives! Any guesses on what each power derivatives! Any guesses on what each means?means?

the third derivative

''( )f x

'''y

2

2

d y

dxsecthe ond derivative

secthe ond derivative of f

And to find them you just take the And to find them you just take the derivative again...and again…if necessary!derivative again...and again…if necessary!

For example to get from f’’(x) to f’’’(x) you For example to get from f’’(x) to f’’’(x) you just take the derivative of f’’(x).just take the derivative of f’’(x).

And to get from f’(x) to fAnd to get from f’(x) to f(4)(4)(x) you would just (x) you would just take the derivative of f’(x) three times.take the derivative of f’(x) three times.

Example AExample A

Find the second derivative of f(x) = xFind the second derivative of f(x) = x44 – 2x – 2x33

'( )f x

212x 12x''( )f x

26x34x

Example BExample B(4)'''( ) 2 1 ( )Given f x x find f x

12'''( ) 2( 1)f x x

1/22(1/ 2)( )1x (1)

(4) 1/2( 1)f x

(4) ( )f x

Example CExample C3( ) 3 9 1, ''( ) 0Given g x x x solve the following equation g x

'( )g x 29x 9

''( )g x 18x

''( ) 0 18 0We want g x so x

0x

Position, Velocity & AccelerationPosition, Velocity & Acceleration

Velocity is the rate of change of position with respect to time.

Acceleration is the rate of change of velocitywith respect to time.

DVelocity

T

VAcceleration

T

Position, Velocity & AccelerationPosition, Velocity & Acceleration

When you’re driving your car…

Warning: Professional driver, do not attempt!

Position, Velocity & AccelerationPosition, Velocity & Acceleration

squeeeeek!

…and you jam on the brakes…

Position, Velocity & AccelerationPosition, Velocity & Acceleration

…and you feel the car slowing down…

Position, Velocity & AccelerationPosition, Velocity & Acceleration

…what you are really feeling…

Position, Velocity & AccelerationPosition, Velocity & Acceleration

…is actually acceleration.

Position, Velocity & AccelerationPosition, Velocity & Acceleration

I felt that acceleration.

Position, Velocity & AccelerationPosition, Velocity & Acceleration

A) Where is the crab after 2 seconds?

B) How fast is it moving at that instant (2 seconds)?

Example D: A crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given byP (t ) = t 2 + t.

Position, Velocity & AccelerationPosition, Velocity & AccelerationA crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given byP (t ) = t 2 + t.

A) Where is the crab after 2 seconds?

22 2 2P

2 6P feet

Position, Velocity & AccelerationPosition, Velocity & AccelerationA crab is crawling along the edge of your desk. Its location (in feet) at time t (in seconds) is given byP (t ) = t 2 + t.

2 P t t t

' V t P t

feet per second

B) How fast is it moving at that instant (2 seconds)?

2 1t

' 2 2 2 1 P

' 2 5P

Velocity function

Velocity is the rate of change of position.

Position, Velocity & AccelerationPosition, Velocity & AccelerationExample E:A disgruntled calculus student

hurls his calculus book in the air.

Position, Velocity & AccelerationPosition, Velocity & AccelerationThe position of the calculus book:

216 96p t t t 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 & AccelerationPosition, Velocity & AccelerationA) What is the maximum height attained by the book?

216 96p t t t

v t p t 32 96t

Velocity function

0 32 96t 32 96t

3t seconds

23 16 3 96 3p

3 144 288p

3 144p feet

The book attains its maximum height when its velocity is 0.

Position, Velocity & AccelerationPosition, Velocity & AccelerationB) At what time does the book hit the ground?

The book hits the ground when its position is 0.

216 96p t t t 20 16 96t t

0 16 ( 6)t t

16 0t 6 0t 0t 6t sec. sec.

Position, Velocity & AccelerationPosition, Velocity & AccelerationC) How fast is the book moving when it hits the ground?

Good guess: 0 ft/sec This is incorrect.

32 96v t t

6 32 6 96v

6 192 96v

6 192 96v

6 96v ft/sec

Downward direction

Position, Velocity & AccelerationPosition, Velocity & Accelerationthe rate of change of velocity with respect to time.

Acceleration:

32 96v t t

32a t v t ft/sec2

How is the acceleration function related to the position function?

Velocity function

Acceleration function

Acceleration is the second derivative of position.

a t p t

Position, Velocity & AccelerationPosition, Velocity & AccelerationExample 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?

D) When does the car stop?

C) What is the acceleration at the very moment the car is 30 miles away?

Position, Velocity & AccelerationPosition, Velocity & AccelerationA 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?

230 7 t t

20 7 30 t t

0 10 3 t t

10 0 t 3 0 t

10t 3thours

Position, Velocity & AccelerationPosition, Velocity & AccelerationA 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?

' 2 7 V t P t t

' 2 7 V t P t t

' 10 2 10 7 P

' 10 13P Miles per hour

Position, Velocity & AccelerationPosition, Velocity & AccelerationA 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?

' 2 7 V t P t t

'' 2 A t P t Miles per hour2

Position, Velocity & AccelerationPosition, Velocity & AccelerationA 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?

' 2 7 V t P t t

0 2 7 t

7 2 t

3.5t hours

ConclusionConclusion

The height/distance of an object can be The height/distance of an object can be given by a position function.given by a position function.

Velocity measures the rate of change of Velocity measures the rate of change of position with respect to time.position with respect to time.

The velocity function is found by taking the The velocity function is found by taking the derivative of the position function.derivative of the position function.

ConclusionConclusion

In order for an object traveling upward to obtain In order for an object traveling upward to obtain maximum position, its instantaneous velocity must equal maximum position, its instantaneous velocity must equal 0.0.

As an object hits the ground, its velocity is As an object hits the ground, its velocity is notnot 0, its 0, its height is 0.height is 0.

Acceleration measures the rate of change of velocity Acceleration measures the rate of change of velocity with respect to time.with respect to time.

The acceleration function is found by taking the The acceleration function is found by taking the derivative of the velocity function.derivative of the velocity function.