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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?. - PowerPoint PPT Presentation

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?

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

And to find them you just take the derivative again...and againif 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 AFind the second derivative of f(x) = x4 2x3

Example B

Example C

Position, Velocity & AccelerationVelocity is the rate of change of position with respect to time.Acceleration is the rate of change of velocitywith respect to time.

Position, Velocity & AccelerationWhen youre driving your carWarning: Professional driver, do not attempt!

Position, Velocity & Accelerationsqueeeeek!and you jam on the brakes

Position, Velocity & Accelerationand you feel the car slowing down

Position, Velocity & Accelerationwhat you are really feeling

Position, Velocity & Accelerationis actually acceleration.

Position, Velocity & AccelerationI felt that acceleration.

Position, Velocity & AccelerationA) 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 & 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?feet

Position, 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.feet per secondB) How fast is it moving at that instant (2 seconds)?Velocity functionVelocity is the rate of change of position.

Position, Velocity & AccelerationExample E:A disgruntled calculus studenthurls his calculus book in the air.

Position, Velocity & AccelerationThe position of the calculus book:t is in seconds and p(t) is in feetA) 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 & AccelerationA) What is the maximum height attained by the book?Velocity functionsecondsfeetThe book attains its maximum height when its velocity is 0.

Position, Velocity & AccelerationB) At what time does the book hit the ground?The book hits the ground when its position is 0.sec.sec.

Position, Velocity & AccelerationC) How fast is the book moving when it hits the ground?Good guess: 0 ft/secThis is incorrect.ft/secDownward direction

Position, Velocity & Accelerationthe rate of change of velocity with respect to time.Acceleration:ft/sec2How is the acceleration function related to the position function?Velocity functionAcceleration functionAcceleration is the second derivative of position.

Position, 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 & 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?hours

Position, 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?Miles per hour

Position, 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?Miles per hour2

Position, 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?hours

ConclusionThe 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.

ConclusionIn 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.