mark lesmeister dawson high school physics. selected text and problems based on peggy bertrand,...

Calculus for AP Physics: Derivatives

Mark LesmeisterDawson High School Physics

Selected text and problems based on Peggy Bertrand, “1-D Kinematics”, http://prettygoodphysics.wikispaces.com/pgp-Kinematics

Selected problems from Richard White, “AP Physics Problem a Day”, retrieved from LearnAPPhysics.com ©2009

Acknowledgements

http://prettygoodphysics.wikispaces.com/pgp-Kinematics



The average velocity is the rate of change of displacement with respect to time.

The average velocity depends on the interval we choose to measure it.◦ If we choose a very small time interval, we can

get close to the instantaneous velocity.

Review: Average Velocity

if

ifAV tt

xx

t

xv

The instantaneous velocity at time t is the “average velocity” where our two endpoints are very close to the same time t, so that ∆t is very close to 0 .

This quantity is called the derivative of position with respect to time.

Instantaneous Velocity

t

xv

t

0

limdt

dxv

if

ifAV tt

xx

t

xv

Instantaneous Velocity: Example 1

25.1)( ttx

if

ifAV tt

tt

t

xv

22 5.15.1

)(5.1))((

5.1)(5.1 22

ifif

ifif

if

ifAV tt

tt

tttt

tt

tt

t

xv

If then

To find the instantaneous velocity, we make the time interval very small, i.e. make .

We have found the derivative of x with respect to t.

Instantaneous Velocity: Example 1

25.1 tx )(5.1 ifAV ttv

ttt if

ttttttv if 3)2(5.1)(5.1)(5.1

tdt

dxvtx 3 then 5.1 If 2

There are shortcuts for finding derivatives of certain functions.

The first shortcut we will use is called the

power rule:

Finding Derivatives: The Power Rule

. )(then

constant, a is where)( If

1

n

n

nCtdt

dxtv

CCttx

Calculate the instantaneous velocity as a function of time and at the specific time indicated for each of the following:

Practice Problems

s 4 3)(

s 3 5)(

s 3 4)(

s 2 3)(

2

4

tttx

ttx

tttx

tttx

To find the derivative of a polynomial, we just take the derivative of each term.

Derivative of a Polynomial

dt

dB

dt

dA

dt

dx

tBAx

of functions are B andA where

Find the derivatives of the following polynomials:

Derivative of a Polynomial

constants. are and , where)(

s 2 6523)(

002

21

00

23

avxattvxtx

tttttx

Problem from Peggy Bertrand

Find the instantaneous velocity function for each equation, then evaluate it at t = 2 sec.

Exercise 1:

23

3

4

23 .3

2 .2

5 .1

ttx

tx

tx

32

1

32 .5

1 Use:Hint

4 .4

ttx

ttt

x

Average acceleration is the rate of change of velocity with respect to time.

Just as with velocity, the average acceleration depends on the time interval chosen to calculate it.

Average Acceleration

t

vaAV

The instantaneous acceleration at time t is the average acceleration calculated over a very small interval, where our two endpoints are very close to time t, so that ∆t is very close to 0 .

This quantity is called the derivative of velocity with respect to time.

Instantaneous acceleration

t

va

t

0

limdt

dva

Instantaneous acceleration:Example 1

tdt

dxvtx 3 then 5.1 If 2

333*1 011 ttdt

dva

Since a is the derivative of a derivative, it is sometimes called a second derivative.

The “2” exponents do not refer to squares. They just mean to take the derivative twice.

Second derivative

dtdtdx

d

dt

dva

2

2

dt

xda

Sample problem 1: A particle travels from A to B following the function x(t) = 3.0 – 6t + 3t2.

a) What are the functions for velocity and acceleration as a function of time?

b) What is the instantaneous velocity at 6 seconds?

c) What is the initial velocity?

Problem from Peggy Bertrand.

Sample problem 2: A particle travels from A to B following the function x(t) = 2.0 – 4t + 3t2 – t3.

a) What are the functions for velocity and acceleration as a function of time?

b) What is the instantaneous acceleration at 6 seconds?


Sample problem 3: A particle follows the function

2

4.21.5 5x t

t

a) Find the velocity and acceleration functions.

b) Find the instantaneous velocity and acceleration at 2.0 seconds.


Calculate the acceleration function for each of the following, then find a at t=2.

Exercise 2

23

3

4

23 .3

2 .2

5 .1

ttx

tx

tx

32

1

32 .5

1 Use:Hint

4 .4

ttx

ttt

x

Problems from Peggy Bertrand

The average velocity is the slope.

Average Velocity from a Graph

t

x

A

B

x

tt

xvAV

Instantaneous Velocity from a Graph

t

x

Remember that the average velocity between the time at A and the time at B is the slope of the connecting line.

A

B


t

x

What happens if A and B become closer to each other?

A

B


t

x


AB

Instanteous Velocity from a Graph

t

x

A

B



t

x

A

B



t

x

A

B

The line “connecting” A and B is a tangent line to the curve. The velocity at that instant of time is the slope of this tangent line.

A and B are effectively the same point. The time difference is effectively zero.


t

x

A

B

The derivative function evaluated at a point gives the slope of the tangent line.

Average and Instantaneous Acceleration

t

v

Average acceleration is represented by the slope of a line connecting two points on a v/t graph.

Instantaneous acceleration is represented by the slope of a tangent to the curve on a v/t graph.

A

B

C

Derivatives Practice Question 1

If we know the velocity function, we can “work backwards” to find the position function.

For example, if the velocity is given by v=6t, then the position function must be of the form x = 3t2, since

Finding the position function

ttdt

dxv

tx

63*2

3

12

2

However, any function of the form will work, since the derivative of a constant is zero.

The function is called the antiderivative of , since it is the function whose derivative is the given function.

Finding the position function

Ctx 23

Ctx 23tv 6

The value of the constant C is determined by the conditions specified in the problems.◦ Usually the conditions given are the initial

conditions, that is, the position and velocity at t=0.

Power Rule for Antiderivatives

Cn

Atx

AnAtdt

dxv

n

n

1then

constant, a is and 1 , If

1

Use the antiderivative to find the position or velocity functions indicated. Apply the initial condition to find the value of the constant.

Practice Problems

2,1 ,3)( if )( and )( Find

2 , 6 if )( Find

3 , 2 if )( Find

00

02

0

vxttatxtv

xtvtx

xtvtx