1. find the instantaneous velocity for the following 1. find the instantaneous velocity for the...

1. Find the instantaneous velocity for the following 1. Find the instantaneous velocity for the following time—distance function at time = 2 seconds . time—distance function at time = 2 seconds .

2)2(2)( tts

Quiz 10-1Quiz 10-1

42)( 2 xxxf

2. Find the function that represents the slope at any 2. Find the function that represents the slope at any location on the following function:. location on the following function:.

h

xfhxfxf h

)()(lim)( 0

'

2

)2()(lim)( 2

'

t

ststs t

HOMEWORKHOMEWORK

Section 10-2 Section 10-2

(page 810)(page 810)

(evens) 2-8, 12, 14, 18,(evens) 2-8, 12, 14, 18,

22-42 even, 5022-42 even, 50

(18 problems)(18 problems)

10-2Limits and Motion

The Area problem.

What you’ll learn about• Computing Distance Traveled (Constant Velocity)• Computing Distance Traveled (Changing Velocity)• Limits at Infinity• The Connection to Areas• The Definite Integral

… and whyLike the tangent line problem, the area problem has

applications throughout science, engineering, economics and historicy.

Computing Distance Traveled

A car travels at an average rate of 56 miles per hour for 3 hours. How far does the car travel?

trD * Distance traveled = rate (speed) * timeDistance traveled = rate (speed) * time

1

5*

60 hr

hr

miD milesD .300

tt

ss

*

OROR: using the definition of average velocity from section 10-1:: using the definition of average velocity from section 10-1:

mileshrshr

miless .300.5*

.1

60

The Velocity as a function of time plot:

0 1 2 3 4 5 0 1 2 3 4 5 Time (hrs)Time (hrs)

mphtV 60)(

60605050404030302020101000

VVEELLOOCCIITTYY

trD *

Distance traveledDistance traveled is the is the area underarea under the v(t) functionthe v(t) function

for a specific timefor a specific time interval !!interval !!

(constant velocity)(constant velocity)

3005*60 D

Area = 300 milesArea = 300 miles

What if the velocity is NOT constant (velocity is changing):

0 1 2 3 4 5 0 1 2 3 4 5 Time (hrs)Time (hrs)

ttV 12)(

60605050404030302020101000

VVEELLOOCCIITTYY

Distance traveled is the Distance traveled is the area area under the v(t) functionunder the v(t) function for a for a

specific time interval.specific time interval.

This is still pretty easy: This is still pretty easy:

bhA 21

milesA 150)60)(5(21

Velocity is NOT constant

0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

44)5(7.0)( 2 xtV

VVEELLOOCCIITTYY

Ft/secFt/sec

Harder: Harder: How doHow do you find the areayou find the area

under a continuouslyunder a continuously changing curve?changing curve?

5050

4040

3030

2020

1010

00

Why not Why not addadd the the areas of a groupareas of a group

of rectangles (of rectangles (upperupper rightright corner of rectangle corner of rectangle

Is on the curve)? Is on the curve)?

Right Rectangular Approximation Method (Right Rectangular Approximation Method (RRAMRRAM))


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

44)5(7.0)( 2 xtV

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

54321 rectrectrectrectrectarea xfxfxfxfxfarea )5()4()3()2()1(

n

abx

1

5

05

x

1*441*431*411*381*33 area

ftdist

area

199

199

Not accurateNot accurate enough!enough!

(over-estimate)(over-estimate)


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

44)5(7.0)( 2 xtV

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

Not accurate Not accurate enough—enough—underunder

estimates the area.estimates the area.

Left Rectangular Approximation Method (Left Rectangular Approximation Method (LRAMLRAM))


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

44)5(7.0)( 2 xtV

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

Why not make theWhy not make therectangles smaller? rectangles smaller?

BetterBetter, but not , but not accurate enough.accurate enough.

10

1

)(i

i xxfarea

n

abx

5.010

05 x

10

1

)5.0)((i

ixfarea


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

44)5(7.0)( 2 xtV

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

Why not make theWhy not make therectangles even rectangles even smallersmaller? ?

BetterBetter, but , but stillstill not not accurate enough.accurate enough.

20

1

)(i

i xxfarea

n

abx

25.020

05

x

10

1

)25.0)((i

ixfarea


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

Why not make theWhy not make therectangles even rectangles even smallersmaller? ?

44)5(7.0)( 2 xtV

40

1

)(i

i xxfarea

n

abx

125.040

05

x

10

1

)125.0)((i

ixfarea

Even betterEven better, but , but still not accurate still not accurate

enough.enough.


0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

VVEELLOOCCIITTYY

5050

4040

3030

2020

1010

00

Why not make theWhy not make therectangles rectangles infinitesmally wideinfinitesmally wide? ?

Now you’re talkin’!Now you’re talkin’!

44)5(7.0)( 2 xtV

n

iin xxfarea

1

)(lim

Section 9-5 (We didn’t have time to cover this)

Series: The sum of a sequence of numbers.

Section 9-4 Section 9-4 SequencesSequences: A : A listlist of numbers of numbers

ka Where (and k = 1,2,3,…) Where (and k = 1,2,3,…) 12 kak

5

1kka

ka = 2, 5, 10, 17, 26= 2, 5, 10, 17, 26

= 2 + 5 + 10 + 17 + 26= 2 + 5 + 10 + 17 + 26 = 65= 65

““The The summationsummation of ‘ of ‘a’ sub ‘ka’ sub ‘k’ for ‘’ for ‘k’ = 1 to 5k’ = 1 to 5””

We’re going to use this idea of a summation to find out the exact area under the curve.

0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

VVEELLOOCCIITTYY

4040

3030

2020

1010

00

We will make the width of We will make the width of the rectangle the rectangle infinitesimallyinfinitesimally

small.small.

We’ll call that We’ll call that InfinitesimallyInfinitesimally small width: small width: x

The The heightheight of the of the rectangle is just therectangle is just the output value of the output value of the function. function. )(xf

The The distance traveleddistance traveled (area under the curve)(area under the curve) is the sum of all of the is the sum of all of the small rectangles.small rectangles.

Width: (Width: (infinitesimallyinfinitesimally small) small)

We’re going to use this idea of a summation to find out the exact area under the curve.

0 1 2 3 4 5 0 1 2 3 4 5 Time (sec)Time (sec)

VVEELLOOCCIITTYY

4040

3030

2020

1010

00

x

)( ixf

The The velocityvelocity (area under the curve) (area under the curve) is the sum of is the sum of all of the small rectangles.all of the small rectangles.

n

iin xxfarea

1

)(lim

Height: (Height: (function valuefunction value for for the left upper corner ofthe left upper corner ofEach of these slivers (thereEach of these slivers (there are an infinite # of them) are an infinite # of them)

n

abx

Where “b” and “a” Where “b” and “a” are the right and leftare the right and left ends of the interval.ends of the interval.

Definite Integral

1

1

Let be a function on [ , ] and let ( ) be defined as above.

The definite integral of over [ , ], denoted ( ) , is given by

( ) lim ( ) , provided the limit exists.

If t

n

ii

b

a

b a

in ia

f a b f x x

f a b f x dx

f x dx f x x

he limit exists, we say is integrable on [ , ].f a b

n

iin xxfarea

1

)(lim

n

iin xxfareadxxf

1

5

0

)(lim..)(

Limits at Infinity (Informal)

When we write "lim ( ) ," we mean that ( ) gets arbitrarily close

to as gets arbitrarily large.xf x L f x

L x

Calculating the Integral of a function (gives the area under the curve for any specified

interval)

Find: (the area under the function for the Find: (the area under the function for the interval x = [1,5]interval x = [1,5]

5

12xdx

n

iin xxfdxxf

1

5

1

)(lim)(

n

ii xxapprox

1

)1)(54321(22 30

n

abx

b = largest input valueb = largest input value

a = smalles input valuea = smalles input value

14

15

x

n = # of rectangles (bigger is more accurate) n = # of rectangles (bigger is more accurate)

Calculating the Integral of a function (gives the area under the curve for any specified

interval)


5

12xdx

n

iin xxfdxxf

1

5

1

)(lim)(

n

ii xxapprox

1

)5.0)(55.445.335.225.115.0(22 5.27

n

abx

b = largest input valueb = largest input value

a = smalles input valuea = smalles input value

5.08

15

x

n = # of rectangles (bigger is more accurate) n = # of rectangles (bigger is more accurate)

nasarea .. Where does the infinite series “converge”?Where does the infinite series “converge”?

Calculating the Integral of a function


5

12xdx

n

iin xxfdxxf

1

5

1

)(lim)(1010

55

5511

Area Area underunder the line (in the line (in the interval x = [1,5]the interval x = [1,5]

f(x) = 2xf(x) = 2x

Area = area large triangle – area small triangleArea = area large triangle – area small triangle

ssLLunder hbhbarea2

1

2

1

)2)(1(2

1)10)(5(

2

1underarea

24underarea

1. find the instantaneous velocity for the following 1. find the instantaneous velocity for the...

Documents