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IntroductionIntroduction

Dr. Bruce McLean (call me Dr. Mac)Dr. Bruce McLean (call me Dr. Mac) Tallest, fattest, oldest, smartestTallest, fattest, oldest, smartest 2 children your age - 2 your parents’ 2 children your age - 2 your parents’

ageage Percussion & RacquetballPercussion & Racquetball University of Kentucky (1971)University of Kentucky (1971) Topologist – Abstract geometerTopologist – Abstract geometer Bowling Green State University (1965)Bowling Green State University (1965) Ohio Northern University (1963)Ohio Northern University (1963)

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VinculumVinculum

Vinculum - Binding USB TechnologiesVinculum - Binding USB TechnologiesVinculum is the brand name for the new family Vinculum is the brand name for the new family

of USB Host Controller ICs devices from of USB Host Controller ICs devices from FTDI.  . 

 The Vinculum family of devices allow the  The Vinculum family of devices allow the implementation of USB host controller implementation of USB host controller

functionality within products, thus saving functionality within products, thus saving development time and costs by using FTDI's development time and costs by using FTDI's tried and tested tried and tested firmware programmed into programmed into

internal, easily upgradeable Flash memory.   internal, easily upgradeable Flash memory.   

FTDI VNC1LFTDI VNC1LUSB Host Controller ICUSB Host Controller IC

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Add FirstAdd First

vin·cu·lumvin·cu·lum    vɪŋkyələm –    vɪŋkyələm – [[vingving-ky-kyuhuh-l-luhuhm]m]

––noun, plural noun, plural -la -la   -lə -   -lə -

1.a bond signifying union or unity; tie. 1.a bond signifying union or unity; tie.

2.2.MathematicsMathematics. a stroke or brace drawn . a stroke or brace drawn “over” a quantity consisting of several “over” a quantity consisting of several members or terms, as members or terms, as ,, in order to show in order to show that they are to be considered together. that they are to be considered together. [Origin: 1655–65; < L: fetter, equiv. to [Origin: 1655–65; < L: fetter, equiv. to vincvinc((īreīre) to bind + ) to bind + -ulum-ulum -ule] ]

_______

( )a b c a b c

3 9 16 3(5) 15

4 100 26

8 100 27

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vinculum=add vinculum=add firstfirst

n.   n.   pl.pl. vin·cu·lumsvin·cu·lums or or vin·cu·lavin·cu·la (-lə) (-lə)

MathematicsMathematics A bar drawn over two or more algebraic terms to A bar drawn over two or more algebraic terms to indicate that they are to be treated as a single term. indicate that they are to be treated as a single term.

AnatomyAnatomy A ligament that limits the movement of an organ or A ligament that limits the movement of an organ or part. part.

A bond or tie. A bond or tie. [Latin, [Latin, bond, tiebond, tie, from vincīre, , from vincīre, to tieto tie.].]

vin·cu·lumvin·cu·lum (vngky-lm) (vngky-lm)n.n. pl.pl. vin·cu·lumsvin·cu·lums or or vin·cu·lavin·cu·la (-l) (-l)

A uniting band or bandlike structure, such as a frenum or A uniting band or bandlike structure, such as a frenum or ligament.ligament.

The American Heritage® Stedman's Medical DictionaryThe American Heritage® Stedman's Medical DictionaryCopyright © 2002, 2001, 1995 by Houghton Mifflin Copyright © 2002, 2001, 1995 by Houghton Mifflin Company. Published by Houghton Mifflin Company.Company. Published by Houghton Mifflin Company.

4 100 26

8 100 27

3 9 16 3(5) 15

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Domain = HomeDomain = Home

All x’s or t’s that can be considered in All x’s or t’s that can be considered in a a

function f.function f.

Domain = [0, 120]Domain = [0, 120]

The set of all x such thatThe set of all x such that

{ | 0 120}x x

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Range of fRange of f

All f(x)’s or f(t)’s that can All f(x)’s or f(t)’s that can

be realized in a function, be realized in a function,

called f.called f.

Range = [0, 38] U (62, 100]Range = [0, 38] U (62, 100]

= = { | 0 38} { | 62 100}x x x x

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Domain = HomeDomain = Home

When working out to obtain cardio When working out to obtain cardio benefit, a student must increase benefit, a student must increase their heart rate to 85% of their their heart rate to 85% of their maximum heart rate.maximum heart rate.

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What must this student What must this student increase their heart rate to increase their heart rate to in order to obtain a cardio in order to obtain a cardio

benefit?benefit? A) 200A) 200 B) 185B) 185 C) 170C) 170 D) 160D) 160 E) 85E) 85

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If a student had a resting heart If a student had a resting heart rate of rate of 6565 beats per minute, beats per minute,

and works out for and works out for 2020 minutes, minutes, what is the domain of s?what is the domain of s?

A) [0, 40]A) [0, 40] B) [0, 20]B) [0, 20] C) [0, 65]C) [0, 65] D) [65, 200]D) [65, 200] E) [65, 170]E) [65, 170]

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If this student has a resting If this student has a resting heart rate of heart rate of 6565 beats per beats per

minute, and works out for minute, and works out for 2020 minutes, what is the range of minutes, what is the range of

s?s? A) [65, 185]A) [65, 185] B) [0, 20]B) [0, 20] C) [0, 65]C) [0, 65] D) [65, 200]D) [65, 200] E) [65, 170]E) [65, 170]

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Domain of Domain of ss::

What is the definition of maximum What is the definition of maximum heart rate?heart rate?

Can an individual reach their Can an individual reach their maximum heart rate as a limit?maximum heart rate as a limit?

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Domain and Domain and RangeRange of of ss: :

DomainDomain of s = of s = [0, end of work out] = [0, 30] in [0, end of work out] = [0, 30] in

minutesminutes RangeRange of s = of s = [Rest, 185] in beats/min[Rest, 185] in beats/min = [60, 185] in beats/min= [60, 185] in beats/min

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Definition - An individual’s Definition - An individual’s maximum heart rate is 220 – maximum heart rate is 220 –

age.age. If x is their age, g(x) =If x is their age, g(x) = g(x) = 220 – xg(x) = 220 – x What is the maximum What is the maximum

heart rate for a student heart rate for a student

that is 20 years old?that is 20 years old?

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g(x) = 220 – xg(x) = 220 – x g(20) = 220 – 20g(20) = 220 – 20 g(20) =220 – 20= 200 beats/min.g(20) =220 – 20= 200 beats/min.

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What is the maximum heart What is the maximum heart rate for a student that is rate for a student that is 18.518.5

years old?years old? g(g(xx) = 220 – ) = 220 – xx g(18.5) = 220 – 18.5g(18.5) = 220 – 18.5 g(18.5) =220 – 18.5= 201.5 g(18.5) =220 – 18.5= 201.5

beats/min.beats/min.

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What is the maximum heart What is the maximum heart rate for a rate for a 7070 year old? year old? g(g(xx) = ) =

220 – 220 – xx A) 200A) 200 B) 170B) 170 C) 150C) 150 D) 85D) 85 E) 70E) 70

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Suppose f is the cardio function Suppose f is the cardio function and g is the maximum heart and g is the maximum heart

rate function. Thus f(x) = .85 rate function. Thus f(x) = .85 g(x)g(x)

Domain of f = Domain of gDomain of f = Domain of g = {x | 0 x 105}= {x | 0 x 105} = [0, 105]= [0, 105]

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RangeRange

All y’s or f(x)’s or f(t)’s that can be All y’s or f(x)’s or f(t)’s that can be produced by a functionproduced by a function

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What is the range of f?What is the range of f?A) [97.75, 187]A) [97.75, 187]

B) [115, 220]B) [115, 220]

C) [97.75, 115]C) [97.75, 115]

D) [0, 97.75]D) [0, 97.75]

E) [0, 220]E) [0, 220]

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Range of fRange of f

Range is the set of cardio valuesRange is the set of cardio values

Range of f = [97.75, 187]Range of f = [97.75, 187]

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What is the range of g?What is the range of g?A) [97.75, 187]A) [97.75, 187]

B) [115, 220]B) [115, 220]

C) [97.75, 115]C) [97.75, 115]

D) [0, 97.75]D) [0, 97.75]

E) [0, 220]E) [0, 220]

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Range of gRange of g

g(x) = 220 – xg(x) = 220 – x Range is the set of Range is the set of

maximum valuesmaximum values Range of g = [115, 220]Range of g = [115, 220] What is the slope of f ?What is the slope of f ?

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SlopeSlope f(f(xx) = 0.85(220 – x) = 187 ) = 0.85(220 – x) = 187 – 0.85– 0.85 xx

SlopeSlope = =

SlopeSlope = = 97.75 1

5

87

10

(105) (

105

0)

0

f f

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What is the slope of g?What is the slope of g? g(g(xx) = 220 – x = 220 ) = 220 – x = 220 –1–1xx slope = slope =

A) 2A) 2

B) 1B) 1

C) -2C) -2

D) -1.5D) -1.5

E) -1E) -1

(105) (

105

0)

0

g g

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Domain AlgebraicallyDomain Algebraically

The denominator can not be zeroThe denominator can not be zero No negatives under the square rootNo negatives under the square root

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DomainDomain

R R RealsReals

(-oo, +oo)(-oo, +oo)

( ) 3( 1)( 1)( 3)x x xy f x

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RangeRange

R R RealsReals

(-oo, +oo)(-oo, +oo)

( ) 3( 1)( 1)( 3)x x xy f x

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DomainDomainR – {-2, 2} R – {-2, 2}

Reals except -2 and 2.Reals except -2 and 2.

(-oo, -2) U (-2, 2) U (2, +oo)(-oo, -2) U (-2, 2) U (2, +oo)

2

2

4( )

4

xy f x

x

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RangeRangeRemove (-1, 1] from the realsRemove (-1, 1] from the reals

R - (-1, 1].R - (-1, 1].

(-oo, -1] U (1, +oo)(-oo, -1] U (1, +oo)

2

2

4( )

4

xy f x

x

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DomainDomain

Reals greater than or equal to Reals greater than or equal to 4.4.

[4, +oo)[4, +oo)

( ) 4f x xy

0 4 4ox xr

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RangeRange

Nonnegative reals.Nonnegative reals.

[0, +oo)[0, +oo)

( ) 4f x xy

0 y

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DomainDomain

The denominator can not be zeroThe denominator can not be zero No negatives under the square rootNo negatives under the square root

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What is the domain of What is the domain of f(x) = 1/x? f(x) = 1/x?

A) R = realsA) R = reals

B) Z = integersB) Z = integers

C) nonnegative realsC) nonnegative reals

D) positive realsD) positive reals

E) all reals except x = 0E) all reals except x = 0

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DomainDomain

The denominator can not be zeroThe denominator can not be zero No negatives under the square rootNo negatives under the square root

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What is the domain of What is the domain of f(x) = ? f(x) = ?

A) R = realsA) R = reals

B) [-2, 2]B) [-2, 2]

C)C)

D) positive realsD) positive reals

E)E)

2 4x

( , 2) (2, )

( , 2] [2, )

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What is the domain ofWhat is the domain of

f(x) = ? f(x) = ?A) R = realsA) R = reals

B) [-2, 2]B) [-2, 2]

C)C)

D) positive realsD) positive reals

E)E)

2 4

2

x

x

( , 2] (2, )

( , 2] [2, )

04/20/2304/20/23 3737

CalculusCalculus

LimitsLimits DerivativesDerivatives IntegralsIntegrals