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Transcript of Dr. Hugh Blanton ENTC 3331. Dr. Blanton - ENTC 3331 - Math Review 2.
Dr. Hugh Blanton
ENTC 3331
Dr. Blanton - ENTC 3331 - Math Review 2
Dr. Blanton - ENTC 3331 - Math Review 3
Measurement UnitsMeasurement Units
• The System of International Units (SI units) was adopted in 1960.• The use of older systems still persists,
but it is always possible to convert non-standard measurements to SI units.
Dr. Blanton - ENTC 3331 - Math Review 4
SI (International Standard) Base UnitsSI (International Standard) Base Units
• meter (m) = about a yard• kilogram (kg) = about 2.2 lbs• liter (l) = about a quart• liter (l) = 1000 mL
Dr. Blanton - ENTC 3331 - Math Review 5
Fundamental UnitsFundamental Units
Physical Property Unit (abbreviation)
length meter (m)
mass kilogram (kg)
time second (s)
electric current ampere (A)
temperature kelvin (K)
number of atoms or molecules mol (mol)
light Intensity candela (cd)
Seven Fundamental physical phenomena.
Dr. Blanton - ENTC 3331 - Math Review 6
Unit ConversionsUnit Conversions
•When converting physical values between one system of units and another, it is useful to think of the conversion factor as a mathematical equation.
• In solving such equations, one must only multiply or divide both sides of the equation by the same factor to keep the equation consistent.
Dr. Blanton - ENTC 3331 - Math Review 7
ft 23ft 3
yd 1ft 23
Same
Quantityyd 3
23
ft 3
yd 1ft 23
Unit ConversionsUnit Conversions
•Example: 23 feet = ? yards
yd 7.6ft 3
yd 1ft 23
Dr. Blanton - ENTC 3331 - Math Review 8
2yd 5
Unit ConversionsUnit Conversions
•Example: 5 yd2 = ? ft2
yd 1
ft 3 yd 5 2
2
2
yd 1
ft 3 yd 5
2
22
yd 1
ft 9 yd 5 2
2
22 ft 45
yd 1
ft 9 yd 5
Dr. Blanton - ENTC 3331 - Math Review 9
UnitsUnits
• Fundamental Units• The SI system recognizes that there are
only a few truly fundamental physical properties that need basic (and arbitrary) units of measure, and that all other units can be derived from them.
Dr. Blanton - ENTC 3331 - Math Review 10
• Derived Units• The funadmental units are used as the
basis of numerous derived SI units.• Note that derived SI units are
sometimes named after famous physicists.
Dr. Blanton - ENTC 3331 - Math Review 11
Derived UnitsDerived UnitsMeasureme
ntUnit Unit Description
force newton (N) Force required to accelerate a mass of 1 kg at 1 m/s2
pressure pascal (Pa) Pressure that exerts a force of 1 newton per m2 of surface area
frequency hertz (Hz) Number of cycles of periodic activity per second
energy joule (J) Energy expended in moving a resistive force of 1 newton over 1 m.
power watt (W) Rate of energy expenditure of 1 joule per second.
electrical charge
coulomb (C)
Charge that passes a point in an electrical circuit if 1 ampere of current flows for 1 second.
electrical resistance
ohm Ratio of the voltage divided by the current in an electrical circuit.
Dr. Blanton - ENTC 3331 - Math Review 12
Unit Multiplication FactorsUnit Multiplication Factors
• An additional letter that denotes a multiplying factor may prefix fundamental or derived units.• The more common multiplying factors
increase or decrease the unit by powers of ten.
Dr. Blanton - ENTC 3331 - Math Review 13
Unit Multiplication FactorsUnit Multiplication Factors
• An additional letter that denotes a multiplying factor may prefix fundamental or derived units.• The more common
multiplying factors increase or decrease the unit by powers of ten.
tera (T) 1012
giga (G) 109
mega (M) 106
kilo (k) 103
hecto (h) 102
deca (da) 10
deci (d) 10-1
centi (c) 10-2
milli (m) 10-3
micro 10-6
nano (n) 10-9
pico (p) 10-12
femto 10-15
Dr. Blanton - ENTC 3331 - Math Review 14
Powers of Ten (big)Powers of Ten (big)
•101 = 10•103 = 1000 (thousand)•106 = 1,000,000 (million)•109 = 1,000,000,000 (billion)
Dr. Blanton - ENTC 3331 - Math Review 15
Powers of Ten (small)Powers of Ten (small)
•100 = 1•10-3 = 0.001 (thousandth)•10-6 = 0.000001 (millionth)•10-9 = 0.000000001 (billionth)
Dr. Blanton - ENTC 3331 - Math Review 16
Scientific NotationScientific Notation
• 7,000,000,000
• = 7 billion
• = 7 109
• 7,000,000
• = 7 million
• = 7 106
Dr. Blanton - ENTC 3331 - Math Review 17
Scientific NotationScientific Notation
• 7,240,000
• = 7.24 million
• = 7.24 106
3 significant digits
Dr. Blanton - ENTC 3331 - Math Review 18
Very Large QuantitiesVery Large Quantities
• 7,240,000 = 7.24 106
6 decimal places
Dr. Blanton - ENTC 3331 - Math Review 19
Very Small QuantitiesVery Small Quantities
• 0.0000123 = 1.23 10-5
5 decimal places
Dr. Blanton - ENTC 3331 - Math Review 20
Engineering NotationEngineering Notation
• Exponents = 3, 6, 9, 12, . . .• Instead of 5.32 107
• we write• 53.2 106
• Decimal part got bigger
Exponent got
smaller
Dr. Blanton - ENTC 3331 - Math Review 21
Adding and SubtractingAdding and Subtracting
•Exponents must be the same!•(1.2 106) + (2.3 105)
•change to•(1.2 106) + (0.23 106)
•= 1.43 106
Dr. Blanton - ENTC 3331 - Math Review 22
MultiplyingMultiplying
•Exponents Add•(3.1 106)(2.0 102)
•= 6.2 108
Dr. Blanton - ENTC 3331 - Math Review 23
DividingDividing
•Exponents Subtract
•(3.8 106)•(2.0 102)
•= 1.9 104
Dr. Blanton - ENTC 3331 - Math Review 24
1
5
2
5
Adding FractionsAdding Fractions
•You can only add like to like• Same Denominators
1
5
2
5
3
5
Dr. Blanton - ENTC 3331 - Math Review 25
Different DenominatorsDifferent Denominators
•Make them the same• find a common denominator
•The product of all denominators is always a common denominator• But not always the least common denominator
Dr. Blanton - ENTC 3331 - Math Review 26
Finding the LCDFinding the LCD
•Example:
1
12
4
15
Dr. Blanton - ENTC 3331 - Math Review 27
Factor the DenominatorsFactor the Denominators
15 3 5 12 2 2 3
Dr. Blanton - ENTC 3331 - Math Review 28
Assemble LCDAssemble LCD
15 3 5 12 2 2 3
2 2 3 5 60
Dr. Blanton - ENTC 3331 - Math Review 29
Build up Denominators to LCDBuild up Denominators to LCD
1
12
4
15
×5
×5
×4
×41
12
4
15
5
60
16
60
Dr. Blanton - ENTC 3331 - Math Review 30
Add NumeratorsAdd Numerators
5
60
16
60
5
60
16
60
21
60
5
60
16
60
21
60
7
20
And Reduce if Needed
Dr. Blanton - ENTC 3331 - Math Review 31
Rational ExpressionsRational Expressions
•Example:
x
x
x
x x
1
1
2
2 12 2
Dr. Blanton - ENTC 3331 - Math Review 32
Factor the DenominatorsFactor the Denominators
x x x2 1 1 1 ( )( )
x xx x
2 2 11 1
( )( )
Dr. Blanton - ENTC 3331 - Math Review 33
Assemble LCDAssemble LCD
( )( )x x 1 1
( )( )x x 1 1
( )( )( )x x x 1 1 1
DE
NO
MIN
AT
OR
S
Dr. Blanton - ENTC 3331 - Math Review 34
Build up Fractions to LCDBuild up Fractions to LCD
x
x x
x
x x
1
1 1
2
1 1( )( ) ( )( )
LCD x x x ( )( )( )1 1 1
x ( )1
x ( )1)( x( )1
x( )1FACTORED
Dr. Blanton - ENTC 3331 - Math Review 35
Add NumeratorsAdd Numerators
( )( ) ( )
( )( )( )
x x x x
x x x
1 1 2 1
1 1 1
Dr. Blanton - ENTC 3331 - Math Review 36
x x x x
x x x
2 22 1 2 2
1 1 1
( )( )( )
Simplify NumeratorSimplify Numerator
3 1
1 1 1
2x
x x x
( )( )( )
( )( ) ( )
( )( )( )
x x x x
x x x
1 1 2 1
1 1 1
Dr. Blanton - ENTC 3331 - Math Review 37
RadicalsRadicals
xRadicand
Radical
n
Index
Dr. Blanton - ENTC 3331 - Math Review 38
MeaningMeaning
x y
y x
n
n
if and only if
Dr. Blanton - ENTC 3331 - Math Review 39
ExampleExample
8 2
2 8
3
3
because
Dr. Blanton - ENTC 3331 - Math Review 40
An AmbiguityAn Ambiguity
25 5
5 252
because•but it’s also true that. . .
Dr. Blanton - ENTC 3331 - Math Review 41
It’s also true thatIt’s also true that
( ) 5 252
•So why not say
25 5 •?
Dr. Blanton - ENTC 3331 - Math Review 42
Two Answers?Two Answers?
•Roots with an even index always have both a positive and a negative root
•Because squaring either a negative or a positive gives the same result
Dr. Blanton - ENTC 3331 - Math Review 43
Principal RootPrincipal Root
•To avoid confusion we define the principal root to be the positive root, so:
25 5 5 (not )
Dr. Blanton - ENTC 3331 - Math Review 44
The Negative RootThe Negative Root
•If we want the negative root we use a minus sign:
25 5
Dr. Blanton - ENTC 3331 - Math Review 45
Negative RadicandsNegative Radicands
•Do Not Confuse
25•With 25 •!!!
25 •Does not exist
Dr. Blanton - ENTC 3331 - Math Review 46
Negative RadicandsNegative Radicands
•You cannot take an even root of a negative number
•Because you cannot square any number and get a negative result
Dr. Blanton - ENTC 3331 - Math Review 47
Odd Roots of Negative RadicandsOdd Roots of Negative Radicands
•You can take odd roots of negative numbers:
8 2
2 2 2 8
3 because
( )( )( )
Dr. Blanton - ENTC 3331 - Math Review 48
Some Square Root IdentitiesSome Square Root Identities
x x2
x x2
•for all non-negative x
•for all non-negative x
•for all x
xx 2
Dr. Blanton - ENTC 3331 - Math Review 49
A Common ErrorA Common Error
a b a b •for example, you cannot say
3 4 72 2 (WRONG!)•What is the correct result?
Dr. Blanton - ENTC 3331 - Math Review 50
First Evaluate InsideFirst Evaluate Inside
3 4
9 16
25
5
2 2
Dr. Blanton - ENTC 3331 - Math Review 51
ProductsProducts
•You can “split up” a radical when it contains a product (not a sum!):
ab a b•(as long as a and b are non-negative)
Dr. Blanton - ENTC 3331 - Math Review 52
ExampleExample
400 16 25
16 25
4 5 20
Dr. Blanton - ENTC 3331 - Math Review 53
Perfect SquaresPerfect Squares
•Perfect squares are numbers that have whole number square roots: 4, 9, 16, 25, 36, 49, 64, etc.
•All other numbers have irrational roots
Dr. Blanton - ENTC 3331 - Math Review 54
NumbersNumbers
• Natural Numbers: 1, 2, 3, . . .
• Whole Numbers: 0, 1, 2, 3, . . .
• Integers: . . . , -2, -1, 0, 1, 2, . . .
Dr. Blanton - ENTC 3331 - Math Review 55
NumbersNumbers
• Rational Numbers
• a/b (a,b integers, b not zero)
• Irrational Numbers Cannot be a ratio of integers Decimals never repeat or end. (decimals of rationals do)
Dr. Blanton - ENTC 3331 - Math Review 56
Rational and IrrationalRational and Irrational
45454545.011
5
75.04
3
41421356.12
Rational(Terminates)
Rational(Repeats)
Irrational
Dr. Blanton - ENTC 3331 - Math Review 57
NumbersNumbers
• Real Numbers Rationals + Irrationals All points on number line All signed distances
The Number Line
Dr. Blanton - ENTC 3331 - Math Review 58
Imaginary NumbersImaginary Numbers
•Square root of a negative number•We Define:
1 i Math, Physics
1 j Engineering, Electronics
Dr. Blanton - ENTC 3331 - Math Review 59
Properties of jProperties of j
2 1j By Definition
3j j Because j 3= j 2j = (-1)j
4 1j Because j 4= j 2j 2 = (-1)(-1)
5j j Because j 5= j 4j = (1)j
Dr. Blanton - ENTC 3331 - Math Review 60
Expressing Square Roots of Negative NumbersExpressing Square Roots of Negative Numbers
4 ( 1)4
4 1 4
4 2 2j j
Dr. Blanton - ENTC 3331 - Math Review 61
Expressing Square Roots of Negative NumbersExpressing Square Roots of Negative Numbers
3 ( 1)3
3 1 3
3 3j
Dr. Blanton - ENTC 3331 - Math Review 62
Complex NumbersComplex Numbers
6 2 j
•Real Part + Imaginary Part•Example:
Real Part = 6
Dr. Blanton - ENTC 3331 - Math Review 63
Complex NumbersComplex Numbers
6 2 j
•Real Part + Imaginary Part•Example:
Imaginary Part = 2
Dr. Blanton - ENTC 3331 - Math Review 64
Adding and Subtracting Complex NumbersAdding and Subtracting Complex Numbers
•Likes stay with likes• Re + Re = Re• Im + Im = Im
•Just collecting like terms
Dr. Blanton - ENTC 3331 - Math Review 65
Adding and Subtracting Complex NumbersAdding and Subtracting Complex Numbers
•Example:
(6 2 ) (2 3 )j j
8 j
Dr. Blanton - ENTC 3331 - Math Review 66
Adding and Subtracting Complex NumbersAdding and Subtracting Complex Numbers
•Example:
(6 2 ) (2 3 )j j
8 j
Dr. Blanton - ENTC 3331 - Math Review 67
Adding and Subtracting Complex NumbersAdding and Subtracting Complex Numbers
•Example:
(6 2 ) (2 3 )j j
8 j
Dr. Blanton - ENTC 3331 - Math Review 68
MultiplyingMultiplying
•Remember that j 2 = -1
Dr. Blanton - ENTC 3331 - Math Review 69
MultiplyingMultiplying
(6 2 )(2 3 )j j 212 18 4 6j j j
)1(61412 j
j1418
Dr. Blanton - ENTC 3331 - Math Review 70
DividingDividing
•Complex Conjugate• Reverse sign of imaginary part
6 2 jConjugate of
6 2 jis
Dr. Blanton - ENTC 3331 - Math Review 71
DividingDividing
• Write as fraction• Multiply numerator and denominator by
the complex conjugate of denominator• Multiply and simplify
Dr. Blanton - ENTC 3331 - Math Review 72
DividingDividing
(6 2 ) (2 3 )j j
(6 2 )
(2 3 )
j
j
(2 3 )
(2 3 )
j
j
Dr. Blanton - ENTC 3331 - Math Review 73
DividingDividing
(6 2 )
(2 3 )
j
j
(2 3 )
(2 3 )
j
j
2
2
12 18 4 6
4 6 6 9
j j j
j j j
Dr. Blanton - ENTC 3331 - Math Review 74
DividingDividing
2
2
12 18 4 6
4 6 6 9
j j j
j j j
12 22 6
4 9
j
Dr. Blanton - ENTC 3331 - Math Review 75
DividingDividing
12 22 6
4 9
j
6 22 6 22
13 13 13
jj
Dr. Blanton - ENTC 3331 - Math Review 76
Graphing Complex NumbersGraphing Complex Numbers
•Real part is x-coordinate
•Im. part is y-coordinate
Dr. Blanton - ENTC 3331 - Math Review 77
Graphing Complex NumbersGraphing Complex Numbers
•Example: 3 + 2j (3, 2)
Re
Im
Dr. Blanton - ENTC 3331 - Math Review 78
Polar FormPolar Form
•Example: 3 + 2j 3.633.4°
Re
Im
r
Dr. Blanton - ENTC 3331 - Math Review 79
Polar FormPolar Form
•Re + j Im rrej
2 2Re Imr
1 Imtan
Re
Re cosr
Im sinr
Dr. Blanton - ENTC 3331 - Math Review 80
Trigonometric FormTrigonometric Form
•r (cos + j sin )•Start with Re + j Im
•Substitute•Re = r cos •Im = r sin
Dr. Blanton - ENTC 3331 - Math Review 81
Trigonometric FormTrigonometric Form
•Start with Re + j Im
•Substitute
•r cos + j r sin
•r (cos + j sin )
Dr. Blanton - ENTC 3331 - Math Review 82
sincos je j
Euler’s Identity
sincos jrre j
Dr. Blanton - ENTC 3331 - Math Review 83
Complex ArithmeticComplex Arithmetic
•Addition & Subtraction• Easiest in rectangular form
•Multiplication & Division• Easiest in polar form
Dr. Blanton - ENTC 3331 - Math Review 84
Multiplication in Polar FormMultiplication in Polar Form
•(r11) (r22)
•= r1r2 (1+2)
Dr. Blanton - ENTC 3331 - Math Review 85
Division in Polar FormDivision in Polar Form
•(r11) / (r22)
•= r1 / r2 (1-2)
Dr. Blanton - ENTC 3331 - Math Review 86
Vectors & ScalersVectors & Scalers
• There is a fundamental distinction between two types of quantity:• Scalers and• Vectors
• Scalers possess a magnitude, whereas vectors have both magnitude and direction.
• Properties such as mass and temperature clearly have no directionality and are examples of scalers.
• A complete description of force would be impossible without specifying both the magnitude and direction of the quantity.
Dr. Blanton - ENTC 3331 - Math Review 87
VectorsVectors
•Represent magnitude and direction•Example: Displacement
• “go 2 miles East”
Dr. Blanton - ENTC 3331 - Math Review 88
Vector QuantitiesVector Quantities
•Force•Velocity•Magnetic Field
Dr. Blanton - ENTC 3331 - Math Review 89
Vector NotationVector Notation
•Vector: Bold or arrow over
•Scalar: Italic, no arrow
F
F
Dr. Blanton - ENTC 3331 - Math Review 90
Numerical DescriptionNumerical Description
•Polar Form• Magnitude and angle
•Rectangular Form• x- and y-components
A vector can be represented in:
Dr. Blanton - ENTC 3331 - Math Review 91
Polar FormPolar Form
Mag
nitud
e
Angle
V
V = (r, )V =(53, 65°)
V = rV = 5365°
Dr. Blanton - ENTC 3331 - Math Review 92
Rectangular FormRectangular Form
V
V
Vx
Vy
Vx=V cos
Vy=V sin
Dr. Blanton - ENTC 3331 - Math Review 93
Rectangular to PolarRectangular to Polar
V
V
Vx
Vy
2 2x yV V V
1tan y
x
V
V
Dr. Blanton - ENTC 3331 - Math Review 94
Vector AdditionVector Addition
•Resultant vector•Not the sum of the magnitudes•Vectors add head-to-tail
Dr. Blanton - ENTC 3331 - Math Review 95
Vector Addition ExampleVector Addition Example
•Go 3 miles East,•then 4 Miles North
3
4
R
R = 5 miles at 53°
Dr. Blanton - ENTC 3331 - Math Review 96
Adding Nonperpendicular VectorsAdding Nonperpendicular Vectors
•x-components add to givex-component of resultant•y-components add to givey-component of resultant
Dr. Blanton - ENTC 3331 - Math Review 97
Adding Nonperpendicular VectorsAdding Nonperpendicular Vectors
Rx = Ax + Bx Ry = Ay + By
R = A + B
A
BR
Dr. Blanton - ENTC 3331 - Math Review 98
Adding Nonperpendicular VectorsAdding Nonperpendicular Vectors
AB
R
Ax Bx
Ay
By
Rx
Ry
Dr. Blanton - ENTC 3331 - Math Review 99
Trigonometric FunctionsTrigonometric Functions
•Right Triangles Only!
hypotenuse
oppo
site
adjacent
Dr. Blanton - ENTC 3331 - Math Review 100
Trigonometric FunctionsTrigonometric Functions
hypotenuse
adja
cent
opposite
Dr. Blanton - ENTC 3331 - Math Review 101
Similar TrianglesSimilar Triangles
Same Angle
Dr. Blanton - ENTC 3331 - Math Review 102
Similar TrianglesSimilar Triangles
Same Ratiosof Sides
Dr. Blanton - ENTC 3331 - Math Review 103
Similar TrianglesSimilar Triangles
•Ratios of sides depend ONLY on •So the ratio is a function of
Dr. Blanton - ENTC 3331 - Math Review 104
Ratios of SidesRatios of Sides
•Six Possible
hypotenuse
oppo
site
adjacent
sin = opp hyp
cos = adj hyp
tan = opp adj
Dr. Blanton - ENTC 3331 - Math Review 105
Ratios of SidesRatios of Sides
sin = opp hyp
cos = adj hyp
tan = opp adj
csc = hyp opp
sec = hyp adj
cot = adj opp
Dr. Blanton - ENTC 3331 - Math Review 106
The Main 3 Trig FunctionsThe Main 3 Trig Functions
sin = opp hyp
cos = adj hyp
tan = opp adj
S O H C A H T O A
Dr. Blanton - ENTC 3331 - Math Review 107
Solving TrianglesSolving Triangles
•Find all 3 sides and 3 angles•Need: 1 side plus 2 more items
• Only one more thing if it is given that one angle is 90°
Dr. Blanton - ENTC 3331 - Math Review 108
Right TrianglesRight Triangles
•Need 2 sides•OR
•1 side and 1 angle
Dr. Blanton - ENTC 3331 - Math Review 109
Tool KitTool Kit
•The Trig functions• (sin, cos, tan)
•The inverse Trig functions• (sin-1, cos -1, tan -1)
•The Pythagorean Theorem•Sum of angles is 180°
Dr. Blanton - ENTC 3331 - Math Review 110
The Trig FunctionsThe Trig Functions
•Find a side•Given 1 side and 1 angle
sinopp
hyp
cosadj
hyp
tanopp
adj
Dr. Blanton - ENTC 3331 - Math Review 111
The Inverse Trig FunctionsThe Inverse Trig Functions
•Find an angle•Given 2 sides
Dr. Blanton - ENTC 3331 - Math Review 112
The Pythagorean TheoremThe Pythagorean Theorem
•Find a side•Given 2 sides
Dr. Blanton - ENTC 3331 - Math Review 113
Angles add to 180°Angles add to 180°
•Find an angle•Given the other angle
Dr. Blanton - ENTC 3331 - Math Review 114
Vector MultiplicatonVector Multiplicaton
• Three types vector multiplication:• Simple multiplication• Dot Product
• Always yields a scaler answer.
• Cross Product• Always gives a vector result.
Dr. Blanton - ENTC 3331 - Math Review 115
Dot ProductDot Product
ABABBA cos
Dr. Blanton - ENTC 3331 - Math Review 116
x
z
Dr. Blanton - ENTC 3331 - Math Review 117
x
z
A
22332 222 A
22
ˆˆˆ 3z3y2x
A
A