FUNDAMENTALS OF ENGINEERING
SUPPLIED-REFERENCE HANDBOOK
FIFTH EDITION
NATIONAL COUNCIL OF EXAMINERS
FOR ENGINEERING AND SURVEYING
2001 by the National Council of Examiners for Engineering and Surveying®.
N C E E S
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FUNDAMENTALS OF ENGINEERING
SUPPLIED-R EFERENCE H ANDBOOK
FIFTH E DITION
N A T I O N A L C O U N C I L O F E X A M I N E R S
F O R E N G I N E E R I N G A N D S U R V E Y I N G
2001 by the National Council of Examiners for Engineering and Surveying®^.
N C E E S
FUNDAMENTALS OF ENGINEERING
SUPPLIED-R EFERENCE H ANDBOOK
FIFTH E DITION
Prepared by National Council of Examiners for Engineering and Surveying ®^ (NCEES ®) 280 Seneca Creek Road P.O. Box 1686 Clemson, SC 29633- Telephone: (800) 250- Fax: (864) 654- www.ncees.org
iii
FOREWORD
During its August 1991 Annual Business Meeting, the National Council of Examiners for Engineering and Surveying (NCEES) voted to make the Fundamentals of Engineering (FE) examination an NCEES supplied-reference examination. Then during its August 1994 Annual Business Meeting, the NCEES voted to make the FE examination a discipline-specific examination. As a result of the 1994 vote, the FE examination was developed to test the lower-division subjects of a typical bachelor engineering degree program during the morning portion of the examination, and to test the upper-division subjects of a typical bachelor engineering degree program during the afternoon. The lower-division subjects refer to the first 90 semester credit hours (five semesters at 18 credit hours per semester) of engineering coursework. The upper-division subjects refer to the remainder of the engineering coursework.
Since engineers rely heavily on reference materials, the FE Supplied-Reference Handbook will be made available prior to the examination. The examinee may use this handbook while preparing for the examination. The handbook contains only reference formulas and tables; no example questions are included. Many commercially available books contain worked examples and sample questions. An examinee can also perform a self-test using one of the NCEES FE Sample Questions and Solutions books (a partial examination), which may be purchased by calling (800) 250-3196.
The examinee is not allowed to bring reference material into the examination room. Another copy of the FE Supplied-Reference Handbook will be made available to each examinee in the room. When the examinee departs the examination room, the FE Supplied-Reference Handbook supplied in the room shall be returned to the examination proctors.
The FE Supplied-Reference Handbook has been prepared to support the FE examination process. The FE Supplied-Reference Handbook is not designed to assist in all parts of the FE examination. For example, some of the basic theories, conversions, formulas, and definitions that examinees are expected to know have not been included. The FE Supplied-Reference Handbook may not include some special material required for the solution of a particular question. In such a situation, the required special information will be included in the question statement.
DISCLAIMER: The NCEES in no event shall be liable for not providing reference material to support all the questions in the FE examination. In the interest of constant improvement, the NCEES reserves the right to revise and update the FE Supplied-Reference Handbook as it deems appropriate without informing interested parties. Each NCEES FE examination will be administered using the latest version of the FE Supplied- Reference Handbook.
So that this handbook can be reused, PLEASE, at the examination site,
DO NOT WRITE IN THIS HANDBOOK.
v
CONVERSION FACTORS
Multiply By To Obtain Multiply By To Obtain acre 43,560 square feet (ft 2 ) joule (J) (^) 9.478× 10 ñ4^ Btu ampere-hr (A-hr) 3,600 coulomb (C) J 0.7376 ft-lbf Ângstrˆm (≈) (^1) × 10 ñ10^ meter (m) J 1 newton∑m (N∑m) atmosphere (atm) 76.0 cm, mercury (Hg) J / s 1 watt (W) atm, std 29.92 in, mercury (Hg) atm, std 14.70 lbf/in 2 abs (psia) kilogram (kg) 2.205 pound (lbm) atm, std 33.90 ft, water kgf 9.8066 newton (N) atm, std (^) 1.013× 10 5 pascal (Pa) kilometer (km) 3,281 feet (ft) km/hr 0.621 mph bar (^1) × 10 5 Pa kilopascal (kPa) 0.145 lbf/in 2 (psi) barrelsñoil 42 gallonsñoil kilowatt (kW) 1.341 horsepower (hp) Btu 1,055 joule (J) kW 3,413 Btu/hr Btu (^) 2.928× 10 ñ4^ kilowatt-hr (kWh) kW 737.6 (ft-lbf )/sec Btu 778 ft-lbf kW-hour (kWh) 3,413 Btu Btu/hr (^) 3.930× 10 ñ4^ horsepower (hp) kWh 1.341 hp-hr Btu/hr 0.293 watt (W) kWh (^) 3.6× 10 6 joule (J) Btu/hr 0.216 ft-lbf/sec kip (K) 1,000 lbf K 4,448 newton (N) calorie (g-cal) (^) 3.968× 10 ñ3^ Btu cal (^) 1.560× 10 ñ6^ hp-hr liter (L) 61.02 in^3 cal 4.186 joule (J) L 0.264 gal (US Liq) cal/sec 4.186 watt (W) L 10 ñ3^ m^3 centimeter (cm) (^) 3.281× 10 ñ2^ foot (ft) L/second (L/s) 2.119 ft 3 /min (cfm) cm 0.394 inch (in) L/s 15.85 gal (US)/min (gpm) centipoise (cP) 0.001 pascal∑sec (Pa∑s) centistokes (cSt) (^1) × 10 ñ6^ m^2 /sec (m^2 /s) meter (m) 3.281 feet (ft) cubic feet/second (cfs) 0.646317 million gallons/day (mgd)
m 1.094 yard
cubic foot (ft 3 ) 7.481 gallon m/second (m/s) 196.8 feet/min (ft/min) cubic meters (m^3 ) 1,000 Liters mile (statute) 5,280 feet (ft) electronvolt (eV) (^) 1.602× 10 ñ19^ joule (J) mile (statute) 1.609 kilometer (km) mile/hour (mph) 88.0 ft/min (fpm) foot (ft) 30.48 cm mph 1.609 km/h ft 0.3048 meter (m) mm of Hg (^) 1.316× 10 ñ3^ atm ft-pound (ft-lbf) (^) 1.285× 10 ñ3^ Btu mm of H2O (^) 9.678× 10 ñ5^ atm ft-lbf (^) 3.766× 10 ñ7^ kilowatt-hr (kWh) ft-lbf 0.324 calorie (g-cal) newton (N) 0.225 lbf ft-lbf 1.356 joule (J) N∑m 0.7376 ft-lbf ft-lbf/sec (^) 1.818× 10 ñ3^ horsepower (hp) N∑m 1 joule (J)
gallon (US Liq) 3.785 liter (L) pascal (Pa) (^) 9.869× 10 ñ6^ atmosphere (atm) gallon (US Liq) 0.134 ft 3 Pa 1 newton/m^2 (N/m^2 ) gallons of water 8.3453 pounds of water Pa∑sec (Pa∑s) 10 poise (P) gamma (γ, Γ) 1 × 10 ñ9^ tesla (T)^ pound (lbm,avdp)^ 0.454^ kilogram (kg) gauss (^1) × 10 ñ4^ T lbf 4.448 N gram (g) (^) 2.205× 10 ñ3^ pound (lbm) lbf-ft 1.356 N∑m lbf/in 2 (psi) 0.068 atm hectare (^1) × 10 4 square meters (m 2 ) psi 2.307 ft of H2O hectare 2.47104 acres psi 2.036 in of Hg horsepower (hp) 42.4 Btu/min psi 6,895 Pa hp 745.7 watt (W) hp 33,000 (ft-lbf)/min radian (^180) / π degree hp 550 (ft-lbf)/sec hp-hr 2,544 Btu stokes (^1) × 10 ñ4^ m^2 /s hp-hr (^) 1.98× 10 6 ft-lbf hp-hr (^) 2.68× 10 6 joule (J) therm (^1) × 10 5 Btu hp-hr 0.746 kWh watt (W) 3.413 Btu/hr inch (in) 2.540 centimeter (cm) W (^) 1.341× 10 ñ3^ horsepower (hp) in of Hg 0.0334 atm W 1 joule/sec (J/s) in of Hg 13.60 in of H2O weber/m^2 (Wb/m^2 ) 10,000 gauss in of H2O 0.0361 lbf/in 2 (psi) in of H2O 0.002458 atm
MATHEMATICS
STRAIGHT LINE
The general form of the equation is
Ax + By + C = 0
The standard form of the equation is
y = mx + b ,
which is also known as the slope-intercept form.
The point-slope form is y ñ y 1 = m ( x ñ x 1)
Given two points: slope, m = ( y 2 ñ y 1)/( x 2 ñ x 1)
The angle between lines with slopes m 1 and m 2 is
α = arctan [( m 2 ñ m 1)/(1 + m 2∑ m 1)]
Two lines are perpendicular if m 1 = ñ1/ m 2
The distance between two points is
d = ( y 2 − y 1 ) 2 +( x 2 − x 1 )^2
QUADRATIC EQUATION
ax^2 + bx + c = 0
a
b b ac Roots 2
− ±^2 − 4
CONIC SECTIONS
e = eccentricity = cos θ/(cos φ)
[Note: X ′ and Y ′, in the following cases, are translated axes.]
Case 1. Parabola e = 1:
( y ñ k )^2 = 2 p ( x ñ h ); Center at ( h , k )
is the standard form of the equation. When h = k = 0, Focus: ( p/ 2,0); Directrix: x = ñ p/ 2
Case 2. Ellipse e < 1:
( ) ( ) ( )
( )
( ae, ) x a/e
b a e
e b a c/a
h k ,
h,k b
y k a
x h
Focus: 0 ; Directrix :
Eccentricity: 1
isthestandardformoftheequation.When 0
1 ; Centerat
2
2 2
2
2 2
2
Case 3. Hyperbola e > 1:
( ) ( ) ( )
( )
( ae, ) x a/e
b a e
e b a c/a
h k ,
h,k b
y k a
x h
Focus: 0 ; Directrix :
Eccentricity: 1
isthestandardformoftheequation.When 0
1 ; Centerat
2
2 2
2
2 2
2
- Brink, R.W., A First Year of College Mathematics , Copyright © 1937 by D. Appleton-Century Co., Inc. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.
MATHEMATICS (continued)
TRIGONOMETRY
Trigonometric functions are defined using a right triangle.
sin θ = y/r , cos θ = x/r
tan θ = y/x , cot θ = x/y
csc θ = r/y , sec θ = r/x
Law of Sines C
c B
b A
a sin sin sin
Law of Cosines a^2 = b^2 + c^2 ñ 2 bc cos A b^2 = a^2 + c^2 ñ 2 ac cos B c^2 = a^2 + b^2 ñ 2 ab cos C
Identities
csc θ = 1 / sin θ
sec θ = 1 / cos θ
tan θ = sin θ / cos θ
cot θ = 1 / tan θ
sin 2 θ + cos^2 θ = 1
tan^2 θ + 1 = sec 2 θ
cot^2 θ + 1 = csc 2 θ
sin (α + β) = sin α cos β + cos α sin β
cos (α + β) = cos α cos β ñ sin α sin β
sin 2α = 2 sin α cos α
cos 2α = cos^2 α ñ sin 2 α = 1 ñ 2 sin 2 α = 2 cos^2 α ñ 1
tan 2α = (2 tan α) / (1 ñ tan^2 α)
cot 2α = (cot^2 α ñ 1) / (2 cot α)
tan (α + β) = (tan α + tan β) / (1 ñ tan α tan β)
cot ( α + β) = (cot α cot β ñ 1) / (cot α + cot β)
sin ( α ñ β) = sin α cos β ñ cos α sin β
cos (α ñ β) = cos α cos β + sin α sin β
tan (α ñ β) = (tan α ñ tan β) / (1 + tan α tan β)
cot (α ñ β) = (cot α cot β + 1) / (cot β ñ cot α)
sin (α / 2) = ± ( 1 −cosα) 2
cos (α / 2) = ± ( 1 +cosα) 2
tan (α / 2) = ± ( 1 −cosα) ( 1 +cosα)
cot (α / 2) = ± ( 1 +cosα) ( 1 −cosα)
sin α sin β = (1 / 2)[cos (α ñ β) ñ cos (α + β)] cos α cos β = (1 / 2)[cos (α ñ β) + cos (α + β)] sin α cos β = (1 / 2)[sin (α + β) + sin (α ñ β)] sin α + sin β = 2 sin (1 / 2)(α + β) cos (1 / 2)(α ñ β) sin α ñ sin β = 2 cos (1 / 2)(α + β) sin (1 / 2)(α ñ β) cos α + cos β = 2 cos (1 / 2)(α + β) cos (1 / 2)(α ñ β) cos α ñ cos β = ñ 2 sin (1 / 2)(α + β) sin (1 / 2)(α ñ β)
COMPLEX NUMBERS
Definition i = − 1 ( a + ib ) + ( c + id ) = ( a + c ) + i ( b + d ) ( a + ib ) ñ ( c + id ) = ( a ñ c ) + i ( b ñ d ) ( a + ib )( c + id ) = ( ac ñ bd ) + i ( ad + bc ) ( )( ) ( )( )
( ) ( ) c^2 d^2
ac bd ibc ad c id c id
a ib c id c id
a ib
( a + ib ) + ( a ñ ib ) = 2 a ( a + ib ) ñ ( a ñ ib ) = 2 ib ( a + ib )( a ñ ib ) = a^2 + b^2
Polar Coordinates x = r cos θ; y = r sin θ; θ = arctan ( y/x )
r = x + iy = x^2 + y^2
x + iy = r (cos θ + i sin θ) = rei θ [ r 1 (cos θ 1 + i sin θ 1 )][ r 2 (cos θ 2 + i sin θ 2 )] = r 1 r 2 [cos (θ 1 + θ 2 ) + i sin (θ 1 + θ 2 )] ( x + iy ) n^ = [ r (cos θ + i sin θ)] n = r n (cos n θ + i sin n θ) ( ) ( )
[ ( 1 2 ) ( 1 2 )] 2
1 2 2 2
(^1 1) cos sin cos sin
cos sin = θ −θ + θ −θ θ + θ
θ + θ i r
r r i
r i
Euler's Identity ei θ^ = cos θ + i sin θ e^ − i θ^ = cos θ ñ i sin θ
i
e i ei ei ei 2
, sin 2
cos
θ −θ θ− − θ θ=
θ=
Roots If k is any positive integer, any complex number (other than zero) has k distinct roots. The k roots of r (cos θ + i sin θ) can be found by substituting successively n = 0, 1, 2, …, ( k ñ 1) in the formula
ú
ú û
ù
ê
ê ë
é ÷ ÷ ø
ö ç ç è
æ
θ ÷÷+ ø
ö ç ç è
æ
θ
k
n k
i k
n k
w kr
o 360 o sin
360 cos
MATHEMATICS (continued)
MATRICES
A matrix is an ordered rectangular array of numbers with m rows and n columns. The element aij refers to row i and column j.
Multiplication
If A = ( aik ) is an m × n matrix and B = ( bkj ) is an n × s matrix, the matrix product AB is an m × s matrix
( ) (^) ÷ ø
ö ç è
æ = = å =
n l
ci (^) j ailblj 1
C
where n is the common integer representing the number of columns of A and the number of rows of B ( l and k = 1, 2, …, n ).
Addition
If A = ( aij ) and B = ( bij ) are two matrices of the same size m × n , the sum A + B is the m × n matrix C = ( cij ) where cij = aij + b (^) ij.
Identity
The matrix I = ( aij ) is a square n × n identity matrix where aii = 1 for i = 1, 2, …, n and aij = 0 for i ≠≠≠≠ j.
Transpose
The matrix B is the transpose of the matrix A if each entry bji in B is the same as the entry aij in A and conversely. In equation form, the transpose is B = A T.
Inverse
The inverse B of a square n × n matrix A is
( ) (^1) , where adj A
A
B = A − =
adj( A ) = adjoint of A (obtained by replacing A T^ elements with their cofactors, see DETERMINANTS ) and
A = determinant of A.
DETERMINANTS
A determinant of order n consists of n^2 numbers, called the elements of the determinant, arranged in n rows and n columns and enclosed by two vertical lines. In any determinant, the minor of a given element is the determinant that remains after all of the elements are struck out that lie in the same row and in the same column as the given element. Consider an element which lies in the h th column and the k th row. The cofactor of this element is the value of the minor of the element (if h + k is even ), and it is the negative of the value of the minor of the element (if h + k is odd ).
If n is greater than 1, the value of a determinant of order n is the sum of the n products formed by multiplying each element of some specified row (or column) by its cofactor. This sum is called the expansion of the determinant [according to the elements of the specified row (or column)]. For a second-order determinant:
12 21 1 2
1 2 ab ab b b
a a = −
For a third-order determinant:
12 3 2 31 312 3 21 213 13 2
1 2 3
1 2 3
1 2 3 abc abc abc abc abc ab c
c c c
b b b
a a a = + + − − −
VECTORS
A = ax i + ay j + az k Addition and subtraction: A + B = ( ax + bx ) i + ( ay + by ) j + ( az + bz ) k A ñ B = ( ax ñ bx ) i + ( ay ñ by ) j + ( az ñ bz ) k The dot product is a scalar product and represents the projection of B onto A times A . It is given by A ∑ B = ax bx + ay by + az bz = A B cos θ = B ∑ A The cross product is a vector product of magnitude B A sin θ which is perpendicular to the plane containing A and B. The product is
B A
i j k
A × B = =− ×
x y z
x y z b b b
a a a
The sense of A × B is determined by the right-hand rule. A × B = A B n sin θ, where n = unit vector perpendicular to the plane of A and B.
j
i k
MATHEMATICS (continued)
Taylor's Series
( ) ( )
( ) ( )
( ) ( )
( ) (^) ( ) +K + ( − ) +K
−
′′ − +
′ = +
n n x a n
f a
x a
f a x a
f a f x f a
!
1! 2!
2
is called Taylor's series , and the function f ( x ) is said to be expanded about the point a in a Taylor's series.
If a = 0, the Taylor's series equation becomes a Maclaurin's series.
PROBABILITY AND STATISTICS
Permutations and Combinations
A permutation is a particular sequence of a given set of objects. A combination is the set itself without reference to order.
- The number of different permutations of n distinct objects taken r at a time is
( ) ( )!
! n r
n P n,r −
=
- The number of different combinations of n distinct objects taken r at a time is
( )
( ) [ !( )!]
! ! (^) r n r
n r
Pn,r C n,r −
= =
- The number of different permutations of n objects taken n at a time , given that ni are of type i , where i = 1, 2,…, k and Σ ni = n , is
( ) !!!
! 1 2
1 2 k
k n n n
n P n;n,n, n K
K =
Laws of Probability
Property 1. General Character of Probability
The probability P ( E ) of an event E is a real number in the range of 0 to 1. The probability of an impossible event is 0 and that of an event certain to occur is 1.
Property 2. Law of Total Probability
P ( A + B ) = P ( A ) + P ( B ) ñ P ( A , B ), where
P ( A + B ) = the probability that either A or B occur alone or that both occur together,
P ( A ) = the probability that A occurs,
P ( B ) = the probability that B occurs, and
P ( A , B ) = the probability that both A and B occur simultaneously.
Property 3. Law of Compound or Joint Probability If neither P ( A ) nor P ( B ) is zero, P ( A , B ) = P ( A ) P ( B | A ) = P ( B ) P ( A | B ), where P ( B | A ) = the probability that B occurs given the fact that A has occurred, and P ( A | B ) = the probability that A occurs given the fact that B has occurred. If either P ( A ) or P ( B ) is zero, then P ( A , B ) = 0.
Probability Functions A random variable x has a probability associated with each of its values. The probability is termed a discrete probability if x can assume only the discrete values x = X 1 , X 2 , …, X (^) i , …, X (^) N The discrete probability of the event X = xi occurring is defined as P ( X (^) i ).
Probability Density Functions If x is continuous, then the probability density function f ( x ) is defined so that = the probability that x lies between x 1 and x 2. The probability is determined by defining the equation for f ( x ) and integrating between the values of x required.
Probability Distribution Functions The probability distribution function F ( X (^) n ) of the discrete probability function P ( X (^) i ) is defined by
( ) ( ) ( (^) i n )
n k
F Xn = (^) å PXk = PX ≤ X = 1 When x is continuous, the probability distribution function F ( x ) is defined by
F ( ) x = (^) ò−^ x ∞^ f ( ) tdt
which implies that F ( a ) is the probability that x ≤≤≤≤ a. The expected value g ( x ) of any function is defined as
E { g ( ) x } = (^) ò−^ x ∞^ g ( ) t f ( ) tdt
Binomial Distribution P ( x ) is the probability that x will occur in n trials. If p = probability of success and q = probability of failure = 1 ñ p , then
( ) ( ) ( )
x nx pxqnx x n x
n P x Cn,xpq − − −
= = !!
! , where
x = 0, 1, 2, …, n , C ( n , x ) = the number of combinations, and n , p = parameters.
ò 12 ( )
x x f xdx
MATHEMATICS (continued)
Normal Distribution (Gaussian Distribution)
This is a unimodal distribution, the mode being x = μ, with two points of inflection (each located at a distance σ to either side of the mode). The averages of n observations tend to become normally distributed as n increases. The variate x is said to be normally distributed if its density function f ( x ) is given by an expression of the form
( ) (^ )^
2 22 2
(^1) − −μ σ σ π
f x = e x , where
μ = the population mean,
σ = the standard deviation of the population, and
ñ∞ ≤ x ≤ ∞
When μ = 0 and σ^2 = σ = 1, the distribution is called a standardized or unit normal distribution. Then
f ( ) x ex^ , −∞≤ x ≤∞. π
= −^ where 2
(^1 )
A unit normal distribution table is included at the end of this section. In the table, the following notations are utilized:
F ( x ) = the area under the curve from ñ∞ to x ,
R ( x ) = the area under the curve from x to ∞, and
W ( x ) = the area under the curve between ñ x and x.
Dispersion, Mean, Median, and Mode Values
If X 1 , X 2 , …, Xn represent the values of n items or observations, the arithmetic mean of these items or observations, denoted , is defined as
= ( )( + + + ) =( )å
n i
X n X X Xn n Xi 1
1 1 2 K 1
X → μ for sufficiently large values of n.
The weighted arithmetic mean is
å
å
i
i i w w
wX X , where
X (^) w = the weighted arithmetic mean,
Xi = the values of the observations to be averaged, and
wi = the weight applied to the Xi value.
The variance of the observations is the arithmetic mean of the squared deviations from the population mean. In symbols, X 1 , X 2 , …, Xn represent the values of the n sample observations of a population of size N. If μ is the arithmetic mean of the population, the population variance is defined by
( ) ( ) ( ) ( )
( ) ( )
2 1
2 2 2
2 1
2
1
1 [ ]
= å −μ
σ = −μ + −μ + + −μ
=
N i i
N
/N X
/ N X X K X
The standard deviation of a population is
σ =^ ( 1 N )å^ (^ Xi −μ)^2
The sample variance is
[ ( )] ( )
2
1
(^2) = 1 − (^1) å −
n i
s n Xi X
The sample standard deviation is
úå^ (^ − ) û
ù êë
é −
=
n i
Xi X n
s 1
2 1
The coefficient of variation = CV = s/ X
The geometric mean = n^ X (^) 1 X 2 X 3 K Xn
The root-mean-square value = ( 1 n )å Xi^2
The median is defined as the value of the middle item when the data are rank-ordered and the number of items is odd. The median is the average of the middle two items when the rank-ordered data consists of an even number of items. The mode of a set of data is the value that occurs with greatest frequency.
t-Distribution The variate t is defined as the quotient of two independent variates x and r where x is unit normal and r is the root mean square of n other independent unit normal variates ; that is, t = x/r. The following is the t -distribution with n degrees of freedom:
( )
[( )] ( ) (^) ( 1 2 )(^1 )^2
1 2
1 2
Γ π +
Γ + = (^) n n n t n
n f t
where ñ ∞ ≤ t ≤≤≤≤ ∞. A table at the end of this section gives the values of t α , n for values of α and n. Note that in view of the symmetry of the t -distribution, t 1 −α ,n = ñ t α ,n. The function for α follows:
α =ò t ∞ (^) α ,nf ( ) tdt
A table showing probability and density functions is included on page 149 in the INDUSTRIAL ENGINEERING SECTION of this handbook.
X
MATHEMATICS (continued)
UNIT NORMAL DISTRIBUTION TABLE
MATHEMATICS (continued) t -DISTRIBUTION TABLE VALUES OF t αα αα ,n n αααα = 0.10 αααα = 0.05 αααα = 0.025 αααα = 0.01 αααα = 0.005 n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
3. inf.
- UNITS TABLE OF C ONTENTS
- CONVERSION F ACTORS
- M ATHEMATICS
- S TATICS
- DYNAMICS
- M ECHANICS OF M ATERIALS
- F LUID M ECHANICS
- THERMODYNAMICS
- HEAT TRANSFER
- TRANSPORT P HENOMENA
- C HEMISTRY
- M ATERIALS S CIENCE/S TRUCTURE OF M ATTER
- ELECTRIC C IRCUITS
- C OMPUTERS, M EASUREMENT, AND C ONTROLS
- ENGINEERING ECONOMICS
- ETHICS
- C HEMICAL ENGINEERING
- C IVIL ENGINEERING
- ENVIRONMENTAL ENGINEERING
- ELECTRICAL AND C OMPUTER ENGINEERING
- INDUSTRIAL ENGINEERING
- M ECHANICAL ENGINEERING
- INDEX
- x f ( x ) F ( x ) R ( x ) 2 R ( x ) W ( x )
- Fractiles
- inf.
MATHEMATICS (continued)
DIFFERENTIAL CALCULUS
The Derivative
For any function y = f ( x ),
the derivative = D (^) x y = dy/dx = y ′
[ ( ) ( )]
{ f ( x x ) f ( ) x ( x )}
y y x
x
x = +∆ − ∆
′= ∆ ∆
∆→
∆→ [ ]
0 limit
limit
y ′ = the slope of the curve f ( x ).
Test for a Maximum
y = f ( x ) is a maximum for x = a , if f ′( a ) = 0 and f ″( a ) < 0.
Test for a Minimum
y = f ( x ) is a minimum for x = a , if f ′( a ) = 0 and f ″( a ) > 0.
Test for a Point of Inflection
y = f ( x ) has a point of inflection at x = a ,
if f ″( a ) = 0, and
if f ″( x ) changes sign as x increases through
x = a.
The Partial Derivative
In a function of two independent variables x and y , a derivative with respect to one of the variables may be found if the other variable is assumed to remain constant. If y is kept fixed , the function
z = f ( x , y )
becomes a function of the single variable x , and its derivative (if it exists) can be found. This derivative is called the partial derivative of z with respect to x. The partial derivative with respect to x is denoted as follows:
x
f x,y x
z ∂
∂
∂
∂
The Curvature of Any Curve
♦
The curvature K of a curve at P is the limit of its average curvature for the arc PQ as Q approaches P. This is also expressed as: the curvature of a curve at a given point is the rate-of-change of its inclination with respect to its arc length.
ds
d s
K
α
∆
∆α
∆ s→ 0
limit
Curvature in Rectangular Coordinates
[ 1 ( y )^2 ] 32
y K +^ ′
When it may be easier to differentiate the function with respect to y rather than x , the notation x ′ will be used for the derivative. x ′ = dx/dy
[ ( )]
2 32 1 x
x K +^ ′
The Radius of Curvature The radius of curvature R at any point on a curve is defined as the absolute value of the reciprocal of the curvature K at that point.
[ ( )]
0
1
232 ′′ ≠ ′′
- ′ =
= ≠
y y
y R
K K
R
L'Hospital's Rule (L'HÙpital's Rule) If the fractional function f ( x ) /g ( x ) assumes one of the indeterminate forms 0 / 0 or ∞ / ∞ (where α is finite or infinite), then
f ( ) x g ( ) x
x→ α
limit
is equal to the first of the expressions
g ( ) x
f x , g x
f x , g x
f x x x x ′′′
′′′ ′′
′′ ′
′ →α →α →α
limit limit limit
which is not indeterminate, provided such first indicated limit exists.
INTEGRAL CALCULUS The definite integral is defined as:
å (^ )∆^ =ò ( )
→∞ =
n i
b n f xi xi af xdx 1
limit
Also, ∆ x (^) i → 0 forall i.
A table of derivatives and integrals is available on page 15. The integral equations can be used along with the following methods of integration: A. Integration by Parts (integral equation #6), B. Integration by Substitution, and C. Separation of Rational Fractions into Partial Fractions.
♦ Wade, Thomas L., Calculus , Copyright © 1953 by Ginn & Company. Diagram reprinted by permission of Simon & Schuster Publishers.
MATHEMATICS (continued) DERIVATIVES AND INDEFINITE INTEGRALS
In these formulas, u , v , and w represent functions of x. Also, a , c , and n represent constants. All arguments of the trigonometric functions are in radians. A constant of integration should be added to the integrals. To avoid terminology difficulty, the following definitions are followed: arcsin u = sin ñ1^ u , (sin u ) ñ1^ = 1 / sin u.
dc/dx = 0
dx/dx = 1
d ( cu ) /dx = c du/dx
d ( u + v ñ w ) /dx = du/dx + dv/dx ñ dw/dx
d ( uv ) /dx = u dv/dx + v du/dx
d ( uvw ) /dx = uv dw/dx + uw dv/dx + vw du/dx
( ) v^2
vdudx udvdx dx
d uv −
d ( un ) /dx = nun ñ1^ du/dx
d [ f ( u )] /dx = { d [ f ( u )] / du } du/dx
du/dx = 1 / ( dx/du )
( ) ( ) dx
du u
e dx
d (^) au 1 log
log = a
( ) dx
du dx u
d ln u 1
( ) ( ) dx
du aa dx
d au u = ln
- d ( eu ) /dx = eu^ du/dx
- d ( uv ) /dx = vuv ñ1^ du/dx + (ln u ) uv^ dv/dx
- d (sin u ) /dx = cos u du/dx
- d (cos u ) /dx = ñsin u du/dx
- d (tan u ) /dx = sec^2 u du/dx
- d (cot u ) /dx = ñcsc^2 u du/dx
- d (sec u ) /dx = sec u tan u du/dx
- d (csc u ) /dx = ñcsc u cot u du/dx
( ) ( 2 sin 2 )
sin (^11)
1 −π ≤ ≤ π −
= −
− u dx
du dx u
d u
( ) ( ≤ ≤π) −
= − −
− u dx
du dx u
d u 1 2
1 0 cos 1
cos 1
( ) ( 2 tan 2 ) 1
tan (^11) 2
1 −π < < π
= −
− u dx
du dx u
d u
( ) ( < <π)
= − −
− u dx
du dx u
d u 1 2
1 0 cot 1
cot 1
( )
( 0 sec 2 )( sec 2 )
1
sec 1
1 1
2
1
≤ <π −π≤ <− π
−
=
− −
−
u u
dx
du dx u u
d u
( )
( 0 csc 2 )( csc 2 )
1
csc 1
1 1
2
1
< ≤π −π< ≤− π
−
=−
− −
−
u u
dx
du dx u u
d u
- ò d f ( x ) = f ( x )
- ò dx = x
- ò a f ( x ) dx = a ò f ( x ) dx
- ò [ u(x ) ± v ( x )] dx = ò u ( x ) dx ± ò v ( x ) dx
- ( 1 ) 1
1 ≠ −
ò =
m m
x x dx
m m
- ò u ( x ) dv ( x ) = u ( x ) v ( x ) ñ ò v ( x ) du ( x )
- (^) ò = +
ax b ax b a
dx ln
1
- (^) ò = x x
dx 2
- ò ax^ dx = a
a x ln
- ò sin x dx = ñ cos x
- ò cos x dx = sin x
- (^) ò = − 4
sin 2 2
sin 2 x x xdx
- (^) ò = + 4
sin 2 2
cos 2
x x xdx
ò x sin x dx = sin x ñ x cos x
ò x cos x dx = cos x + x sin x
ò sin x cos x dx = (sin 2 x ) / 2
( ) ( )
( ) ( )
( 2 2 ) 2
cos 2
cos sin cos a b a b
a bx a b
a bx ò ax^ bxdx ≠
− −
− =−
- ò tan x dx = ñlncos x = ln sec x
- ò cot x dx = ñln csc x = ln sin x
- ò tan^2 x dx = tan x ñ x
- ò cot^2 x dx = ñcot x ñ x
- ò eax^ dx = (1 /a ) eax
- ò xeax^ dx = ( eax/a^2 )( ax ñ 1)
- ò ln x dx = x [ln ( x ) ñ 1] ( x > 0)
- tan ( 0 )
ò (^2) + 2 = ≠
− (^) a a
x a x a
dx
- tan ( 0 0 ) (^1 ) ò (^2) ÷ > >
÷
ø
ö ç
ç è
æ
− (^) , a ,c c
a x ax c ac
dx
27a.
( 4 0 )
tan 4
2
2
1 (^22)
ò −
−
ac b
ac b
ax b ax bx c ac b
dx
27b.
( 4 0 )
ln 4
2
2
2 (^22)
ò
- −
b ac
ax b b ac
ax b b ac ax bx c b ac
dx
27c. ( 4 0 ) 2
ò (^) ax (^2) + bx + c =− ax + b, b − ac =
dx