ALGEBRA 1
Man-hours (is always assumed constant)
LOGARITHM
(Wor ker s1 )(time1 ) (Wor ker s 2 )(time2 )
=
quantity.of .work1
quantity.of .work 2
x = log b N → N = b x
Properties
ALGEBRA 2
log( xy ) = log x + log y
x
log  = log x − log y
 y
log x n = n log x
log x
log b x =
log b
log a a = 1
UNIFORM MOTION PROBLEMS
REMAINDER AND FACTOR THEOREMS
Traveling against the wind or upstream:
S = Vt
Traveling with the wind or downstream:
Vtotal = V1 + V2
Given:
f ( x)
(x − r)
Vtotal = V1 − V2
DIGIT AND NUMBER PROBLEMS
Remainder Theorem: Remainder = f(r)
Factor Theorem: Remainder = zero
100h + 10t + u → 2-digit number
QUADRATIC EQUATIONS
where:
Ax 2 + Bx + C = 0
Root =
− B ± B 2 − 4 AC
2A
h = hundred’s digit
t = ten’s digit
u = unit’s digit
CLOCK PROBLEMS
Sum of the roots = - B/A
Products of roots = C/A
MIXTURE PROBLEMS
Quantity Analysis: A + B = C
Composition Analysis: Ax + By = Cz
where:
WORK PROBLEMS
Rate of doing work = 1/ time
Rate x time = 1 (for a complete job)
Combined rate = sum of individual rates
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x = distance traveled by the
minute hand in minutes
x/12 = distance traveled by the hour
hand in minutes
PROGRESSION PROBLEMS
a1
an
am
d
S
=
=
=
=
=
HARMONIC PROGRESSION (HP)
first term
nth term
any term before an
common difference
sum of all “n” terms
-
a sequence of number in which their reciprocals
form an AP
calcu function: LINEAR (LIN)
Mean – middle term or terms between two terms
in the progression.
ARITHMETIC PROGRESSION (AP)
COIN PROBLEMS
-
Penny = 1 centavo coin
Nickel = 5 centavo coin
Dime = 10 centavo coin
Quarter = 25 centavo coin
Half-Dollar = 50 centavo coin
difference of any 2 no.’s is constant
calcu function: LINEAR (LIN)
a n = a m + ( n − m) d
nth term
d = a2 − a1 = a3 − a2 ,...etc
S=
n
(a1 + an )
2
S=
n
[2 a1 + (n − 1) d ]
2
Common
difference
DIOPHANTINE EQUATIONS
If the number of equations is less than the
number of unknowns, then the equations are
called “Diophantine Equations”.
Sum of ALL
terms
ALGEBRA 3
GEOMETRIC PROGRESSION (GP)
Fundamental Principle:
“If one event can occur in m different ways, and
after it has occurred in any one of these ways, a
second event can occur in n different ways, and
then the number of ways the two events can
occur in succession is mn different ways”
-
PERMUTATION
Sum of ALL
terms
RATIO of any 2 adj, terms is always constant
Calcu function: EXPONENTIAL (EXP)
an = a m r n−m
nth term
Permutation of n objects taken r at a time
nPr =
r=
a 2 a3
=
a1 a2
n!
(n − r )!
ratio
a ( r n − 1)
S= 1
→ r >1
r −1
a1 (1 − r n )
S=
→ r <1
1− r
Permutation of n objects taken n at a time
Sum of ALL
terms, r >1
nPn = n!
Permutation of n objects with q,r,s, etc. objects
are alike
P=
Sum of ALL
terms, r < 1
n!
q!r!s!...
Permutation of n objects arrange in a circle
S=
a1
1− r
→ r < 1& n = ∞
Sum of ALL
terms,
r<1,n=∞
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P = ( n − 1)!
COMBINATION
PROBABILITY
Combination of n objects taken r at a time
Probability of an event to occur (P)
nCr =
n!
(n − r )!r!
Combination of n objects taken n at a time
P=
number _ of _ successful _ outcomes
total _ outcomes
Probability of an event not to occur (Q)
nCn = 1
Q=1–P
Combination of n objects taken 1, 2, 3…n at a
time
C = 2n − 1
MULTIPLE EVENTS
Mutually exclusive events without a common
outcome
PA or B = PA + PB
BINOMIAL EXPANSION
Properties of a binomial expansion: (x + y)n
Mutually exclusive events with a common
outcome
1. The number of terms in the resulting expansion is
PA or B = PA + PB – PA&B
equal to “n+1”
2. The powers of x decreases by 1 in the
successive terms while the powers of y increases
Dependent/Independent Probability
by 1 in the successive terms.
PAandB =PA × PB
3. The sum of the powers in each term is always
equal to “n”
4. The first term is xn while the last term in yn both
of the terms having a coefficient of 1.
th
r term in the expansion (x + y)
term involving yr in the expansion (x + y)n
y term = nCr (x)
n-r
(y)
P = nCr pr qn-r
n
r th term = nCr-1 (x)n-r+1 (y)r-1
r
REPEATED TRIAL PROBABILITY
r
sum of coefficients of (x + y)n
Sum = (coeff. of x + coeff. of y) n
p = probability that the event happen
q = probability that the event failed
VENN DIAGRAMS
Venn diagram in mathematics is a diagram
representing a set or sets and the logical,
relationships between them. The sets are drawn
as circles. The method is named after the British
mathematician and logician John Venn.
sum of coefficients of (x + k)n
Sum = (coeff. of x + k)n – (k)n
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PLANE
TRIGONOMETRY
ANGLE, MEASUREMENTS &
CONVERSIONS
1 revolution = 360 degrees
1 revolution = 2π radians
1 revolution = 400 grads
1 revolution = 6400 mils
1 revolution = 6400 gons
Relations between two angles (A & B)
Complementary angles → A + B = 90°
Supplementary angles → A + B = 180°
Explementary angles → A + B = 360°
TRIGONOMETRIC IDENTITIES
sin 2 A + cos2 A = 1
1 + cot 2 A = csc 2 A
1 + tan 2 A = sec 2 A
sin( A ± B) = sin A cos B ± cos A sin B
cos( A ± B) = cos A cos B m sin A sin B
tan A ± tan B
tan( A ± B) =
1 m tan A tan B
cot A cot B m 1
cot( A ± B) =
cot A ± cot B
sin 2 A = 2 sin A cos B
cos 2 A = cos 2 A − sin 2 A
2 tan A
tan 2 A =
1 − tan 2 A
cot 2 A − 1
cot 2 A =
2 cot A
SOLUTIONS TO OBLIQUE TRIANGLES
SINE LAW
Angle (θ)
REFLEX
Measurement
θ = 0°
0° < θ < 90°
θ = 90°
90° < θ < 180°
θ =180°
180° < θ < 360°
FULL OR PERIGON
θ = 360°
NULL
ACUTE
RIGHT
OBTUSE
STRAIGHT
a
b
c
=
=
sin A sin B sin C
COSINE LAW
a2 = b2 + c2 – 2 b c cos A
b2 = a2 + c2 – 2 a c cos B
c2 = a2 + b2 – 2 a b cos C
Pentagram – golden triangle (isosceles)
36°
AREAS OF TRIANGLES AND
QUADRILATERALS
72° 72°
TRIANGLES
1. Given the base and height
Area =
1
bh
2
2. Given two sides and included angle
Area =
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1
ab sin θ
2
3. Given three sides
4. Quadrilateral circumscribing in a circle
Area = s ( s − a )(s − b)( s − c)
s=
Area = rs
a+b+c
2
Area = abcd
a+b+c+d
s=
2
4. Triangle inscribed in a circle
Area =
abc
4r
THEOREMS IN CIRCLES
5. Triangle circumscribing a circle
Area = rs
6. Triangle escribed in a circle
Area = r ( s − a )
QUADRILATERALS
1. Given diagonals and included angle
Area =
1
d1d 2 sin θ
2
2. Given four sides and sum of opposite angles
Area = ( s − a)( s − b)(s − c )(s − d ) − abcd cos 2 θ
A+C B + D
=
2
2
a+b+c+d
s=
2
θ=
3. Cyclic quadrilateral – is a quadrilateral
inscribed in a circle
Area = ( s − a)( s − b)(s − c )(s − d )
a+b+c+d
2
(ab + cd )(ac + bd )(ad + bc )
r=
4( Area )
s=
d1 d 2= ac + bd → Ptolemy’s Theorem
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SIMILAR TRIANGLES
2
2
Number of diagonal lines (N):
2
A1  A   B   C   H 
=  =  =  = 
A2  a   b   c   h 
N=
2
SOLID GEOMETRY
Area of a regular polygon inscribed in a circle of
radius r
Area =
POLYGONS
3 sides – Triangle
4 sides – Quadrilateral/Tetragon/Quadrangle
5 sides – Pentagon
6 sides – Hexagon
7 sides – Heptagon/Septagon
8 sides – Octagon
9 sides – Nonagon/Enneagon
10 sides – Decagon
11 sides – Undecagon
12 sides – Dodecagon
15 sides – Quidecagon/ Pentadecagon
16 sides – Hexadecagon
20 sides – Icosagon
1000 sides – Chillagon
 180° 
Area = nr 2 tan 

 n 
Area of a regular polygon having each side
measuring x unit length
Area =
1 2
 180° 
nx cot 

4
 n 
PLANE GEOMETRIC FIGURES
CIRCLES
πd 2
= πr 2
4
Circumference = πd = 2πr
A=
Sector of a Circle
Sum of interior angles:
A=
S = n θ = (n – 2) 180°
A=
Value of each interior angle
( n − 2)(180°)
n
Value of each exterior angle
α = 180° − θ =
1 2  360° 
nr sin 

2
 n 
Area of a regular polygon circumscribing a
circle of radius r
Let: n = number of sides
θ = interior angle
α = exterior angle
θ=
n
( n − 3)
2
360°
n
360°
s = rθ ( rad ) =
πrθ (deg)
180°
Segment of a Circle
A segment = A sector – A triangle
ELLIPSE
A=πab
Sum of exterior angles:
S = n α = 360°
1
1
rs = r 2θ
2
2
2
πr θ (deg)
PARABOLIC SEGMENT
A=
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2
bh
3
PYRAMID
TRAPEZOID
1
A = ( a + b) h
2
PARALLELOGRAM
A = ab sin α
A = bh
1
A = d1d 2 sin θ
2
1
Bh
3
A( lateral ) = ∑ Afaces
V =
A( surface) = A( lateral ) + B
Frustum of a Pyramid
V =
RHOMBUS
1
A = d1d 2 = ah
2
A = a 2 sin α
h
( A1 + A2 + A1 A2 )
3
A1 = area of the lower base
A2 = area of the upper base
PRISMATOID
V =
SOLIDS WITH PLANE SURFACE
h
( A1 + A2 + 4 Am )
6
Lateral Area = (No. of Faces) (Area of 1 Face)
Am = area of the middle section
Polyhedron – a solid bounded by planes. The bounding
planes are referred to as the faces and the intersections
of the faces are called the edges. The intersections of the
edges are called vertices.
REGULAR POLYHEDRON
PRISM
V = Bh
a solid bounded by planes whose faces are congruent
regular polygons. There are five regular polyhedrons
namely:
A.
B.
C.
D.
E.
A(lateral) = PL
A(surface) = A(lateral) + 2B
where: P = perimeter of the base
L = slant height
B = base area
Truncated Prism


∑ heights

V = B
number
of
heights


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Tetrahedron
Hexahedron (Cube)
Octahedron
Dodecahedron
Icosahedron
Octahedron
Triangle
Icosahedron
Hexahedron
Square
Triangle
Tetrahedron
Triangle
Pentagon Dodecahedron
Name
Type of
FACE
h
( A1 + A2 + A1 A2
3
A( lateral ) = π ( R + r ) L
V =
SPHERES AND ITS FAMILIES
No. of
FACES
4
6
8
12
20
6
12
12
30
30
SPHERE
No. of
EDGES
4 3
πr
3
A( surface ) = 4πr 2
V =
12
V = 2.18 x 3
20
V = 7.66 x 3
6
2 3
x
3
V=
8
V = x3
2 3
x
12
4
SPHERICAL LUNE
V=
No. of
VERTICES
Formulas for
VOLUME
FRUSTUM OF A CONE
Where: x = length of one edge
SOLIDS WITH CURVED SURFACES
CYLINDER
is that portion of a spherical surface bounded by the
halves of two great circles
A( surface ) =
πr 2θ (deg)
90°
SPHERICAL ZONE
is that portion of a spherical surface between two parallel
planes. A spherical zone of one base has one bounding
plane tangent to the sphere.
A( zone ) = 2π r h
V = Bh = KL
SPHERICAL SEGMENT
A(lateral) = PkL = 2 π r h
is that portion of a sphere bounded by a zone and the
planes of the zone’s bases.
2
A(surface) = A(lateral) + 2B
Pk = perimeter of right section
K = area of the right section
B = base area
L= slant height
V =
πh
(3r − h)
3
πh
(3a 2 + h 2 )
6
πh
V =
(3a 2 + 3b 2 + h 2 )
6
V =
CONE
1
Bh
3
A(lateral ) = πrL
V =
SPHERICAL WEDGE
is that portion of a sphere bounded by a lune and the
planes of the half circles of the lune.
V =
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πr 3θ (deg)
270°
SPHERICAL CONE
PARABOLOID
is a solid formed by the revolution of a circular sector
about its one side (radius of the circle).
a solid formed by rotating a parabolic segment about its
axis of symmetry.
1
A( zone) r
3
A( surface) = A( zone) + A( lateralofcone )
1
V = πr 2 h
2
V =
SPHERICAL PYRAMID
is that portion of a sphere bounded by a spherical
polygon and the planes of its sides.
3
πr E
V =
540°
SIMILAR SOLIDS
3
3
3
2
2
2
V1  H   R   L 
=  =  = 
V2  h   r   l 
A1  H   R   L 
=  =  = 
A2  h   r   l 
2
E = [(n-2)180°]
A 
 V1 
  =  1 
 A2 
 V2 
E = Sum of the angles
E = Spherical excess
n = Number of sides of the given spherical polygon
3
ANALYTIC
GEOMETRY 1
SOLIDS BY REVOLUTIONS
RECTANGULAR COORDINATE SYSTEM
TORUS (DOUGHNUT)
a solid formed by rotating a circle about an axis not
passing the circle.
V = 2π2Rr2
A(surface) = 4 π2Rr
ELLIPSOID
4
V = πabc
3
x = abscissa
y = ordinate
Distance between two points
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
Slope of a line
m = tan θ =
OBLATE SPHEROID
a solid formed by rotating an ellipse about its minor axis.
It is a special ellipsoid with c = a
4
V = πa 2b
3
y2 − y1
x2 − x1
Division of a line segment
x=
x1 r2 + x 2 r1
r1 + r2
y=
y1r2 + y 2 r1
r1 + r2
y=
y1 + y 2
2
PROLATE SPHEROID
a solid formed by rotating an ellipse about its major axis.
It is a special ellipsoid with c=b
4
V = πab 2
3
Location of a midpoint
x=
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x1 + x 2
2
STRAIGHT LINES
Distance between two parallel lines
General Equation
d =
Ax + By + C = 0
C1 − C2
A2 + B 2
Slope relations between parallel lines:
m1 = m2
Point-slope form
y – y1 = m(x – x1)
Two-point form
y − y1 =
y2 − y1
( x − x1 )
x2 − x1
Slope and y-intercept form
Line 1 → Ax + By + C1 = 0
Line 2 → Ax + By + C2 = 0
Slope relations between perpendicular lines:
m1m2 = –1
Line 1 → Ax + By + C1 = 0
Line 2 → Bx – Ay + C2 = 0
y = mx + b
PLANE AREAS BY COORDINATES
Intercept form
A=
x y
+ =1
a b
Slope of the line, Ax + By + C = 0
m=−
A
B
Angle between two lines
 m − m1 

θ = tan −1  2
1
m
m
+

1 2 
Note: Angle θ is measured in a counterclockwise
direction. m2 is the slope of the terminal side while m1 is
the slope of the initial side.
Distance of point (x1,y1) from the line
Ax + By + C = 0;
d=
1 x1 , x2 , x3 ,....xn , x1
2 y1 , y2 , y3 ,.... yn , y1
Note: The points must be arranged in a counter clockwise
order.
LOCUS OF A MOVING POINT
The curve traced by a moving point as it moves in a
plane is called the locus of the point.
SPACE COORDINATE SYSTEM
Length of radius vector r:
r = x2 + y2 + z2
Distance between two points P1(x1,y1,z1) and
P2(x2,y2,z2)
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2 + ( z 2 − z1 ) 2
Ax1 + By1 + C
± A2 + B 2
Note: The denominator is given the sign of B
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Standard Equation:
ANALYTIC
GEOMETRY 2
(x – h)2 + (y – k)2 = r2
General Equation:
CONIC SECTIONS
a two-dimensional curve produced by slicing a plane
through a three-dimensional right circular conical surface
Ways of determining a Conic Section
1.
2.
3.
4.
x2 + y2 + Dx + Ey + F = 0
Center at (h,k):
h=−
By Cutting Plane
Eccentricity
By Discrimination
By Equation
Radius of the circle:
General Equation of a Conic Section:
r 2 = h2 + k 2 −
Ax2 + Cy2 + Dx + Ey + F = 0 **
Cutting plane
Eccentricity
Parallel to base
e→0
Parabola
Parallel to element
e = 1.0
Ellipse
none
e < 1.0
Parallel to axis
e > 1.0
Discriminant
Equation**
Circle
B2 - 4AC < 0, A = C
A=C
Parabola
B2 - 4AC = 0
Ellipse
B2 - 4AC < 0, A ≠ C
Circle
Hyperbola
Hyperbola B2 - 4AC > 0
D
E
; k =−
2A
2A
A≠C
same sign
Sign of A
opp. of B
A or C = 0
F
1
D2 + E 2 − 4F
or r =
A
2
PARABOLA
a locus of a moving point which moves so that it’s always
equidistant from a fixed point called focus and a fixed line
called directrix.
where: a = distance from focus to vertex
= distance from directrix to vertex
AXIS HORIZONTAL:
Cy2 + Dx + Ey + F = 0
Coordinates of vertex (h,k):
k=−
CIRCLE
A locus of a moving point which moves so that its
distance from a fixed point called the center is constant.
E
2C
substitute k to solve for h
Length of Latus Rectum:
LR =
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D
C
AXIS VERTICAL:
ELLIPSE
Ax2 + Dx + Ey + F = 0
Coordinates of vertex (h,k):
h=−
D
2A
a locus of a moving point which moves so that the sum of
its distances from two fixed points called the foci is
constant and is equal to the length of its major axis.
d = distance of the center to the directrix
substitute h to solve for k
Length of Latus Rectum:
E
LR =
A
STANDARD EQUATIONS:
Major axis is horizontal:
( x − h)2 ( y − k ) 2
+
=1
a2
b2
STANDARD EQUATIONS:
Opening to the right:
Major axis is vertical:
2
( x − h) 2 ( y − k ) 2
+
=1
2
2
b
a
(y – k) = 4a(x – h)
Opening to the left:
(y – k)2 = –4a(x – h)
General Equation of an Ellipse:
Ax2 + Cy2 + Dx + Ey + F = 0
Opening upward:
(x – h) 2 = 4a(y – k)
h=−
Opening downward:
(x – h) 2 = –4a(y – k)
Latus Rectum (LR)
a chord drawn to the axis of symmetry of the curve.
LR= 4a
Coordinates of the center:
D
E
;k = −
2A
2C
If A > C, then: a2 = A; b2 = C
If A < C, then: a2 = C; b2 = A
KEY FORMULAS FOR ELLIPSE
for a parabola
Length of major axis: 2a
Eccentricity (e)
the ratio of the distance of the moving point from the
focus (fixed point) to its distance from the directrix (fixed
line).
Length of minor axis: 2b
Distance of focus to center:
e=1
for a parabola
c = a 2 − b2
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Length of latus rectum:
2
2b
LR =
a
(y – k) = m(x – h)
Transverse axis is horizontal:
m=±
Eccentricity:
e=
Equation of Asymptote:
c a
=
a d
b
a
Transverse axis is vertical:
m=±
HYPERBOLA
a locus of a moving point which moves so that the
difference of its distances from two fixed points called the
foci is constant and is equal to length of its transverse
axis.
a
b
KEY FORMULAS FOR HYPERBOLA
Length of transverse axis: 2a
Length of conjugate axis: 2b
Distance of focus to center:
c = a 2 + b2
d = distance from center to directrix
a = distance from center to vertex
c = distance from center to focus
Length of latus rectum:
2b 2
LR =
a
STANDARD EQUATIONS
Transverse axis is horizontal
( x − h)
(y − k)
−
=1
a2
b2
2
2
Eccentricity:
e=
c a
=
a d
Transverse axis is vertical:
( y − k ) 2 ( x − h) 2
=1
−
a2
b2
POLAR COORDINATES SYSTEM
x = r cos θ
GENERAL EQUATION
Ax2 – Cy2 + Dx + Ey + F = 0
Coordinates of the center:
h=−
D
E
; k =−
2A
2C
If C is negative, then: a2 = C, b2 = A
If A is negative, then: a2 = A, b2 = C
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y = r sin θ
r=
x2 + y2
tan θ =
y
x
SPHERICAL
TRIGONOMETRY
Napier’s Rules
1. The sine of any middle part is equal to the
product of the cosines of the opposite parts.
Co-op
Important propositions
1. If two angles of a spherical triangle are equal,
the sides opposite are equal; and conversely.
2. The sine of any middle part is equal to the
product of the tangent of the adjacent parts.
Tan-ad
2. If two angels of a spherical triangle are
unequal, the sides opposite are unequal, and
the greater side lies opposite the greater
angle; and conversely.
Important Rules:
3. The sum of two sides of a spherical triangle is
greater than the third side.
2. When the hypotenuse of a right spherical
triangle is less than 90°, the two legs are of the
same quadrant and conversely.
a+b>c
4. The sum of the sides of a spherical triangle is
less than 360°.
1. In a right spherical triangle and oblique angle
and the side opposite are of the same quadrant.
3. When the hypotenuse of a right spherical
triangle is greater than 90°, one leg is of the first
quadrant and the other of the second and
conversely.
0° < a + b + c < 360°
QUADRANTAL TRIANGLE
5. The sum of the angles of a spherical triangle is
greater that 180° and less than 540°.
180° < A + B + C < 540°
6. The sum of any two angles of a spherical
triangle is less than 180° plus the third angle.
is a spherical triangle having a side equal to 90°.
SOLUTION TO OBLIQUE TRIANGLES
Law of Sines:
sin a sin b sin c
=
=
sin A sin B sin C
A + B < 180° + C
SOLUTION TO RIGHT TRIANGLES
NAPIER CIRCLE
Sometimes called Neper’s circle or Neper’s pentagon, is
a mnemonic aid to easily find all relations between the
angles and sides in a right spherical triangle.
Law of Cosines for sides:
cos a = cos b cos c + sin b sin c cos A
cos b = cos a cos c + sin a sin c cos B
cos c = cos a cos b + sin a sin b cos C
Law of Cosines for angles:
cos A = − cos B cos C + sin B sin C cos a
cos B = − cos A cos C + sin A sin C cos b
cos C = − cos A cos B + sin A sin B cos c
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AREA OF SPHERICAL TRIANGLE
π R2 E
A=
180°
R = radius of the sphere
E = spherical excess in degrees,
E = A + B + C – 180°
TERRESTRIAL SPHERE
Radius of the Earth = 3959 statute miles
Prime meridian (Longitude = 0°)
Equator (Latitude = 0°)
Latitude = 0° to 90°
Longitude = 0° to +180° (eastward)
= 0° to –180° (westward)
1 min. on great circle arc = 1 nautical mile
1 nautical mile = 6080 feet
= 1852 meters
1 statute mile = 5280 feet
= 1760 yards
1 statute mile = 8 furlongs
= 80 chains
Derivatives
dC
=0
dx
d
du dv
+
(u + v ) =
dx
dx dx
d
dv
du
+v
(uv ) = u
dx
dx
dx
dv
du
−u
v
d u
dx
  = dx 2
dx  v 
v
du
d n
(u ) = nu n −1
dx
dx
du
d
u = dx
dx
2 u
du
−c
d c
dx
 =
2
dx  u 
u
d u
du
(a ) = a u ln a
dx
dx
d u
du
(e ) = e u
dx
dx
du
log a e
d
dx
(ln a u ) =
dx
u
du
d
(ln u ) = dx
dx
u
d
du
(sin u ) = cos u
dx
dx
d
du
(cos u ) = − sin u
dx
dx
d
du
(tan u ) = sec 2 u
dx
dx
d
du
(cot u ) = − csc 2 u
dx
dx
d
du
(sec u ) = sec u tan u
dx
dx
d
du
(csc u ) = − csc u cot u
dx
dx
d
du
1
(sin −1 u ) =
2
dx
1 − u dx
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− 1 du
d
(cos −1 u ) =
dx
1 − u 2 dx
d
1 du
(tan −1 u ) =
dx
1 + u 2 dx
d
− 1 du
(cot −1 u ) =
dx
1 + u 2 dx
d
du
1
(sec −1 u ) =
2
dx
u u − 1 dx
d
− 1 du
(csc −1 u ) =
dx
u u 2 − 1 dx
du
d
(sinh u ) = cosh u
dx
dx
d
du
(cosh u ) = sinh u
dx
dx
d
du
(tanh u ) = sec h 2 u
dx
dx
d
du
(coth u ) = − csc h 2 u
dx
dx
d
du
(sec hu ) = − sec hu tanh u
dx
dx
d
du
(csc hu ) = − csc hu coth u
dx
dx
d
1
du
(sinh −1 u ) =
dx
u 2 + 1 dx
d
(cosh −1 u ) =
dx
1
du
u 2 − 1 dx
d
1 du
(tanh −1 u ) =
dx
1 − u 2 dx
d
− 1 du
(sinh −1 u ) = 2
dx
u − 1 dx
d
− 1 du
(sec h −1u ) =
dx
u 1 − u 2 dx
d
− 1 du
(csc h −1u ) =
dx
u 1 + u 2 dx
DIFFERENTIAL
CALCULUS
LIMITS
Indeterminate Forms
0
,
0
∞
, (0)(∞), ∞ - ∞, 0 0 , ∞ 0 , 1∞
∞
L’Hospital’s Rule
f ( x)
f ' ( x)
f "( x)
= Lim
= Lim
.....
x → a g ( x)
x → a g ' ( x)
x → a g" ( x)
Lim
Shortcuts
Input equation in the calculator
TIP 1: if x → 1, substitute x = 0.999999
TIP 2: if x → ∞ , substitute x = 999999
TIP 3: if Trigonometric, convert to RADIANS then
do tips 1 & 2
MAXIMA AND MINIMA
Slope (pt.)
Y’
MAX
0
Y”
(-) dec
Concavity
down
MIN
0
(+) inc
up
INFLECTION
-
No change
-
HIGHER DERIVATIVES
nth derivative of xn
dn
(x n ) = n !
n
dx
nth derivative of xe n
dn
n
X
xe
x
n
e
(
)
=
(
+
)
dx n
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TIME RATE
the rate of change of the variable with respect to time
+
dx
= increasing rate
dt
dx
−
dt = decreasing rate
APPROXIMATION AND ERRORS
∫ tan udu = ln sec u + C
∫ cot udu = ln sin u + C
∫ sec udu = ln sec u + tan u + C
∫ csc udu = ln csc u − cot u + C
du
∫ a − u = sin
2
u
+C
a
−1
2
1
du
u
∫ a + u = a tan a + C
2
−1
2
If “dx” is the error in the measurement of a quantity x,
then “dx/x” is called the RELATIVE ERROR.
∫ u u − a = a sec a + C
RADIUS OF CURVATURE
∫ 2au − u = cos 1 − a  + C
[1 + ( y ' ) 2 ] 2
R=
y"
INTEGRAL
CALCULUS 1
∫ du = u + C
∫ adu = au + C
∫ [ f (u) + g (u )]du = ∫ f (u )du + ∫ g (u)du
u n+1
∫ u du = n + 1 + C..............(n ≠ 1)
du
∫ u = ln u + C
au
u
a
du
=
+C
∫
ln a
∫ e du = e + C
∫ sin udu = − cos u + C
∫ cos udu = sin u + C
∫ sec udu = tan u + C
∫ csc udu = − cot u + C
∫ sec u tan udu = sec u + C
∫ csc u cot udu = − csc u + C
u
u
2
2
−1
u
2
du
3
n
1
du
−1

u
2
∫ sinh udu = cosh u + C
∫ cosh udu = sinh u + C
∫ sec h udu = tanh u + C
∫ csc h udu = − coth u + C
∫ sec hu tanh udu = − sec hu + C
∫ csc hu coth udu = − csc hu + C
∫ tanh udu = ln cosh u + C
∫ coth udu = ln sinh u + C
2
2
du
∫ u + a = sinh
2
−1
2
du
u
+C
a
u
∫ u − a = cosh a + C
−1
2
2
du
1
a
∫ u u + a = − a sinh u + C
2
−1
2
du
1
−1
1
−1
u
∫ a − u = a tanh a + C.............. u < a
2
2
du
u
∫ a − u = a coth a + C.............. u > a
2
2
∫ udv = uv − ∫ vdu
2
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PLANE AREAS
CENTROIDS
Plane Areas bounded by a curve and the coordinate
axes:
Half a Parabola
x2
A = ∫ y ( curve) dx
x1
y2
A = ∫ x( curve ) dy
y1
Plane Areas bounded by a curve and the coordinate
axes:
x2
A = ∫ ( y( up ) − y( down ) )dx
x1
3
x= b
8
2
y= h
5
Whole Parabola
2
y= h
5
Triangle
y2
A = ∫ ( x( right ) − x(left ) )dy
y1
Plane Areas bounded by polar curves:
θ
1 2 2
A = ∫ r dθ
2 θ1
1
2
x = b= b
3
3
1
2
y= h= h
3
3
LENGTH OF ARC
CENTROID OF PLANE AREAS
(VARIGNON’S THEOREM)
Using a Vertical Strip:
x2
A • x = ∫ dA • x
x1
x2
y
A • y = ∫ dA •
2
x1
Using a Horizontal Strip:
y2
x
A • x = ∫ dA •
2
y1
x2
2
 dy 
S = ∫ 1 +   dx
 dx 
x1
y2
2
z2
 dx   dy 
  +   dz
 dz   dz 
 dx 
S = ∫ 1 +   dy
 dy 
y1
S=∫
y2
A • y = ∫ dA • y
y1
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z1
2
2
INTEGRAL
CALCULUS 2
CENTROIDS OF VOLUMES
x2
V • x = ∫ dV • x
x1
y2
TIP 1: Problems will usually be of this nature:
• “Find the area bounded by”
• “Find the area revolved around..”
V • y = ∫ dV • y
TIP 2: Integrate only when the shape is IRREGULAR,
otherwise use the prescribed formulas
WORK BY INTEGRATION
Work = force × distance
y1
VOLUME OF SOLIDS BY REVOLUTION
x2
y2
x1
y1
W = ∫ Fdx = ∫ Fdy ; where F = k x
Circular Disk Method
x2
V = π ∫ R 2 dx
Work done on spring
Cylindrical Shell Method
k = spring constant
x1 = initial value of elongation
x2 = final value of elongation
1
2
2
W = k ( x2 − x1 )
2
x1
y2
V = 2π ∫ RL dy
y1
Work done in pumping liquid out of the
container at its top
Circular Ring Method
x2
V = π ∫ ( R − r )dx
2
Work = (density)(volume)(distance)
2
x1
Force = (density)(volume) = ρv
PROPOSITIONS OF PAPPUS
Specific Weight:
First Proposition: If a plane arc is revolved about a
γ =
coplanar axis not crossing the arc, the area of the surface
generated is equal to the product of the length of the arc
and the circumference of the circle described by the
centroid of the arc.
Weight
Volume
A = S • 2π r
γwater = 9.81 kN/m2 SI
γwater = 45 lbf/ft2 cgs
A = ∫ dS • 2π r
Density:
Second Proposition: If a plane area is revolved
about a coplanar axis not crossing the area, the volume
generated is equal to the product of the area and the
circumference of the circle described by the centroid of
the area.
V = A • 2π r
V = ∫ dA • 2π r
ρ=
mass
Volume
ρwater = 1000 kg/m3 SI
ρwater = 62.4 lb/ft3 cgs
ρsubs = (substance) (ρwater)
1 ton = 2000lb
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MOMENT OF INERTIA
Moment of Inertia about the x- axis:
x2
I x = ∫ y 2 dA
x1
Moment of Inertia about the y- axis:
Ellipse
πab3
Ix =
4
πa 3b
Iy =
4
y2
I y = ∫ x 2 dA
FLUID PRESSURE
Parallel Axis Theorem
F = ∫ wh dA
F = wh A = γ h A
y1
The moment of inertia of an area with respect to any
coplanar line equals the moment of inertia of the area
with respect to the parallel centroidal line plus the area
times the square of the distance between the lines.
I x = Ixo = Ad 2
F = force exerted by the fluid on one side of
the area
h = distance of the c.g. to the surface of liquid
w = specific weight of the liquid (γ)
A = vertical plane area
Moment of Inertia for Common Geometric
Figures
Specific Weight:
Square
γ =
bh3
Ix =
3
I xo =
3
bh
12
γwater = 9.81 kN/m2 SI
γwater = 45 lbf/ft2 cgs
Triangle
bh 3
Ix =
12
I xo =
Weight
Volume
bh3
36
Circle
πr 4
I xo =
4
Half-Circle
πr 4
Ix =
8
Quarter-Circle
πr 4
Ix =
16
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MECHANICS 1
CABLES
PARABOLIC CABLES
the load of the cable of distributed horizontally along the
span of the cable.
VECTORS
Dot or Scalar product
Uneven elevation of supports
P • Q = P Q cosθ
P • Q = Px Q x + Py Q y + Pz Q z
2
2
wx
wx
H= 1 = 2
2d1
2d 2
Cross or Vector product
P × Q = P Q sin θ
i
j
k
P × Q = Px
Qx
Py
Qy
Pz
Qz
EQUILIBRIUM OF COPLANAR FORCE
SYSTEM
Conditions to attain Equilibrium:
∑F
∑F
∑M
( x − axis )
=0
( y − axis )
=0
( po int)
=0
Friction
Ff = μN
tanφ = μ
φ = angle of friction
if no forces are applied except for the weight,
φ=θ
T1 = ( wx1 ) 2 + H 2
T2 = ( wx2 ) 2 + H 2
Even elevation of supports
L
> 10
d
wL2
H=
8d
2
 wL 
2
T= 
 +H
 2 
8d 2 32d 4
S = L+
−
3L
5 L3
L = span of cable
d = sag of cable
T = tension of cable at support
H = tension at lowest point of cable
w = load per unit length of span
S = total length of cable
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CATENARY
MECHANICS 2
the load of the cable is distributed along the entire length
of the cable.
Uneven elevation of supports
RECTILINEAR MOTION
Constant Velocity
T1 = wy1
T2 = wy 2
S = Vt
H = wc
Constant Acceleration: Horizontal Motion
y1 = S1 + c
2
2
2
y2 = S 2 + c 2
2
2
 S + y1 
x1 = c ln  1

 c 
 S + y2 
x 2 = c ln  2

 c 
Span = x1 + x 2
1 2
at
2
V = V0 ± at
S = V0 t ±
V 2 = V0 ± 2aS
2
Total length of cable = S1 + S2
Even elevation of supports
T = wy
Constant Acceleration: Vertical Motion
H = wc
y = S +c
2
2
+ (sign) = body is speeding up
– (sign) = body is slowing down
2
± H = V0t −
1 2
gt
2
S + y
x = c ln 

 c 
Span = 2 x
V = V0 ± gt
Total length of cable = 2S
+ (sign) = body is moving down
– (sign) = body is moving up
V 2 = V0 ± 2 gH
2
Values of g,
general
estimate
exact
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SI (m/s2)
9.81
9.8
9.806
English (ft/s2)
32.2
32
32.16
Variable Acceleration
dS
dt
dV
a=
dt
V=
PROJECTILE MOTION
ROTATION (PLANE MOTION)
Relationships between linear & angular
parameters:
V = rω
a = rα
V = linear velocity
ω = angular velocity (rad/s)
a = linear acceleration
α = angular acceleration (rad/s2)
r = radius of the flywheel
Linear Symbol
Angular Symbol
S
V
A
t
θ
ω
α
t
Distance
Velocity
Acceleration
Time
Constant Velocity
x = (V0 cos θ )t
± y = (V0 sin θ )t −
± y = x tan θ −
1 2
gt
2
gx 2
2V0 cos 2 θ
2
Maximum Height and Horizontal Range
V0 sin θ
max ht
2g
2
y=
2
V sin 2θ
x= 0
g
2
Maximum Horizontal Range
Assume: Vo = fixed
θ = variable
2
V
Rmax = 0 ⇔ θ = 45°
g
θ = ωt
Constant Acceleration
1
θ = ω 0 t ± αt 2
2
ω = ω 0 ± αt
ω 2 = ω 0 ± 2αθ
2
+ (sign) = body is speeding up
– (sign) = body is slowing down
D’ALEMBERT’S PRINCIPLE
“Static conditions maybe produced in a body possessing
acceleration by the addition of an imaginary force called
reverse effective force (REF) whose magnitude is
(W/g)(a) acting through the center of gravity of the body,
and parallel but opposite in direction to the acceleration.”
W 
REF = ma =  a
g
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UNIFORM CIRCULAR MOTION
motion of any body moving in a circle with a constant
speed.
mV 2 WV 2
Fc =
=
r
gr
V2
ac =
r
Fc = centrifugal force
V = velocity
m = mass
W = weight
r = radius of track
ac = centripetal acceleration
g = standard gravitational acceleration
BANKING ON HI-WAY CURVES
BOUYANCY
A body submerged in fluid is subjected by an
unbalanced force called buoyant force equal to
the weight of the displaced fluid
Fb = W
Fb = γVd
Fb = buoyant force
W = weight of body or fluid
γ = specific weight of fluid
Vd = volume displaced of fluid or volume of
submerged body
Specific Weight:
γ =
Weight
Volume
γwater = 9.81 kN/m2 SI
γwater = 45 lbf/ft2 cgs
Ideal Banking: The road is frictionless
V2
tan θ =
gr
Non-ideal Banking: With Friction on the road
V2
tan(θ + φ ) =
tan φ = µ
;
gr
V = velocity
r = radius of track
g = standard gravitational acceleration
θ = angle of banking of the road
φ = angle of friction
μ = coefficient of friction
Conical Pendulum
T = W secθ
1
2π
g
h
IMPULSE AND MOMENTUM
Impulse = Change in Momentum
F∆t = mV − mV0
F = force
t = time of contact between the body and the force
m = mass of the body
V0 = initial velocity
V = final velocity
Impulse, I
F V2
tan θ =
=
W gr
f =
ENGINEERING
MECHANICS 3
Momentum, P
frequency
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I = F∆t
P = mV
Work, Energy and Power
LAW OF CONSERVATION OF MOMENTUM
“In every process where the velocity is changed, the
momentum lost by one body or set of bodies is equal to
the momentum gain by another body or set of bodies”
Work
W = F ⋅S
Force
Newton (N)
Dyne
Pound (lbf)
Distance
Meter
Centimeter
Foot
Work
Joule
ft-lbf
erg
Potential Energy
PE = mgh = Wh
Momentum lost = Momentum gained
Kinetic Energy
KE linear =
m1V1 + m2V2 = m1V1' + m2V2'
m1 = mass of the first body
m2 = mass of the second body
V1 = velocity of mass 1 before the impact
V2 = velocity of mass 2 before the impact
V1’ = velocity of mass 1 after the impact
V2’ = velocity of mass 2 after the impact
KE rotational =
1
mV 2
2
1 2
Iω → V = rω
2
I = mass moment of inertia
ω = angular velocity
Coefficient of Restitution (e)
e=
Type of collision
ELASTIC
INELASTIC
PERFECTLY
INELASTIC
Mass moment of inertia of rotational
INERTIA for common geometric figures:
V −V
V1 − V2
'
2
e
100%
conserved
Not 100%
conserved
Max Kinetic
Energy Lost
'
1
Solid sphere: I =
Kinetic Energy
0 < e >1
e=0
e =1
Special Cases
2 2
mr
5
Thin-walled hollow sphere: I =
Solid disk: I =
1 2
mr
2
1 2
mr
2
1
2
2
Hollow Cylinder: I = m( router − rinner )
2
Solid Cylinder: I =
m = mass of the body
r = radius
e=
hr
hd
2 2
mr
3
e = cot θ tan β
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Latent Heat is the heat needed by the body to change
POWER
its phase without changing its temperature.
rate of using energy
P=
W
= F ⋅V
t
1 watt = 1 Newton-m/s
1 joule/sec = 107 ergs/sec
1 hp = 550 lb-ft per second
= 33000 lb-ft per min
= 746 watts
LAW ON CONSERVATION OF ENERGY
“Energy cannot be created nor destroyed, but it
can be change from one form to another”
Q = ±mL
Q = heat needed to change phase
m = mass
L = latent heat (fusion/vaporization)
(+) = heat is entering (substance melts)
(–) = heat is leaving (substance freezes)
Latent heat of Fusion – solid to liquid
Latent heat of Vaporization – liquid to gas
Values of Latent heat of Fusion and
Vaporization,
Kinetic Energy = Potential Energy
WORK-ENERGY RELATIONSHIP
The net work done on an object always
produces a change in kinetic energy of the
object.
Work Done = ΔKE
Positive Work – Negative Work = ΔKE
Total Kinetic Energy = linear + rotation
HEAT ENERGY AND CHANGE IN PHASE
Sensible Heat is the heat needed to change the
temperature of the body without changing its phase.
Q = mcΔT
Lf = 144 BTU/lb
Lf = 334 kJ/kg
Lf ice = 80 cal/gm
Lv boil = 540 cal/gm
Lf = 144 BTU/lb
= 334 kJ/kg
Lv = 970 BTU/lb
= 2257 kJ/kg
1 calorie = 4.186 Joules
1 BTU = 252 calories
= 778 ft–lbf
LAW OF CONSERVATION OF HEAT
ENERGY
When two masses of different temperatures are
combined together, the heat absorbed by the lower
temperature mass is equal to the heat given up by the
higher temperature mass.
Q = sensible heat
m = mass
c = specific heat of the substance
ΔT = change in temperature
Specific heat values
Cwater
Cwater
Cwater
Cice
Csteam
= 1 BTU/lb–°F
= 1 cal/gm–°C
= 4.156 kJ/kg
= 50% Cwater
= 48% Cwater
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Heat gained = Heat lost
THERMAL EXPANSION
For most substances, the physical size increase with an
increase in temperature and decrease with a decrease in
temperature.
ΔL = LαΔT
ΔL = change in length
L = original length
α = coefficient of linear expansion
ΔT = change in temperature
ΔV = VβΔT
ΔV = change in volume
V = original volume
β = coefficient of volume expansion
ΔT = change in temperature
Note: In case β is not given; β = 3α
THERMODYNAMICS
In thermodynamics, there are four laws of very
general validity. They can be applied to systems
about which one knows nothing other than the
balance of energy and matter transfer.
ZEROTH LAW OF THERMODYNAMICS
stating that thermodynamic equilibrium is an
equivalence relation.
If two thermodynamic systems are in thermal
equilibrium with a third, they are also in thermal
equilibrium with each other.
FIRST LAW OF THERMODYNAMICS
about the conservation of energy
The increase in the energy of a closed system
is equal to the amount of energy added to the
system by heating, minus the amount lost in the
form of work done by the system on its
surroundings.
SECOND LAW OF THERMODYNAMICS
about entropy
The total entropy of any isolated
thermodynamic system tends to increase over
time, approaching a maximum value.
THIRD LAW OF THERMODYNAMICS,
about absolute zero temperature
As a system asymptotically approaches
absolute zero of temperature all processes
virtually cease and the entropy of the system
asymptotically approaches a minimum value.
This law is more clearly stated as: "the entropy
of a perfectly crystalline body at absolute zero
temperature is zero."
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STRENGTH OF
MATERIALS
SI
N/m2 = Pa
SIMPLE STRESS
Stress =
Units of σ
Force
Area
mks/cgs
English
Kg/cm2
lbf /m2 = psi
kN/m2 = kPa
103 psi = ksi
Axial Stress
MN/m2 = MPa
103 lbf = kips
the stress developed under the action of the force acting
axially (or passing the centroid) of the resisting area.
GN/m2 = Gpa
P
σ axial = axial
A
N/mm2 = MPa
Standard Temperature and Pressure (STP)
Paxial ┴ Area
σaxial = axial/tensile/compressive stress
P = applied force/load at centroid of x’sectional area
A = resisting area (perpendicular area)
101.325 kPa
Shearing stress
the stress developed when the force is applied parallel to
the resisting area.
=
=
=
=
=
=
=
14.7 psi
1.032 kgf/cm2
780 torr
1.013 bar
1 atm
780 mmHg
29.92 in
P
σs =
A
Thin-walled Pressure Vessels
Pappliedl ║ Area
σT =
σs = shearing stress
P = applied force or load
A = resisting area (sheared area)
A. Tangential stress
B. Longitudinal stress (also for Spherical)
Bearing stress
the stress developed in the area of contact (projected
area) between two bodies.
σb =
ρ r ρD
=
t
2t
σL =
ρ r ρD
=
2t
4t
P P
=
A dt
P ┴ Abaering
σb = bearing stress
P = applied force or load
A = projected area (contact area)
d,t = width and height of contact, respectively
σT = tangential/circumferential/hoop stress
σL = longitudinal/axial stress, used in spheres
r = outside radius
D = outside diameter
ρ = pressure inside the tank
t = thickness of the wall
F = bursting force
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SIMPLE STRAIN / ELONGATION
Types of elastic deformation:
Strain – ratio of elongation to original length
a. Due to axial load
HOOKE’S LAW ON AXIAL DEFORMATION
“Stress is proportional to strain”
σαε
σ = Yε
Young ' s Modulus of Elasticity
σ = Eε
Modulus of Elasticity
σ s =E sε s
Modulus in Shear
σ V =EV εV
Bulk Modulus of Elasticity
1
Ev
compressibility
δ =
ε=
ε = strain
δ = elongation
L = original length
δ
L
Elastic Limit – the range beyond which the material
WILL NOT RETURN TO ITS ORIGINAL SHAPE when
unloaded but will retain a permanent deformation
Yield Point – at his point there is an appreciable
elongation or yielding of the material without any
corresponding increase in load; ductile materials and
continuous deformation
Ultimate Strength – it is more commonly called
ULTIMATE STRESS; it’s the hishes ordinate in the curve
Rupture Strength/Fracture Point – the stress at failure
PL
AE
δ = elongation
P = applied force or load
A = area
L = original length
E = modulus of elasticity
σ = stress
ε = strain
b. Due to its own mass
ρgL2 mgL
δ =
=
2E
2 AE
δ = elongation
ρ = density or unit mass of the body
g = gravitational acceleration
L = original length
E = modulus of elasticity or Young’s modulus
m = mass of the body
c. Due to changes in temperature
δ = Lα (T f − Ti )
δ = elongation
α = coefficient of linear expansion of the body
L = original length
Tf = final temperature
Ti = initial temperature
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d. Biaxial and Triaxial Deformation
µ=−
εy
εx
=−
εz
εx
Power delivered by a rotating shaft:
P = Tω
Prpm = 2πTN
rps
TORSIONAL SHEARING STRESS
2πTN
60
2πTN
Php =
550
2πTN
Php =
3300
Torsion – refers to twisting of solid or hollow
rotating shaft.
T = torque
N = revolutions/time
Solid shaft
HELICAL SPRINGS
μ = Poisson’s ratio
μ = 0.25 to 0.3 for steel
= 0.33 for most metals
= 0.20 for concrete
μmin = 0
μmax = 0.5
Prpm =
ft − lb
sec
ft − lb
min
16T
πd 3
τ =
16 PR 
d 
1
+


πd 3  4 R 
16TD
π (D 4 − d 4 )
τ =
16 PR  4m − 1 0.615 
+


m 
πd 3  4m − 4
τ=
Hollow shaft
τ =
rpm
τ = torsional shearing stress
T = torque exerted by the shaft
D = outer diameter
d = inner diameter
where,
Maximum twisting angle of the shaft’s fiber:
elongation,
θ=
TL
JG
θ = angular deformation (radians)
T = torque
L = length of the shaft
G = modulus of rigidity
J = polar moment of inertia of the cross
m=
Dmean Rmean
=
d
r
64 PR 3 n
δ =
Gd 4
τ = shearing stress
δ = elongation
R = mean radius
d = diameter of the spring wire
n = number of turns
G = modulus of rigidity
πd 4
J=
→ Solid shaft
32
π (D 4 − d 4 )
J=
→ Hollow shaft
32
Gsteel = 83 GPa;
Esteel = 200 GPa
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Mode of Interest
Annually
Semi-Annually
Quarterly
Semi-quarterly
Monthly
Semi-monthly
Bimonthly
Daily
ENGINEERING
ECONOMICS 1
SIMPLE INTEREST
I = Pin
F = P (1 + in)
P = principal amount
F = future amount
I = total interest earned
i = rate of interest
n = number of interest periods
m
1
2
4
8
12
24
6
360
Shortcut on Effective Rate
Ordinary Simple Interest
n=
days
360
n=
months
12
Exact Simple Interest
days
→ ordinary year
365
days
n=
→ leap year
366
n=
ANNUITY
Note: interest must be effective rate
COMPOUND INTEREST
F = P(1 + i ) n
Nominal Rate of Interest
i=
NR
⇔ n = mN
m
Effective Rate of Interest
ER = (1 + i ) − 1
m
Ordinary Annuity
A [(1 + i ) n − 1]
F=
i
A [(1 + i ) n − 1]
P=
(1 + i ) n i
m
 NR 
ER = 1 +
 −1
m 

ER ≥ NR ; equal if Annual
i = rate of interest per period
NR = nominal rate of interest
m = number of interest periods per year
n = total number of interest periods
N = number pf years
ER = effective rate of interest
A = uniform periodic amount or annuity
Perpetuity or Perpetual Annuity
P=
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A
i
LINEAR / UNIFORM GRADIENT SERIES
ENGINEERING
ECONOMICS 2
DEPRECIATION
P = PA + PG
Straight Line Method (SLM)
d=
G  (1 + i) n − 1
n 
PG = 
−

i  i(1 + i) n
(1 + i) n 
FG =
C0 − Cn
n
Dm = md
G  (1 + i ) n − 1 
− n

i
i 

Cm = C0 – Dm
d = annual depreciation
C0 = first cost
Cm = book value
Cn = salvage or scrap value
n = life of the property
Dm = total depreciation after m-years
m = mth year
1

n
AG = G  −

n
 i (1 + i) − 1
Perpetual Gradient
G
PG = 2
i
Sinking Fund Method (SFM)
UNIFORM GEOMETRIC GRADIENT
d=
(C 0 − C n )i
(1 + i ) n − 1
d [(1 + i ) m − 1]
Dm =
i
Cm = C0 – Dm
 (1 + q) n (1 + i ) − n − 1
P = C

q−i


if q ≠ i
 (1 + q) n − (1 + i ) n 
F = C

q−i


if q ≠ i
Cn
P=
1+ q
q=
Cn(1 + i) n
P=
1+ q
i = standard rate of interest
Sum of the Years Digit (SYD) Method
 2(n − m + 1) 
d m = (C 0 − C n ) 

 n(n + 1) 
if q = i
sec ond
−1
first
C = initial cash flow of the geometric gradient series
which occurs one period after the present
q = fixed percentage or rate of increase
 (2n − m + 1)m 
Dm = (C 0 − C n ) 

 n( n + 1) 
SYD =
n(n + 1)
2
Cm = C0 – Dm
SYD = sum of the years digit
dm = depreciation at year m
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Declining Balance Method (DBM)
k =1− n
BONDS
P = Panuity = Pcpd int erest
Cn
Co
Zr[(1 + i ) n − 1]
C
P=
+
n
(1 + i ) i
(1 + i) n
Matheson Formula
C
k =1− m m
Co
C m = C 0 (1 − k ) m
d m = kC 0 (1 − k ) m −1
Dm = C0 – Cm
k = constant rate of depreciation
P = present value of the bond
Z = par value or face value of the bond
r = rate of interest on the bond per period
Zr = periodic dividend
i = standard interest rate
n = number of years before redemption
C = redemption price of bond
BREAK-EVEN ANALYSIS
Total income = Total expenses
CAPITALIZED AND ANNUAL COSTS
CC = C 0 + P
CC = Capitalized Cost
C0 = first cost
P = cost of perpetual maintenance (A/i)
AC = d + C 0 (i ) + OMC
AC = Annual Cost
d = Annual depreciation cost
i = interest rate
OMC = Annual operating & maintenance cost
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