Math-Formula hacks engineering PDF

Preview

Summary

math made easier with a summary of formulas...

Description

ALGEBRA 1

Man-hours (is always assumed constant)

LOGARITHM

x

log b N

N

(Wor ker s 1 )(time1 ) quantity.of .work 1

bx

Properties

log(xy )

ALGEBRA 2

log x

log y

x y

log x

log y

log x n

n log x

log

(Wor ker s2 )( time2 ) quantity.of .work 2

UNIFORM MOTION PROBLEMS

S

log x log b log a a 1

Traveling with the wind or downstream:

log b x

Vtotal

REMAINDER AND FACTOR THEOREMS

Vtotal

f ( x) (x r )

100h 10t u

QUADRATIC EQUATIONS

where:

Bx C B

V1 V2

DIGIT AND NUMBER PROBLEMS

Remainder Theorem: Remainder = f(r) Factor Theorem: Remainder = zero

Root

V1 V2

Traveling against the wind or upstream:

Given:

Ax2

Vt

0 B 2 4 AC 2A

2-digit number

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

PDF created with pdfFactory trial version www pdffactory com

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

an

difference of any 2 no.’s is constant calcu function: LINEAR (LIN)

( n m) d

am

d

a2

a1

S

n (a1 2

S

n [2a1 2

a3

an )

nth term Common difference

a2 ,...etc

an

(n 1)d ]

Sum of ALL terms

RATIO of any 2 adj, terms is always constant Calcu function: EXPONENTIAL (EXP)

am r n

If the number of equations is less than the number of unknowns, then the equations are called “Diophantine Equations”.

Sum of ALL terms

GEOMETRIC PROGRESSION (GP) -

DIOPHANTINE EQUATIONS

ALGEBRA 3 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

th

m

n term

Permutation of n objects taken r at a time

n!

nPr r

S

S

a2 a1

a3 a2

( n r )!

ratio Permutation of n objects taken n at a time

a1 (r n 1) r 1 a1 (1 r n ) 1 r

r 1

r 1

S um of ALL terms, r >1

nPn

n!

Permutation of n objects with q,r,s, etc. objects are alike

P

S um 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 1.0

Discriminant

Equation**

Circle

B2 - 4AC < 0, A = C

A=C

Parabola

B2 - 4AC = 0

Ellipse

B2 - 4AC < 0, A

Ellipse Hyperbola

Hyperbola B2 - 4AC > 0

E 2A

Radius of the circle:

General Equation of a Conic Section:

Circle

D ; k 2A

C

A C same sign Sign of A opp. of B A or C = 0

h2

k2

F A

r

or

1 2

D2

E2

4F

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

PDF created with pdfFactory trial version www pdffactory com

D C

AXIS VERTICAL:

ELLIPSE

Ax2 + Dx + Ey + F = 0 Coordinates of vertex (h,k):

D 2A

h

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:

LR

E A

STANDARD EQUATIONS: Major axis is horizontal:

( x h )2 a2

STANDARD EQUATIONS:

k )2

(y

b2

1

Opening to the right: Major axis is vertical: 2

(x h )2 b2

(y – k) = 4a(x – h) Opening to the left:

(y – k)2 = –4a(x – h)

Ax2 + Cy2 + Dx + Ey + F = 0

(x – h) 2 = 4a(y – k)

Coordinates of the center:

h

Opening downward:

(x – h) 2 = –4a(y – k) Latus Rectum (LR) a chord drawn to the axis of symmetry of the curve.

D ;k 2A

E 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

1

General Equation of an Ellipse:

Opening upward:

LR= 4a

(y k )2 a2

for a parabola

c

PDF created with pdfFactory trial version www pdffactory com

a 2 b2

Length of latus rectum: 2

Equation of Asymptote:

2b a

LR

(y – k) = m(x – h) Transverse axis is horizontal:

Eccentricity:

e

c a

m

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:

a2

c d = distance from center to directrix a = distance from center to vertex c = distance from center to focus

Length of latus rectum:

( x h) 2 a2

( y k )2 b2

2b2 a

LR

STANDARD EQUATIONS Transverse axis is horizontal

b2

Eccentricity:

1

e

c a

a d

Transverse axis is vertical:

( y k )2 a2

( x h) 2 b2

1

POLAR COORDINATES SYSTEM x = r cos

GENERAL EQUATION

y = r sin

Ax 2 – Cy2 + Dx + Ey + F = 0 Coordinates of the center:

h

D ; k 2A

E 2C

If C is negative, then: a2 = C, b2 = A If A is negative, then: a2 = A, b2 = C

PDF created with pdfFactory trial version www pdffactory com

x2

r tan

y2 y x

SPHERICAL TRIGONOMETRY Important propositions 1. If two angles of a spherical triangle are equal, the sides opposite are equal; and conversely.

Napier’s Rules 1. The sine of any middle part is equal to the product of the cosines of the opposite parts.

Co-op 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 A

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.

sin b sin B

sin c sin C

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 acos b sin asin bcos C Law of Cosines for angles:

cos A cos B

cos B cos C sin B sin C cos a cos A cos C sin A sin C cos b

cos C

cos A cos B sin A sin B cos c

PDF created with pdfFactory trial version www pdffactory com

AREA OF SPHERICAL TRIANGLE

A

R2E 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 du dv d (uv ) u v dx dx dx du dv v u d u dx dx 2 dx v v d n 1 du ( u ) nu n dx dx du d dx u dx 2 u du c d c dx 2 dx u u d u du ( a ) a u ln a dx dx du d u (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 ) cosu dx dx d du (cos u) sin u dx dx d du (tan u ) sec 2 u dx dx d du 2 (cot u) csc u dx dx d du (sec u ) sec u tan u dx dx d du (csc u) csc u cot u dx dx 1 d du (sin 1 u) 2 dx 1 u dx

PDF created with pdfFactory trial version www pdffactory com

d 1 (cos u ) dx d 1 (tan u) dx d (cot 1 u) dx d 1 (sec u) dx d 1 (csc u) dx

du 1 u dx 1 du 1 u2 dx 1 du 2 1 u dx du 1 2 u u 1 dx 1

1

du u u 1 dx d du (sinh u ) cosh u dx dx d du (cosh u) sinh u dx dx d du (tanh u) sec h2 u dx dx d du 2 (coth u ) csc h u dx dx d du (sec hu) sec hu tanh u dx dx d du (csc hu ) csc hu cothu dx dx 1 d du 1 (sinh u ) 2 dx u 1 dx d 1 (cosh u) dx

2

1

du u 1 dx 1 du d 1 (tanh u ) 1 u 2 dx dx 1 du d (sinh 1 u ) 2 dx u 1 dx 1 d du 1 (sec h u ) 2 dx u 1 u dx d 1 (csc h u ) dx

DIFFERENTIAL CALCULUS

2

LIMITS Indeterminate Forms

0 , 0

, (0)( ),

du u 1 u2 dx

0

, 1

L’Hospital’s Rule

Lim x

a

f ( x) g (x )

Lim x

a

f ' ( x) g '(x)

Lim x

a

f "(x ) ..... g " (x )

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 MIN

2

1

- , 00 ,

INFLECTION

0 -

Y” (-) dec

Concavity down

(+) inc

up -

No change

HIGHER DERIVATIVES nth derivative of xn

dn (xn ) n dx

n!

nth derivative of xe n

dn n ( ) xe n dx PDF created with pdfFactory trial version www pdffactory com

( x n) e X

TIME RATE the rate of change of the variable with respect to time

dx dt

= increasing rate

dx dt = decreasing rate APPROXIMATION AND ERRORS If “dx” is the error in the measurement of a quantity x, then “dx/x” is called the RELATIVE ERROR.

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

du

RADIUS OF CURVATURE 3

R

[1 ( y ' ) 2 ]2 y"

INTEGRAL CALCULUS 1 du

u C

adu

au

du u

f (u )du

un 1 C ..............(n n 1 ln u C

2au u

a du e u du

eu

sin udu cos udu 2 sec udu

C cos u C sin u C tan u C

csc 2 udu

cos

1

cosh udu

sinh u C

sec h 2udu csc h 2udu

g (u )du

C

tanh u C coth u C

sec hu tanh udu

sec hu C

csc hu coth udu

csc hu C

coth udu

ln sinh u

du 1)

u a

1

cosh u C

u2 a2 du 2

a du

2

sinh

1

cosh

1

C C

u a

C

u a

C

a 1 sinh 1 a u

C u u 2 a2 du u 1 C ..............u tanh 1 2 2 a u a a du u 1 C ..............u coth 1 2 2 a u a a udv uv vdu

cot u C

sec u tan udu sec u C cscu cot udu

2

sinh udu

u au C ln a

u

u a

tanh udu ln cosh u

C

[ f (u ) g (u )]du u n du

1

C a2 u2 du u 1 C tan 1 2 2 a a a u u du 1 C sec 1 a u u2 a2 a sin

cscu C

PDF created with pdfFactory trial version www pdffactory com

a a

PLANE AREAS

CENTROIDS

Plane Areas bounded by a curve and the coordinate axes:

Half a Parabola

x2

A

y (curve) dx

x

x( curve) dy

y

x1 y2

A y1

Plane Areas bounded by a curve and the coordinate axes: x2

A

( y( up)

y( down) ) dx

x1

( x( right )

y

x( left ) ) dy x

y1

Plane Areas bounded by polar curves:

A

Whole Parabola

2 h 5

Triangle

y2

A

3 b 8 2 h 5

1 2 2 r d 2 1

y

1 b 3 1 h 3

2 b 3 2 h 3

LENGTH OF ARC

CENTROID OF PLANE AREAS (VARIGNON’S THEOREM) Using a Vertical Strip: x2

A x

A y

x2

dA x

S

x1

x1

x2

y2

y dA 2 x1

S

A x

x dA 2 y1

z

2

S z1

y2

A y

1

1

dx dy dy

2

y1

Using a Horizontal Strip: y2

2

dy dx dx

dA y y1

PDF created with pdfFactory trial version www pdffactory com

dx dz

2

2

dy dz dz

INTEGRAL CALCULUS 2 TIP 1: Problems will usually be of this nature: “Find the area bounded by” “Find the area revolved around..” TIP 2: Integrate only when the shape is IRREGULAR, otherwise use the prescribed formulas

CENTROIDS OF VOLUMES x2

V

x

dV

x

dV

y

x1 y2

V

y y1

WORK BY INTEGRATION Work = force × distance x2

VOLUME OF SOLIDS BY REVOLUTION

W

Circular Disk Method

y2

Fdx x1

Fdy ; where F = k x y1

x2

Work done on spring

R 2dx

V

W

x1

Cylindrical Shell Method y2

V

2

RL dy

1 k (x2 2 2

x12 )

k = spring constant x1 = initial value of elongation x2 = final value of elongation

Work done in pumping liquid out of the container at its top

y1

Circular Ring Method

Work = (density)(volume)(distance)

x2

(R 2 r 2 )dx

V x1

PROPOSITIONS OF PAPPUS

Force = (density)(volume) = v Specific Weight:

Weight Volume

First Proposition: If a plane arc is revolved about a 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.

A A

S 2 r dS 2 r

= 9.81 kN/m 2 SI 2 cgs water = 45 lbf/ft

water

Density:

mass Volume

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 V

A 2 r dA 2 r

= 1000 kg/m 3 SI 3 cgs water = 62.4 lb/ft = (substance) ( water) subs 1 ton = 2000lb water

PDF created with pdfFactory trial version www pdffactory com

MOMENT OF INERTIA Moment of Inertia about the x- axis:

Ix

x2

y 2dA

Ix

Ellipse

x1

Iy

Moment of Inertia about the y- axis:

ab 3 4 a3b 4

y2

x 2dA

Iy

FLUID PRESSURE