Algebra, Trigonometry, and Set Theory PDF

ALGEBRA MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING LAW OF EXPONENTS WORD PROBLEMS CLOCK PROBLEMS Product of Powers 𝒙𝒎 𝒙𝒏 = 𝒙𝒎...

ALGEBRA MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING LAW OF EXPONENTS WORD PROBLEMS CLOCK PROBLEMS Product of Powers 𝒙𝒎 𝒙𝒏 = 𝒙𝒎 𝒏 𝑴𝒊𝒏𝒖𝒕𝒆 𝑯𝒂𝒏𝒅 (𝜽𝟏 ) = 𝟔𝒕 Quotient of Powers 𝒙𝒎 /𝒙𝒏 = 𝒙𝒎 𝒏 𝑯𝒐𝒖𝒓 𝑯𝒂𝒏𝒅 (𝜽𝟐 ) = 𝜽𝑺.𝑯. + 𝟎. 𝟓𝒕 Power of a Power (𝒙𝒎 )𝒏 = 𝒙𝒎𝒏 MONEY PROBLEMS 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 = 𝑪𝒂𝒑𝒊𝒕𝒂𝒍 (𝑹𝒂𝒕𝒆 %) Power of a Product (𝒙𝒚)𝒎 = 𝒙𝒎 𝒚𝒎 𝑷𝒓𝒐𝒇𝒊𝒕 = 𝑺𝒆𝒍𝒍𝒊𝒏𝒈 𝑷𝒓𝒊𝒄𝒆 − 𝑪𝒐𝒔𝒕 𝒙 𝒎 𝒙𝒎 𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕𝒆𝒅 𝑷𝒓𝒊𝒄𝒆 = 𝑶𝒓𝒊𝒈𝒊𝒏𝒂𝒍 𝑷𝒓𝒊𝒄𝒆 (𝟏 − 𝑫%) Power of a Quotient = 𝒚 𝑦 𝟏 Negative Exponent 𝒙 𝒏 = 𝒏 MOTION PROBLEM 𝒙 OBJ. – ENV. OBJ. – OBJ. RELATION RELATION Zero Exponent 𝟎 𝒙 =𝟏 SAME DIRECTION 𝑽𝟏 + 𝑽𝟐 𝑽𝟏 − 𝑽𝟐 OPPOSITE 𝑽𝟏 − 𝑽𝟐 𝑽𝟏 + 𝑽𝟐 DIRECTION LAW OF LOGARITHMS WORK PROBLEM 𝑾𝒐𝒓𝒌 Log of a Product 𝐥𝐨𝐠 𝒂 𝒙𝒚 = 𝒍𝒐𝒈𝒂 𝒙 + 𝐥𝐨𝐠 𝒂 𝒚 𝑹𝒂𝒕𝒆 = 𝑾𝒐𝒓𝒌 = 𝑵𝒎𝒂𝒏 × 𝒉𝒐𝒖𝒓𝒔 𝑻𝒊𝒎𝒆 𝒙 Log of a Quotient 𝐥𝐨𝐠 𝒂 = 𝒍𝒐𝒈𝒂 𝒙 − 𝐥𝐨𝐠 𝒂 𝒚 MIXTURE PROBLEMS 𝒚 Log of a Power 𝐥𝐨𝐠 𝒂 𝒙𝒏 = 𝒏 𝒍𝒐𝒈𝒂 𝒙 Log of 1 𝐥𝐨𝐠 𝒂 𝟏 = 𝟎 Log of the Base 𝐥𝐨𝐠 𝒂 𝒂 = 𝟏 𝑨(%) + 𝑩(%) = (𝑨 + 𝑩)(%) Equivalence 𝒂𝒙 = 𝒚, 𝒕𝒉𝒆𝒏 𝒙 = 𝐥𝐨𝐠 𝒂 𝒚 𝟏 VARIATION PROBLEMS Log of a Reciprocal 𝐥𝐨𝐠 𝒂 = −𝒍𝒐𝒈𝒂 𝒙 Direct Variation Inverse Variation 𝒙 𝒚 = 𝒌𝒙 𝒌 𝒚= 𝒙 LAW OF NATURAL LOGARITHMS Joint Variation Combined Variation 𝒚 = 𝒌𝒙𝒛 𝒌𝒛 𝒚= 𝒙 Ln of a Product 𝒍𝒏 𝒙𝒚 = 𝒍𝒏 𝒙 + 𝒍𝒏 𝒚 𝒙 Ln of a Quotient 𝑙𝑛 𝒚 = 𝑙𝑛 𝒙 − 𝑙𝑛 𝒚 SEQUENCE AND SERIES Ln of a Power 𝑙𝑛 𝒙𝒏 = 𝒏 𝑙𝑛𝒙 ARITHMETIC PROGRESSION Ln of 1 𝑙𝑛 𝟏 = 𝟎 Nth Term 𝒂𝒏 = 𝒂𝟏 + (𝒏 − 𝟏)𝒅 𝒏 Ln of the Base 𝑙𝑛 𝒂 = 𝟏 Summation 𝑺𝒏 = (𝒂𝟏 + 𝒂𝒏 ) 𝟐 𝒙+𝒚 Mean 𝑨𝑴 = Equivalence 𝒂𝒙 = 𝒚, 𝒕𝒉𝒆𝒏 𝒙 = 𝐥𝐨𝐠 𝒂 𝒚 𝟐 𝟏 GEOMETRIC PROGRESSION Ln of a Reciprocal 𝑙𝑛 = −𝒍𝒏 𝒙 𝒙 Nth Term 𝒂𝒏 = 𝒂𝟏 𝒓𝒏 𝟏 𝒂𝟏 (𝟏 − 𝒓𝒏 ) Summation 𝑺𝒏 = SET THEORY (𝟏 − 𝒓) 𝒂𝟏 Infinite 𝑺𝒏 = , |𝑟| < 1.0 UNION (A u B) INTERSECTION (A ∩ B) (𝟏 − 𝒓) Set of all things that are Set of all things that are Mean 𝑮𝑴 = 𝒏 𝒙𝟏 𝒙𝟐 … 𝒙𝒏 members of A or B members of A and B 𝑮𝑴𝟐 = (𝑨𝑴)(𝑯𝑴) HARMONIC PROGRESSION Nth Term Reciprocals form an arithmetic progression 𝟏 𝟏 𝟏 + 𝒙 𝒚 COMPLEMENT (A’ or AC) RELATIVE COMPLEMENT Mean 𝑯𝑴 = 𝟐 Set of all things that are not (A\B or A-B) in A Set of all things the belongs to A but not to B TRIGONOMETRY MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING Sum of Two Angles ANGLE OF MEASUREMENTS sin(𝐴 + 𝐵) = sin 𝐴 cos 𝐵 + cos 𝐴 sin 𝐵 SEXAGESIMAL CENTESIMAL CIRCULAR MILLIRADIAN 𝟏 𝒓𝒆𝒗 = 𝟑𝟔𝟎° = 𝟒𝟎𝟎𝒈𝒓𝒂𝒅𝒔 = 𝟐𝝅 𝒓𝒂𝒅 = 𝟔𝟒𝟎𝟎𝒎𝒊𝒍𝒔 cos(𝐴 + 𝐵) = cos 𝐴 cos 𝐵 − sin 𝐴 sin 𝐵 tan 𝐴 + tan 𝐵 tan(𝐴 + 𝐵) = ANGLE 1 − 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 Acute Angle 𝜽 < 𝟗𝟎° Difference of Two Angles Right Angle 𝜽 = 𝟗𝟎° sin(𝐴 − 𝐵) = sin 𝐴 cos 𝐵 − cos 𝐴 sin 𝐵 Obtuse Angle 𝜽 > 𝟗𝟎° Straight Angle 𝜽 = 𝟏𝟖𝟎° Reflex Angle 𝜽 > 𝟏𝟖𝟎° cos(𝐴 − 𝐵) = cos 𝐴 cos 𝐵 + sin 𝐴 sin 𝐵 Complementary Angle 𝜽𝑨 + 𝜽𝑩 = 𝟗𝟎° tan 𝐴 − tan 𝐵 Supplementary Angle 𝜽𝑨 + 𝜽𝑩 = 𝟏𝟖𝟎° tan(𝐴 − 𝐵) = 1 + 𝑡𝑎𝑛 𝐴 𝑡𝑎𝑛 𝐵 Explementary Angle 𝜽𝑨 + 𝜽𝑩 = 𝟑𝟔𝟎° Double Angle TRIGONOMETRIC IDENTITIES sin(2𝐴) = 2sin 𝐴 cos 𝐴 cos(2𝐴) = cos 𝐴 − sin 𝐴 2 tan 𝐴 tan(2𝐴) = 1 − tan 𝐴 OTHER TRIGONOMETRIC FUNCTIONS Quotient Identities 𝑠𝑖𝑛 𝑐𝑜𝑠 𝑐𝑜𝑡 𝑡𝑎𝑛 = 𝑠𝑖𝑛 = 𝑐𝑜𝑠 = 𝑐𝑜𝑠 𝑐𝑜𝑡 𝑐𝑠𝑐 𝑐𝑠𝑐 𝑠𝑒𝑐 𝑡𝑎𝑛 𝑐𝑜𝑡 = 𝑐𝑠𝑐 = 𝑠𝑒𝑐 = 𝑠𝑒𝑐 𝑡𝑎𝑛 𝑠𝑖𝑛 Product Identities 𝑡𝑎𝑛 = sin 𝑠𝑒𝑐 𝑠𝑖𝑛 = tan 𝑐𝑜𝑠 𝑐𝑜𝑠 = sin 𝑐𝑜𝑡 𝑐𝑜𝑡 = cos 𝑐𝑠𝑐 𝑐𝑠𝑐 = cot 𝑠𝑒𝑐 𝑠𝑒𝑐 = csc 𝑡𝑎𝑛 Reciprocal Identities 1 1 1 𝑡𝑎𝑛 = 𝑠𝑖𝑛 = 𝑐𝑜𝑠 = 𝑐𝑜𝑡 𝑐𝑠𝑐 𝑠𝑒𝑐 Versine vers A = 1 − cos 𝐴 1 1 1 𝑐𝑜𝑡 = 𝑐𝑠𝑐 = 𝑠𝑒𝑐 = 𝑡𝑎𝑛 𝑠𝑖𝑛 𝑐𝑜𝑠 Coversed Sine Reciprocal Identities cvs A = 1 − sin 𝐴 1 1 1 𝑡𝑎𝑛 = 𝑠𝑖𝑛 = 𝑐𝑜𝑠 = 𝑐𝑜𝑡 𝑐𝑠𝑐 𝑠𝑒𝑐 Exsecant exsec A = sec 𝐴 − 1 1 1 1 𝑐𝑜𝑡 = 𝑐𝑠𝑐 = 𝑠𝑒𝑐 = 𝑡𝑎𝑛 𝑠𝑖𝑛 𝑐𝑜𝑠 Excosecant Pythagorean Identities excsc A = csc 𝐴 − 1 sin + cos = 1 Haversine: 1 − cos 𝐴 hav A = tan + 1 = sec 2 1 + cot = csc PLANE TRIGONOMETRY MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING TYPES OF TRIANGLES BASED ON SIDES OBLIQUE TRIANGLES EQUILATERAL ISOSCELES SCALENE Sine Law 𝒂 𝒃 𝒄 = = 𝒔𝒊𝒏 𝑨 𝒔𝒊𝒏 𝑩 𝒔𝒊𝒏 𝑪 Cosine Law 𝒂𝟐 = 𝒃𝟐 + 𝒄𝟐 − 𝟐𝒃𝒄 𝒄𝒐𝒔 𝑨 𝒃𝟐 = 𝒂𝟐 + 𝒄𝟐 − 𝟐𝒂𝒄 𝐜𝐨𝒔 𝑩 TYPES OF TRIANGLES BASED ON ANGLES 𝒄𝟐 = 𝒂𝟐 + 𝒃𝟐 − 𝟐𝒂𝒃 𝒄𝒐𝒔 𝑪 RIGHT OBLIQUE COMPLIMENTARY FUNCTIONS ACUTE OBTUSE 𝒔𝒊𝒏 𝜽 = 𝒄𝒐𝒔 (𝟗𝟎 − 𝜽) 𝒕𝒂𝒏 𝜽 = 𝒄𝒐𝒕 (𝟗𝟎 − 𝜽) 𝒔𝒆𝒄 𝜽 = 𝒄𝒔𝒄 (𝟗𝟎 − 𝜽) WAVE CHARACTERISTICS Area of Triangle 𝟏 Side ⊥ Vertex 𝑨= 𝒃𝒉 𝟐 𝟏 Sides with Included Angle 𝑨= 𝒂𝒃 𝒔𝒊𝒏 𝜽 𝟐 Heron’s Formula 𝑨= 𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄) In Circle 𝑨𝒕 = 𝒓𝒔 𝒂𝒃𝒄 Circumcircle 𝑨𝒕 = 𝟒𝒓 Excircle 𝑨𝒕 𝒓(𝒔 − 𝒂) Amplitude (𝑨) – greatest distance of any point on the graph from the horizontal line. Period (𝑻) – interval over which the graph of a function repeats. RIGHT TRIANGLES Frequency (𝝎) – number of repetitions/cycles per unit of time or 𝟏 𝑻 Phase Shift ( /𝑩 ) – how far the function has shifted horizontally 𝑪 Sine, Cosine, and Tangent Formulas from the usual position. 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 Vertical Shift (𝑫) – how far the function moves up and down. SOH 𝒔𝒊𝒏 𝜽 = 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 CAH 𝒄𝒐𝒔 𝜽 = Function Period Amplitude 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆 𝒐𝒑𝒑𝒐𝒔𝒊𝒕𝒆 𝟐𝝅 TOA 𝒕𝒂𝒏 𝜽 = 𝒚 = 𝑨 𝒔𝒊𝒏 (𝑩𝒙 + 𝑪) + 𝑫 𝑨 𝒂𝒅𝒋𝒂𝒄𝒆𝒏𝒕 𝑩 𝟐𝝅 Pythagorean Theorem 𝒚 = 𝑨 𝒄𝒐𝒔 (𝑩𝒙 + 𝑪) + 𝑫 𝑨 𝑩 𝒂𝟐 + 𝒃𝟐 = 𝒄𝟐 𝝅 𝒚 = 𝑨 𝒕𝒂𝒏 (𝑩𝒙 + 𝑪) + 𝑫 𝑩 PLANE GEOMETRY MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING TERMINOLOGIES RADIUS OF CIRCLES Inscribed Circles  Plane Geometry – deals with the properties of plane figures or 𝑨 𝒓𝒊 = geometrical shapes of two dimensions such as angles, triangles, 𝒔 squares, polygons, and conic sections. Circumscribing Circles  Polygon – a plane figure with three or more angles with straight 𝒂𝒃𝒄 𝒂 𝒃 𝒄 lines as sides. 𝒓𝒄 = = = = 𝟒𝑨 𝟐 𝒔𝒊𝒏 𝑨 𝟐 𝒔𝒊𝒏 𝑩 𝟐 𝒔𝒊𝒏 𝑪  Regular Polygon – a polygon whose angles and sides are all equal. Escribed Circles  Similar Polygon – two polygons are similar if their corresponding 𝑨 angles are equal, and their corresponding sides are 𝒓𝒆 = 𝒔−𝒂 proportional.  Convex Polygon - polygon having each interior angle less than 180°. QUADRILATERALS  Concave Polygon – polygon having an interior angle greater than 180°.  Triangles – a polygon with three sides that are contained in a PERIMETER plane. 𝑷= 𝒂+𝒃+𝒄+𝒅  Quadrilaterals – a portion of a plane bounded by four straight lines, also known as quadrangle or tetragon. AREA 𝟏  Parallelogram (Rhomboid) – a quadrilateral whose opposite 𝑨𝟏 = 𝑫 𝑫 𝐬𝐢𝐧 𝜽 sides are parallel. 𝟐 𝟏 𝟐  Rhombus – a parallelogram with four equal sides. 𝜽+𝜶 𝟐  Rectangle – a parallelogram whose angles are right angles. 𝑨𝟐 = (𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄)(𝒔 − 𝒅) − 𝒂𝒃𝒄𝒅 𝒄𝒐𝒔 𝟐  Square – a rectangle with equal sides.  Trapezoid – a quadrilateral with only two sides parallel. PTOLEMY’S THEOREM  Cyclic Quadrilateral – a quadrilateral whose vertices lie on the 𝒂𝒄 + 𝒃𝒅 = 𝑫𝟏 𝑫𝟐 circumference of a circle. RADIUS OF CIRCUMSCRIBING CIRCLES GENERAL POLYGONS (𝒂𝒃 + 𝒄𝒅)(𝒂𝒄 + 𝒃𝒅)(𝒂𝒅 + 𝒃𝒄) 𝒓𝒄 = 𝟒𝑨 SUM OF INTERIOR ANGLES ∑𝜽 = 𝟏𝟖𝟎°(𝒏 − 𝟐) CIRCLES SUM OF EXTERIOR and CENTRAL ANGLES CIRCUMFERENCE ∑𝜷 = 𝟑𝟔𝟎° 𝑪 = 𝟐𝝅𝒓 = 𝝅𝒅 NUMBER OF DIAGONAL LINES AREA 𝒏(𝒏 + 𝟑) 𝝅 𝟐 𝑫= 𝑨 = 𝝅𝒓𝟐 = 𝒅 𝟐 𝟒 PERIMETER OF REGULAR POLYGONS ARC LENGTH In Degrees: In Radians: 𝑷 = 𝒔𝒏 𝜽 𝒔 = 𝜽𝒓 𝒔= 𝝅𝒓 𝟏𝟖𝟎 AREA 𝒔𝟐 𝒏 AREA OF A SECTOR 𝑨= In Degrees: In Radians: 𝟏𝟖𝟎 𝟒 𝒕𝒂𝒏 𝜽 𝜽 𝟐 𝒏 𝑨𝒔 = 𝝅𝒓𝟐 𝑨𝒔 = 𝒓 𝟑𝟔𝟎 𝟐 TRIANGLES THEOREMS ON CIRCLE AREA Central Angle Thales Theorem 𝟏 𝑨𝟏 = 𝒂𝒃 𝒔𝒊𝒏 𝜽 𝟐 𝟏 𝑨𝟐 = 𝒃𝒉 𝟐 𝑨𝟑 = 𝒔(𝒔 − 𝒂)(𝒔 − 𝒃)(𝒔 − 𝒄) LINES OF A TRIANGLE Median - Centroid Angle Bisector – In Center Cross Chord Theorem Secant – Secant Theorem 𝑨𝑩 = 𝑪𝑫 𝑶𝑨(𝑶𝑩) = 𝑶𝑪(𝑶𝑫) Altitude – Orthocenter Perpendicular Bisector - Circumcenter Secant – Tangent Theorem 𝑶𝑨(𝑶𝑩) = 𝑶𝑪(𝑶𝑫) = (𝑶𝑬)𝟐 SOLID GEOMETRY MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING TERMINOLOGIES SOLIDS WITH CONSTANT CROSS SECTION  Polyhedrons – a three-dimensional figure composed of flat CYLINDER PRISM TRUNCATED PRISM polygonal faces, straight edges, and sharp corners.  Regular Polyhedrons – solids with all their faces as identical polygons.  Prism – a polyhedron with two faces parallel and congruent and whose remaining faces are parallelograms.  Right Prism – a prism that has its lateral faces perpendicular to the base. 𝑽 = 𝝅𝒓𝟐 𝒉 𝑽 = 𝑨𝑩 𝒉 𝑽 = 𝑨𝑩𝒉𝒂𝒗𝒆  Oblique Prism – a prism in which the lateral faces are not 𝑻𝑺𝑨 = 𝟐𝝅𝒓𝟐 + 𝟐𝝅𝒓𝒉 𝑻𝑺𝑨 = 𝟐𝑨𝑩 + 𝑳𝑺𝑨 𝑻𝑺𝑨 = 𝑨𝟏 + 𝑨𝟐 + 𝑳𝑺𝑨 perpendicular to the base.  Cylinder – a prism with two circular bases  Pyramid – a polyhedron that contains triangular lateral faces with a common vertex and a polygonal base. SOLIDS WITH SIMILAR CROSS SECTION  Cone – is formed by a set of line segments connecting the apex to all the points on the circumference of a circular base that is CONE PYRAMID FRUSTUM in a plane that does not contain the apex.  Sphere - a round solid figure with every point on its surface equidistant from its center.  Spherical Segment – the solid defined by cutting a sphere or a ball with a pair of parallel planes.  Spherical Zone – a portion of the surface of a sphere included between two parallel planes. 𝟏 𝟏 𝟏  Spherical Cone – a solid generated by rotating a sector of a 𝑽 = 𝝅𝒓𝟐 𝒉 𝑽 = 𝑨𝑩 𝒉 𝑽= 𝑨 + 𝑨𝟐 + 𝑨𝟏 𝑨𝟐 𝟑 𝟑 𝟑 𝟏 circle about an axis that passes through the center of the circle 𝑻𝑺𝑨 = 𝝅𝒓𝟐 + 𝝅𝒓𝒍 𝑻𝑺𝑨 = 𝑨𝑩 + 𝑳𝑺𝑨 𝑻𝑺𝑨 = 𝑨𝟏 + 𝑨𝟐 + 𝑳𝑺𝑨 but contains no point inside the sector.  Frustum – the remaining portion obtained after removing the top portion when a pyramid or a cone is cut by a plane parallel GENERAL PRISMOIDAL FORMULA to its base. 𝑳 𝑽 = (𝑨𝟏 + 𝟒𝑨𝑴 + 𝑨𝟐 )  Truncated Prism – a portion of a prism formed by passing a 𝟔 plane not parallel to the base and intersecting all the lateral edges, has two nonparallel bases which are polygons of the same number of edges. SPHERES SIMILAR SOLIDS SPHERE LUNE AND WEDGE RATIOS 𝟏/𝟐 𝟏/𝟑 𝑺𝟏 𝑷𝟏 𝑨𝟏 𝑽𝟏 = = = 𝑺𝟐 𝑷𝟐 𝑨𝟐 𝑽𝟐 POLYHEDRONS 𝟒 𝟒 𝟑 𝜽 𝑽 = 𝝅𝒓𝟑 𝑽= 𝝅𝒓 𝟑 𝟑 𝟑𝟔𝟎 𝑺𝑨 = 𝟒𝝅𝒓𝟐 𝜽 𝑺𝑨 = 𝟒𝝅𝒓𝟐 𝟑𝟔𝟎 𝒏𝒇 ONE BASE TWO BASES 𝒗= 𝒎 SPHERICAL SEGMENT SPHERICAL SEGMENT 𝒏𝒇 𝒆= 𝝅𝒉𝟐 𝝅𝒉 𝟐 𝑽= (𝟑𝒓 − 𝒉) 𝑽= (𝟑𝒂𝟐 + 𝟑𝒃𝟐 − 𝒉𝟐 ) 𝟑 𝟔 EULER’S FORMULA 𝑺𝑨 = 𝟐𝝅𝒓𝒉 𝑺𝑨 = 𝟐𝝅𝒓𝒉 𝒗+𝒇−𝒆 = 𝟐 PLASTIC SOLIDS Polyhedra Vertices Faces Edges Sides Volume Cube 8 6 12 Square e3 Tetrahedron 4 4 6 triangle 0.118 e3 Octahedron 6 8 12 Triangle 0.471 e3 Dodecahedron 20 12 30 Pentagon 7.663 e3 Icosahedron 12 20 30 Triangle 2.182 e3 ANALYTIC GEOMETRY MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING DISTANCE FORMULA PARABOLA Locus of a point that moves such that its distance from a fixed Point to a Point 𝒅= (𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐 point called the focus is always equal to its distance from a fixed line called the directrix. 𝑨𝒙 + 𝑩𝒚 + 𝑪 Point to a Line 𝒅= √𝑨𝟐 + 𝑩𝟐 𝑪𝟐 − 𝑪𝟏 Between Parallel Lines 𝒅= √𝑨𝟐 + 𝑩𝟐 LINES MIDPOINT FORMULA 𝒙𝟏 + 𝒙𝟐 𝒚𝟏 + 𝒚𝟐 𝒙= , 𝒚= 𝟐 𝟐 Axis of Symmetry Standard Form General Form DIVISION OF LINE SEGMENT Parallel to Y-Axis (𝑥 − ℎ) = ±4𝑎(𝑦 − 𝑘) 𝐴𝑥 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 𝒙𝟐 − 𝒙 𝒚𝟐 − 𝒚 𝒓𝒙 = , 𝒓𝒚 = Parallel to X-Axis (𝑦 − 𝑘) = ±4𝑎(𝑥 − ℎ) 𝐶𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 𝒙𝟐 − 𝒙𝟏 𝒚𝟐 − 𝒚𝟏 EQUATION OF LINES ELLIPSE Slope Intercept Form 𝒚 = 𝒎𝒙 + 𝒃 𝒚 − 𝒚𝟏 Point-Slope Form 𝒎= 𝒙 − 𝒙𝟏 𝒚𝟐 − 𝒚𝟏 𝒚 − 𝒚𝟏 Two-Point Form = 𝒙𝟐 − 𝒙𝟏 𝒙 − 𝒙𝟏 𝒙 𝒚 Intercept Form + =𝟏 𝒂 𝒃 Parallel Lines 𝑴𝟏 = 𝑴𝟐 𝟏 Perpendicular Lines 𝑴𝟏 = 𝑴𝟐 Major Axis Standard Form General Form (𝑥 − ℎ) (𝑦 − 𝑘) POLAR COORDINATES Parallel to Y-Axis 𝑏 + 𝑎 =1 𝐴𝑥 + 𝐶𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 (𝑥 − ℎ) (𝑦 − 𝑘) Parallel to X-Axis + =1 𝑎 𝑏 Polar coordinates represent points in a Cartesion plane using distance (r) and angle measurements. PROPERTIES 𝒙 = 𝒓𝒄𝒐𝒔 𝜽 𝒚 = 𝒓𝒔𝒊𝒏 𝜽 𝒓𝟐 = 𝒙𝟐 + 𝒚𝟐 𝑎 =𝑏 +𝑐 𝐴 = 𝜋𝑎𝑏 𝑎 +𝑏 𝑑 +𝑑 = 𝑑 +𝑑 𝑃 = 2𝜋 Polar to Cartesian Cartesian to Polar 2 POL(𝑥, 𝑦) REC(𝑟, 𝜃) HYPERBOLA CONIC SECTIONS Conic Sections is a locus of a point that moves such that the ratio of its distance from a fixed point (focus) and fixed line (directrix) is constant. General Form 𝑨𝒙𝟐 + 𝑩𝒙𝒚 + 𝑪𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎 Section Case Eccentricity Circle 𝐴=𝐶 𝑒=0 Parabola 𝐴 = 0 𝑜𝑟 𝐶 = 0 𝑒=1 𝑐 Ellipse 𝐴(𝐶) > 0 𝑒= 1 Parallel to Y-Axis (𝑦 − 𝑘) − (𝑥 − ℎ) =1 𝐴𝑥 − 𝐶𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 𝑎 𝑏 (𝑥 − ℎ) (𝑦 − 𝑘) Parallel to X-Axis − =1 −𝐴𝑥 + 𝐶𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 CIRCLES 𝑎 𝑏 PROPERTIES STANDARD FROM 𝑐 = 𝑎 +𝑏 𝑑 −𝑑 = 𝑑 −𝑑 (𝑥 − ℎ) + (𝑦 − 𝑘) = 𝑟 ASYMPTOTES GENERAL FORM Traverse Parallel to Y Traverse Parallel to X 𝐴𝑥 + 𝐴𝑦 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0 𝑎 𝑏 𝑚=± 𝑚=± 𝑏 𝑎 DIFFERENTIAL CALCULUS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING LIMITS OF FUNCTIONS IMPLICIT DIFFERENTIATION Suppose that f and g are functions such thatlim 𝑓(𝑥) and lim 𝑔(𝑥) A method used to differentiate functions where the dependent → → exist, suppose that k is a constant and suppose that n is a positive variable is not explicitly expressed in terms of the independent integer. Then the following are accepted theorems of limits: 1. lim 𝑘 = 𝑘 variables. → 2. lim 𝑥 = 𝑎 → 1. Differentiate the equation with respect to the 3. lim [𝑘 𝑓(𝑥)] = 𝑘 lim 𝑓(𝑥) → → independent variable. 4. lim [𝑓(𝑥) + 𝑔(𝑥)] = lim 𝑓(𝑥) + lim 𝑔(𝑥) → → → 2. Isolate the derivative to one side of the equation. SOLUTIONS FOR LIMITS: 1. Direct Substitution HIGHER DERIVATIVES 2. Factorization Method 3. Rationalization Method NOTATION 4. In inity Method ∞ The nth derivative of a function is denoted as 𝑓 (𝑥) 𝑜𝑟. To Divide both the numerator and denominator ∞ by the highest power of x. find higher order derivatives, differentiate the function ∞±∞ Write in rational form then use successively n times applying differentiation as needed. L’HOPITALS’S RULE 𝑓(𝑥) 𝑓′(𝑥) RELATIONSHIP WITH FUNCTION BEHAVIOR lim = lim → 𝑔(𝑥) → 𝑔′(𝑥) Higher derivatives help understand the behavior of a function including concavity, inflection points, and rate of change. ASYMPTOTES CONCAVITY VERTICAL ASYMPTOTE The graph curves upward (convex) if 𝑓 (𝑥) > 0, and the graph Value of X such that the denominator will be equal to zero. curves downward (concave) if 𝑓 (𝑥) < 0. 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟 = 0 HORIZONTAL ASYMPTOTE INFLECTION POINTS 𝑎 𝑥 +⋯+𝑎 𝑥 These are points on a curve where the concavity changes. It is 𝑏 𝑥 + ⋯+ 𝑏 𝑥 where the second derivative changes sign 𝑓 (𝑥) = 0 1. 𝑚 = 𝑛 ; 𝑦 = accompanied by change in concavity. 2. 𝑚 < 𝑛 ; 𝑁𝑜 𝐻𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 3. 𝑚 > 𝑛 ; 𝑦 = 0 CRITICAL POINTS Location on a function where the derivative is zero or undefined. SLANTING ASYMPTOTE These points often indicate potential maxima, minima, or points Use Long Division 1. Divide the first term of the dividend by the first term of of inflection. the divisor and write the quotient above the dividend. 2. Multiply the divisor by the quotient and write the product below the dividend. 3. Subtract the product from the dividend and bring down APPLICATION the next term of the dividend. CIRCLE OF CURVATURE BASIC DIFFERENTIATION Radius of Curvature Curvature |𝒚′′| 𝟏 General Power Formula 𝒅 𝑲= 𝟑 𝝆= (𝒖)𝒏 = 𝒏(𝒖)𝒏 𝟏 𝒅𝒖 [𝟏 + (𝒚 )𝟐 ]𝟐 𝒌 𝒅𝒙 Exponential Formula 𝒅 TANGENT AND NORMAL LINES (𝒖)𝒏 = 𝒏(𝒖)𝒏 𝟏 𝒅𝒖 𝒅𝒙 Tangent Line Normal Line 𝒅 𝒖 𝒅𝒚 𝟏 𝒂 = 𝒂𝒖 𝒍𝒏 𝒂 𝒅𝒖 𝑴𝑻𝑳 = 𝑴𝑵𝑳 = − 𝒅𝒙 𝒅𝒙 𝑴𝑻𝑳 Logarithmic Formula 𝒅 𝒅𝒖 TIME RATES 𝒍𝒐𝒈(𝒖) = 𝒅𝒙 𝒖 𝒍𝒏 𝟏𝟎 Velocity Acceleration Flow 𝒅 𝒅𝒖 𝒅𝒔 𝒅𝒗 𝒅𝑽 𝒍𝒏(𝒖) = 𝒗= 𝒂= 𝑸= 𝒅𝒙 𝒖 𝒅𝒕 𝒅𝒕 𝒅𝒕 Product Rule 𝒅 𝒖𝒗 = 𝒖𝒅𝒗 + 𝒗𝒅𝒖 𝒅𝒙 OPTIMIZATION Always express the quantity to be optimized in terms of a single Quotient Rule 𝒅 𝒖 𝒗𝒅𝒖 − 𝒖𝒅𝒗 = 𝒅𝒙 𝒗 𝒗𝟐 variable function. Differentiate the function and set the derivative equal to zero. INTEGRAL CALCULUS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING INTEGRATION TECHNIQUES LENGTH OF CURVES SUBSTITUTION METHOD Parametric Involves substituting a complex expression within the integral with 𝒅𝒙 2 𝒅𝒚 𝟐 a simpler variable to make integration easier. 𝑺 = ∫ √( ) + ( ) 𝒅𝒕 𝒅𝒕 𝒅𝒕 ∫ 𝑓(𝑔(𝑥)𝑔 ′ 𝑥 𝑑𝑥 = ∫ 𝑓(𝑢)𝑑𝑢 , 𝑤ℎ𝑒𝑟𝑒 𝑢 = 𝑔(𝑥) Rectangular 𝒅𝒚 𝟐 𝑺 = ∫ √𝟏 + ( ) 𝒅𝒙 𝒅𝒙 INTEGRATION BY PARTS 𝒅𝒙 𝟐 𝑺 = ∫ √𝟏 + ( ) 𝒅𝒚 This technique is used for integrating the product of two functions 𝒅𝒚 where u and dv are selected. Polar ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢 𝒅𝒓 𝟐 𝑺 = ∫ √𝒓 𝟐 + ( ) 𝒅𝜽 𝒅𝜽 BASIC INTEGRATION SURFACE AREA OF CURVES General Power Formula (𝒖)𝒏+𝟏 ∫(𝒖)𝒏 𝒅𝒖 = +𝑪 𝑨𝒔 = 𝟐𝝅 ∫ 𝒙 𝒅𝑺 𝑜𝑟 𝑨𝒔 = 𝟐𝝅 ∫ 𝒚 𝒅𝑺 𝒏+𝟏 Exponential Formula ∫ 𝒆𝒖 𝒅𝒖 = 𝒆𝒖 + 𝑪 CENTROIDS 𝒂𝒖 ∫ 𝒂𝒖 𝒅𝒖 = +𝑪 Length 𝐥𝐧 𝒂 ̅ = ∫ 𝒙 𝒅𝑺 𝑺𝒙 ̅ = ∫ 𝒚 𝒅𝑺 𝑺𝒚 Logarithmic Formula 𝟏 ∫ 𝒅𝒖 = 𝐥𝐧|𝒖| + 𝑪 Area 𝒖 ̅ = ∫ 𝒙 𝒅𝑨 𝑨𝒙 ̅ = ∫ 𝒚 𝒅𝑨 𝑨𝒚 ∫ 𝐥𝐧 𝒖 𝒅𝒖 = 𝒖 𝐥𝐧 𝒖 − 𝒖 + 𝑪 Volume ̅ = ∫ 𝒙 𝒅𝑽 𝑽𝒙 ̅ = ∫ 𝒚 𝒅𝑽 𝑽𝒚 PLANE AREAS MOMENT OF INERTIA Vertical Strip 𝑨 = ∫(𝒚𝒖𝒑𝒑𝒆𝒓 − 𝒚𝒍𝒐𝒘𝒆𝒓 )𝒅𝒙 About the X-Axis Horizontal Vertical 𝟏 Horizontal Strip 𝑰𝒙 = ∫ 𝒚𝟐 𝒅𝑨 𝑰𝒙 = ∫ 𝒚𝟑 𝒅𝒙 𝑨 = ∫(𝒙𝒓𝒊𝒈𝒉𝒕 − 𝒙𝒍𝒆𝒇𝒕 )𝒅𝒚 𝟑 About the Y-Axis Radial Strip 𝟏 Vertical Horizontal 𝑨 = ∫ 𝒓𝟐 𝒅𝜽 𝟐 𝟏 𝑰𝒚 = ∫ 𝒙𝟐 𝒅𝑨 𝑰𝒚 = ∫ 𝒙𝟑 𝒅𝒚 𝟑 Polar Moment of Inertia VOLUME OF SOLID REVOLUTION Disk Method 𝑱 = ∫ 𝑹𝟐 𝒅𝑨 𝑱 = 𝑰𝒙 + 𝑰𝒚 𝑽 = 𝝅 ∫ 𝑹𝟐 𝒅𝒙 𝟐 𝑽 = 𝝅 ∫ 𝑹 𝒅𝒚 Product of Inertia 𝑰𝒙𝒚 = ∫ 𝒙𝒚𝒅𝑨 Ring Method 𝑽 = 𝝅 ∫(𝑹𝟐 − 𝒓𝟐 ) 𝒅𝒙 𝑽 = 𝝅 ∫(𝑹𝟐 − 𝒓𝟐 ) 𝒅𝒚 WORK GENERAL FORMULA Shell Method 𝑽 = 𝟐𝝅 ∫ 𝒙 𝒅𝑨 𝑾 = ∫ 𝑭(𝒙)𝒅𝒙 𝑽 = 𝟐𝝅 ∫ 𝒚 𝒅𝑨 SPRING 𝑾 = ∫ 𝑭(𝒙) 𝒅𝒙 𝑤ℎ𝑒𝑟𝑒 𝐹 = 𝑘𝑥 PUMP 𝒉𝑳 𝑾 = ∫ 𝜸(𝑨𝒅𝒉)(𝒉𝑻 − 𝒉)𝒅𝒙 𝟎 DIFFERENTIAL EQUATIONS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING A differential equation involves derivatives of an unknown function. It relates the function to its derivatives in one or more APPLICATIONS variables. General Equation [Mode 3 – 5] ORDER AND DEGREE 𝒀 = 𝑪𝒆𝒌(𝒙) ORDER The order is determined by the highest derivative present in the X Y equation. 𝒅𝒚 Cooling Time 𝑇−𝑇 First Order 𝒅𝒙 𝒅𝟐 𝒚 Heating Time 𝑇 −𝑇 Second Order 𝒅𝒙𝟐 Natural Population Growth Time 𝑃 DEGREE Radioactive Decay Time 𝑚 The degree applies to polynomial equations formed when derivatives are eliminated. It is defined for polynomial equations having the highest power of the highest derivative in the equation. CHANGE IN CONCENTRATION 𝑪𝒐𝒖𝒕𝒄𝒐𝒎𝒆 𝒕𝒐𝒖𝒕𝒄𝒐𝒎𝒆 𝒅𝒎 = 𝒅𝒕 METHODS FOR SOLVING 𝑪𝒊𝒏𝒊𝒕𝒊𝒂𝒍 𝑪𝒊𝒏 − 𝑪𝒐𝒖𝒕 𝒕𝒊𝒏𝒊𝒕𝒊𝒂𝒍 VARIABLE SEPARABLE TRIGONOMETRIC FUNCTIONS 1. Express the differential equation in the form 𝑑𝑦 = 𝑓(𝑥)𝑔(𝑦) 𝒅 𝑑𝑥 𝒄𝒐𝒔 𝒖 𝒅𝒖 = 𝒔𝒊𝒏 𝒖 + 𝑪 𝒔𝒊𝒏 𝒖 = 𝒄𝒐𝒔 𝒖 𝒅𝒖 𝒅𝒙 2. Separate the variables. 𝒅 𝑑𝑦 𝒄𝒐𝒔 𝒖 = −𝒔𝒊𝒏 𝒖 𝒅𝒖 𝒔𝒊𝒏 𝒖 𝒅𝒖 = −𝒄𝒐𝒔 𝒖 + 𝑪 = 𝑓(𝑥)𝑑𝑥 𝒅𝒙 𝑔(𝑦) 𝒅 𝟐 𝒕𝒂𝒏 𝒖 = 𝒔𝒆𝒄 𝒖 𝒅𝒖 𝒔𝒆𝒄𝟐 𝒖 𝒅𝒖 = 𝒕𝒂𝒏 𝒖 + 𝑪 3. General Solution: 𝒅𝒙 𝑑𝑦 𝒅 = 𝑓(𝑥)𝑑𝑥 + 𝐶 𝒄𝒐𝒕 𝒖 = −𝒄𝒔𝒄𝟐 𝒖 𝒅𝒖 𝒄𝒔𝒄𝟐 𝒖 𝒅𝒖 = −𝒄𝒐𝒕 𝒖 + 𝑪 𝑔(𝑦) 𝒅𝒙 𝒅 𝒔𝒆𝒄 𝒖 = 𝒔𝒆𝒄 𝒖 𝒕𝒂𝒏 𝒖 𝒅𝒖 𝒔𝒆𝒄 𝒖 𝒕𝒂𝒏 𝒖 𝒅𝒖 = 𝒔𝒆𝒄 𝒖 + 𝑪 EXACT EQUATION 𝒅𝒙 1. Express the differential equation in the form 𝒅 𝒄𝒔𝒄 𝒖 = −𝒄𝒔𝒄 𝒖 𝒄𝒐𝒕 𝒖 𝒅𝒖 𝒄𝒔𝒄 𝒖 𝒄𝒐𝒕 𝒖 𝒅𝒖 = −𝒄𝒔𝒄 𝒖 + 𝑪 𝒅𝒙 𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0 wherein INVERSE TRIGONOMETRIC FUNCTIONS 𝜕𝑀 𝜕𝑁 = 𝜕𝑦 𝜕𝑥 𝒅 𝒅𝒖 𝒅𝒖 𝒖 2. General Solution: 𝒂𝒓𝒄𝒔𝒊𝒏 𝒖 = = 𝒂𝒓𝒄𝒔𝒊𝒏 + 𝑪 𝒅𝒙 √𝟏 − 𝒖𝟐 √𝒂𝟐 − 𝒖𝟐 𝒂 𝑀(𝑥, 𝑦)𝑑𝑥 𝑁(𝑥, 𝑦)𝑑𝑦 𝒅 𝒅𝒖 𝐹(𝑥, 𝑦) = + =𝐶 𝒂𝒓𝒄𝒄𝒐𝒔 𝒖 = − "𝑦 𝑎𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡" "𝑒𝑙𝑖𝑚𝑖𝑛𝑎𝑡𝑒 𝑥" 𝒅𝒙 √𝟏 − 𝒖𝟐 𝒅 𝒅𝒖 𝒅𝒖 𝟏 𝒖 𝒂𝒓𝒄𝒕𝒂𝒏 𝒖 = = 𝒂𝒓𝒄𝒕𝒂𝒏 + 𝑪 𝒅𝒙 𝟏 + 𝒖𝟐 𝒂𝟐 + 𝒖𝟐 𝒂 𝒂 LINEAR EQUATION 𝒅 𝒅𝒖 𝒂𝒓𝒄𝒄𝒐𝒕 𝒖 = − 1. Express the differential equation in the form 𝒅𝒙 𝟏 + 𝒖𝟐 𝒅 𝒅𝒖 𝒅𝒖 𝟏 𝒖 𝑑𝑦 𝒂𝒓𝒄𝒔𝒆𝒄 𝒖 = = 𝒂𝒓𝒄𝒔𝒆𝒄 + 𝑪 + 𝑃(𝑥)𝑦 = 𝑄(𝑥) 𝒅𝒙 𝒖√𝒖𝟐 − 𝟏 𝒖√𝒖𝟐 − 𝒂𝟐 𝒂 𝒂 𝑑𝑥 𝒅 𝒅𝒖 𝒂𝒓𝒄𝒄𝒔𝒄 𝒖 = − 2. General Solution: 𝒅𝒙 𝒖√𝒖𝟐 − 𝟏 𝑦(𝜆) = 𝜆 𝑄(𝑥)𝑑𝑥 + 𝐶 wherein ( ) 𝜆 = 𝑒∫ HOMOGENOUS DIFFERENTIAL EQUATION 1. Express the differential equation in the form 𝑀(𝑥, 𝑦)𝑑𝑥 + 𝑁(𝑥, 𝑦)𝑑𝑦 = 0 wherein 𝑀(𝑥, 𝑦) and 𝑁(𝑥, 𝑦) has the same degree. 2. Introduce a new variable to transform the equation into a separable form. 𝑥 = 𝑣𝑦 𝑦 = 𝑣𝑥 𝑑𝑥 = 𝑦𝑑𝑣 + 𝑣𝑑𝑦 𝑑𝑦 = 𝑥𝑑𝑣 + 𝑣𝑑𝑥 PHYSICS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING TERMINOLOGIES ROTATIONAL MOTION  Dynamics – a branch of physics that deals with the forces and Involves the rotation of an object around its axis and is torques affecting motion. characterized by angular displacement, angular velocity, and  Kinematics – describes the motion of objects without angular acceleration. considering the forces or torques that cause the motion.  Kinetics – deals with the effects of forces and torques on the CONSTANT ANGULAR ACCELERATION 1 motion of bodies having mass. 𝜃 = 𝜔 𝑡 + 𝛼𝑡 𝜔 = 𝜔 + 2𝛼𝜃 𝜔 = 𝜔 + 𝛼𝑡 2  Distance – a scalar quantity that refers to the total length traveled by an object. VARYING ACCELERATION  Displacement – a vector quantity that refers to the change in 𝑑𝜃 𝑑𝜔 𝜔= 𝛼= 𝑑𝜔 = 𝛼 𝑑𝜃 position of an object. 𝑑𝑡 𝑑𝑡  Speed – a scalar quantity that refers to how fast an object is moving.  Velocity – a vector quantity that refers to the rate at which an RELATIONSHIP WITH RECTILINEAR MOTIONS object changes its position. 𝑣 𝑠 = 𝑟𝜃 𝑣 = 𝑟𝜔 𝑎 = 𝑟𝛼 𝑎 =  Acceleration – a vector quantity that is defined as the rate of 𝑟 change of an object with respect to time. ERRATIC MOTION RECTILINEAR MOTION Describes irregular and unpredictable movement with no consistent pattern or direction of an object. The movement of an object along a straight line is characterized by a one-dimensional path with motion occurring in either the ACCELERATION-TIME DIAGRAM positive or negative direction or remaining stationary. ∆𝑣 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑎 − 𝑡 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 ∆𝑠 = 𝑣 𝑡 + 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑚𝑜𝑚𝑒𝑛𝑡 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑎 − 𝑡 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 CONSTANT ACCELERATION 1 VELOCITY-TIME DIAGRAM 𝑠 = 𝑣 𝑡 + 𝑎𝑡 𝑣 = 𝑣 + 2𝑎𝑠 𝑣 = 𝑣 + 𝑎𝑡 2 ∆𝑠 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑣 − 𝑡 𝑑𝑖𝑎𝑔𝑟𝑎𝑚 VARYING ACCELERATION 𝑣= 𝑑𝑠 𝑎= 𝑑𝑣 𝑣 𝑑𝑣 = 𝑎 𝑑𝑠 FORCE AND ACCELERATION 𝑑𝑡 𝑑𝑡 The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force and inversely proportional to the mass of the object. PROJECTILE MOTION ∑𝐹 = 𝑚𝑎 The movement of an object along a curved path that is under the influence of gravity, with an initial velocity. WORK AND ENERGY ENERGY EQUATION 1 1 𝑚𝑔ℎ + 𝑘𝑥 + 𝑚𝑣 2 2 =0 "𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙" "𝑘𝑖𝑛𝑒𝑡𝑖𝑐" WORK-ENERGY THEOREM Work done by the net force on a particle equals the change in the particle’s kinetic energy. 1 1 𝐹𝑑 + 𝑚𝑔ℎ + 𝑘𝑥 = 𝑚(𝑣 − 𝑣 ) 2 2 IMPULSE AND MOMENTUM CURVILINEAR MOTION IMPULSE-MOMENTUM THEOREM The impulse of the net external force on a particle during a time Involves the movement of an object along a curved path, interval equals the change in momentum of that particle. departing from the straight-line trajectory observed in rectilinear 𝐹(∆𝑡) = 𝑚(∆𝑣) motion. IMPACT AND COLLISION 1. Apply Law of Conservation of Momentum 𝑚 (𝑣 ) + 𝑚 (𝑣 ) = 𝑚 (𝑣 ) + 𝑚 (𝑣 ) 2. Apply Coefficient of Restitution “e” 𝑣 −𝑣 𝑒= 𝑣 −𝑣 values of e: 𝑒 = 1: perfectly elastic impact (opposite direction after impact) 𝑒 = 0: perfectly plastic impact (together after impact) VECTORS AND CALCULUS FUNCTIONS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING A vector is a quantity that has both magnitude and direction. It is represented by an arrow, where the length of the arrow TRIGONOMETRIC FUNCTIONS represents the magnitude, and the direction indicates the direction of the vector. 𝒅 𝒔𝒊𝒏 𝒖 = 𝒄𝒐𝒔 𝒖 𝒅𝒖 𝒄𝒐𝒔 𝒖 𝒅𝒖 = 𝒔𝒊𝒏 𝒖 + 𝑪 𝒅𝒙 𝒅 VECTOR OPERATIONS 𝒅𝒙 𝒄𝒐𝒔 𝒖 = −𝒔𝒊𝒏 𝒖 𝒅𝒖 𝒔𝒊𝒏 𝒖 𝒅𝒖 = −𝒄𝒐𝒔 𝒖 + 𝑪 Addition 𝑨+𝑩 = 𝑪 𝒅 𝟐 𝒕𝒂𝒏 𝒖 = 𝒔𝒆𝒄 𝒖 𝒅𝒖 𝒔𝒆𝒄𝟐 𝒖 𝒅𝒖 = 𝒕𝒂𝒏 𝒖 + 𝑪 𝒅𝒙 Subtraction 𝑨−𝑩 = 𝑪 𝒅 𝒄𝒐𝒕 𝒖 = −𝒄𝒔𝒄𝟐 𝒖 𝒅𝒖 𝒄𝒔𝒄𝟐 𝒖 𝒅𝒖 = −𝒄𝒐𝒕 𝒖 + 𝑪 𝒅𝒙 Dot Product (Scalar) 𝑨 ∙ 𝑩 = 𝒙𝟏 𝒙𝟐 + 𝒚𝟏 𝒚𝟐 + 𝒛𝟏 𝒛𝟐 𝒅 𝒔𝒆𝒄 𝒖 = 𝒔𝒆𝒄 𝒖 𝒕𝒂𝒏 𝒖 𝒅𝒖 𝒔𝒆𝒄 𝒖 𝒕𝒂𝒏 𝒖 𝒅𝒖 = 𝒔𝒆𝒄 𝒖 + 𝑪 𝒅𝒙 𝑨 ∙ 𝑩 = |𝑨||𝑩| 𝐜𝐨𝐬 𝜽 𝒅 𝒄𝒔𝒄 𝒖 = −𝒄𝒔𝒄 𝒖 𝒄𝒐𝒕 𝒖 𝒅𝒖 𝒄𝒔𝒄 𝒖 𝒄𝒐𝒕 𝒖 𝒅𝒖 = −𝒄𝒔𝒄 𝒖 + 𝑪 Cross Product (Vector) 𝑨×𝑩≠𝑩 ×𝑨 𝒅𝒙 |𝑨 × 𝑩| = |𝑩 × 𝑨| INVERSE TRIGONOMETRIC FUNCTIONS |𝑨 × 𝑩| = |𝑨||𝑩| 𝐬𝐢𝐧 𝜽 𝒅 𝒅𝒖 𝒅𝒖 𝒖 𝒂𝒓𝒄𝒔𝒊𝒏 𝒖 = = 𝒂𝒓𝒄𝒔𝒊𝒏 + 𝑪 𝒅𝒙 √𝟏 − 𝒖𝟐 √𝒂𝟐 − 𝒖𝟐 𝒂 𝒅 𝒅𝒖 COMPONENT OF VECTORS 𝒂𝒓𝒄𝒄𝒐𝒔 𝒖 = − 𝒅𝒙 √𝟏 − 𝒖𝟐 Vectors can be expressed in terms of their components, which 𝒅 𝒅𝒖 𝒅𝒖 𝟏 𝒖 𝒂𝒓𝒄𝒕𝒂𝒏 𝒖 = = 𝒂𝒓𝒄𝒕𝒂𝒏 + 𝑪 are the parts of the vector that act along specific directions in a 𝒅𝒙 𝟏 + 𝒖𝟐 𝒂𝟐 + 𝒖𝟐 𝒂 𝒂 coordinate system. 𝒅 𝒅𝒖 𝒂𝒓𝒄𝒄𝒐𝒕 𝒖 = − 𝒅𝒙 𝟏 + 𝒖𝟐 Magnitude [Mode 8 – Abs(VctA)] 𝒅 𝒅𝒖 𝒅𝒖 𝟏 𝒖 𝒂𝒓𝒄𝒔𝒆𝒄 𝒖 = = 𝒂𝒓𝒄𝒔𝒆𝒄 + 𝑪 𝒅𝒙 𝒖√𝒖𝟐 − 𝟏 𝒖√𝒖𝟐 − 𝒂𝟐 𝒂 𝒂 𝑨⃗ = 𝒙𝟐 + 𝒚𝟐 + 𝒛𝟐 𝒅 𝒅𝒖 𝒂𝒓𝒄𝒄𝒔𝒄 𝒖 = − UNIT VECTOR 𝒅𝒙 𝒖√𝒖𝟐 − 𝟏 𝑨𝒙 𝑨𝒚 𝑨𝒛 𝑨 𝝀𝑨 = 〈 〉= |𝑨| |𝑨| |𝑨| |𝑨| DIRECTION COSINE 𝑨𝒙 𝑨𝒚 𝑨𝒛 𝒄𝒐𝒔 𝜽𝒙 = 𝒄𝒐𝒔 𝜽𝒚 = 𝒄𝒐𝒔 𝜽𝒛 = |𝑨| |𝑨| |𝑨| FORCE VECTOR 𝑭⃗ = 𝑭𝝀 APPLICATIONS Shortest Distance |𝑨 × 𝑩| 𝒅= 𝑨 Area of Triangle 𝟏 𝑨 = |𝑨 × 𝑩| 𝟐 Area of Parallelogram 𝑨 = |𝑨 × 𝑩| Volume of Parallelepiped 𝑽 = |𝑨 × 𝑩| ∙ 𝑪 Direction Number 𝑫𝑵 = 〈𝑨 𝑩 𝑪〉 = |𝑨 × 𝑩| Work 𝑾 = 𝑭⃗𝒅⃗ Moment 𝑴 = 𝒓 × 𝑭𝝀 ENGINEERING ECONOMICS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING ENGINEERING ECONOMY RULES PERPETUITY RULE 1: Reference refers to the period of want, while source refers A perpetuity is a financial arrangement or investment that to the period of what is given. provides a constant stream of cash flows that continues indefinitely into the future. 𝑊 = 𝐻(1 + 𝑖) Interval = 1: Interval > 1: RULE 2: The start and end refer to the period of Annuities. 𝐴 𝐴 𝑃= 𝑃= 𝑖 (1 + 𝑖) − 1 𝑊=𝐴 (1 + 𝑖) CAPITALIZED COST AND ANNUAL COST RULE 3: The start and end refer to the period of Gradients. CAPITALIZED COST Refers to the total cost incurred to acquire, construct, or upgrade 𝑃= [(𝑦)(1 + 𝑖) ] a long-term asset. This includes the purchase price, installation, transportation, and any additional cost such as maintenance and repair. INTEREST 𝐶𝐶 = 𝐹𝐶 + 𝑂+𝑀 + 𝐹𝐶 − 𝑆 𝑖 (1 + 𝑖) − 1 SIMPLE INTEREST Interest is computed based on Ordinary Interest (one banker’s ANNUAL COST year – 360 days) or based on the exact number of days. Annual cost represents the total cost of owning, operating, and maintaining an asset on an annual basis. 𝐼 = 𝑃𝑟𝑛 𝐹 = 𝑃(1 + 𝑟𝑛) 𝐴𝐶 = 𝐶𝐶(𝑖) COMPOUND INTEREST Interest is computed at the end of each interest period and the interest earned for each period is added to the principal. DEPRECIATION 𝑟 𝑟 GENERAL FORMULAS 𝑖 = 1+ −1 𝐹 = 𝑃 1+ 𝑚 𝑚 𝐵𝑉 = 𝐹𝐶 − 𝐷 𝑑 = 𝐵𝑉 − 𝐵𝑉 For continuous: 𝑖 =𝑒 −1 𝐹 = 𝑃𝑒 𝐷 = 𝑑 𝑑 =𝐷 −𝐷 ANNUTIES STRAIGHT LINE METHOD [MODE 3 – 2] X (𝑚) y (𝐵𝑉 ) Annuity refers to a series of equal payments made at regular 0 𝐹𝐶 intervals. 𝑛 𝑆𝑉 ORDINARY ANNUITY (1 + 𝑖) − 1 SUM OF YEARS DIGIT METHOD [MODE 3 – 3] 𝐹 =𝐴 𝑖 X (𝑚) y (𝐵𝑉 ) 0 𝐹𝐶 ANNUITY DUE 𝑛 𝑆𝑉 𝐹 =𝐹 + 𝐴(1 + 𝑖) 𝑛+1 𝑆𝑉 DECLINING BALANCE METHOD [MODE 3 – 6] GRADIENTS X (𝑚) y (𝐵𝑉 ) 0 𝐹𝐶 X (𝟎 − 𝒔𝒐𝒖𝒓𝒄𝒆) y (𝒄𝒂𝒔𝒉𝒇𝒍𝒐𝒘) 𝑛 𝑆𝑉 𝟎 − 𝒙𝟏 𝑦 DOUBLE DECLINING BALANCE METHOD [MODE 3 – 6] 𝟎 − 𝒙𝟐 𝑦 X (𝑚) y (𝐵𝑉 ) 0 𝐹𝐶 UNIFORM GRADIENT [Mode 3 – 2] 1 2 𝐹𝐶 1 − A constant rate of change in cash flows over a series of periods 𝑛 that increases or decreases at a fixed amount per period. SINKING FUND METHOD (1 + 𝑖) − 1 (1 + 𝑖) − 1 𝐹𝐶 = 𝑆𝑉 + 𝑑 𝐷 =𝑑 GEOMETRIC GRADIENT [Mode 3 – 6] 𝑖 𝑖 Involves a constant ratio of change in cash flows over time that increases or decreases by a fixed percentage in each period. INFLATION Increase in the amount of money needed to purchase same INTEREST RATE FACTORS amount of goods or services. Compound Amount 𝑭/𝑷 For Commodity For Purchasing Power Single Payment 𝑃 Present Worth 𝑷/𝑭 𝐹 = 𝑃 (1 + 𝑓) 𝐹 = (1 + 𝑓) Compound Amount 𝑭/𝑨 Sinking Fund 𝑨/𝑭 Equal Payment Present Worth 𝑷/𝑨 Series BREAK-EVEN ANALYSIS Capital Recovery 𝑨/𝑷 A method to determine when costs exactly equal revenue. Gradient Uniform Series 𝑨/𝑮 Gradient Series Gradient Present Worth 𝑷/𝑮 𝑐𝑜𝑠𝑡 = 𝑟𝑒𝑣𝑒𝑛𝑢𝑒 DATA ANALYTICS MATHEMATICS, SURVEYING, AND TRANSPORTATION ENGINEERING STATISTICS PROBABILITIES MEASURES OF CENTRAL TENDENCY A measure of the likelihood that a particular event will occur. The average of a given set of values 𝑑𝑒𝑠𝑖𝑟𝑒𝑑 𝑜𝑢𝑡𝑐𝑜𝑚𝑒 𝑃(𝐸) = Mean ∑𝑥 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒 𝑥̅ = 𝑛 The middle value in a set of data in order INDEPENDENT EVENT Median If the occurrence of one event does not affect the probability of 𝑃 = 0.5(𝑛 + 1) the other event. Mode The most common value 𝑃(𝐴⋂𝐵) = 𝑃(𝐴) ⋅ 𝑃(𝐵) CONDITIONAL PROBABILITY MEASURES OF DISPERSION The probability of event A given that event B has occurred. Range 𝑅=𝑥 −𝑥 𝑃(𝐴⋂𝐵) 𝑃(𝐴/𝐵) = Interquartile Range 𝐼𝑄𝑅 = 𝑄 − 𝑄 𝑃(𝐵) Population SD BINOMIAL DISTRIBUTION Describes the number of successes in a fixed number of Bernoulli ∑(𝑥 − 𝑥̅ ) 𝜎= trials. Standard 𝑛 𝑃(𝑥) = 𝑛𝐶𝑥(𝑝) (𝑞) Deviation Sample SD ∑(𝑥 − 𝑥̅ ) GEOMETRIC DISTRIBUTION 𝑠= Describes the number of trials needed to achieve the first success 𝑛−1 in a sequence of independent Bernoulli trials. Measures the average squared deviation of each data point from 𝑃(𝑥) = 𝑞 (𝑝) Variance the mean. (𝑠 ) NEGATIVE BINOMIAL DISTRIBUTION Mean Absolute ∑(𝑥 − 𝑥̅ ) Describes the number of trials needed to achieve kth success in a 𝑀𝐴𝐷 = Deviation 𝑛 sequence of independent Bernoulli trials. 𝑃(𝑥) = 𝐶 (𝑝) (𝑞) PERCENTILES 𝑃 = 𝑖(𝑛 + 1) MULTINOMIAL DISTRIBUTION Generalization of the binomial distribution to multiple categories wherein: or outcomes. 𝐷 =𝑃 𝑄 =𝑃 𝑛! 𝑃(𝑥 , 𝑥 , … , 𝑥 ) = 𝑃 ⋅𝑃 ⋅ …⋅ 𝑃 𝑥 !𝑥 !…𝑥 ! COUNTING PRINCIPLES HYPERGEOMETRIC DISTRIBUTION LINEAR PERMUTATION 𝑛 𝐶𝑟 + 𝑛 𝐶𝑟 + ⋯ + 𝑛 𝐶𝑟 𝑃(𝑥) = Distinct Permutation Identical Permutation 𝑛𝐶𝑟 𝒏! 𝒏! 𝒏𝑷𝒓 = 𝑳𝑷 = (𝒏 − 𝒓)! 𝒏𝒂 ! 𝒏𝒃 ! … 𝒏𝒛 ! POISSON DISTRIBUTION 𝜇 𝑒 𝑃(𝑥) = CIRCULAR PERMUTATION 𝑥! If 𝑛 ≠ 𝑟 If 𝑛 = 𝑟 CONTINUOUS PROBABILITY DISTRIBUTION 𝒏𝑷𝒓 𝑪𝑷 = 𝑪𝑷 = (𝒏 − 𝟏)! 𝒓 𝑃(𝑎 < 𝑥 < 𝑏) = 𝑓(𝑥) 𝑑𝑥 = 1 COMBINATION 𝒏! NORMAL DISTRIBUTION [Mode 3 – 1] 𝒏𝑪𝒓 = (𝒏 − 𝒓)! 𝒓! MATHEMATICAL EXPECTATION EXPECTATION [Mode 3 - 1] It is a measure of the central tendency of a random variable and represents the long-term average of a variable’s values in Mean Standard Deviation Z-Score repeated trials. 𝑥−𝜇 𝜇 = 𝑛𝑝 𝜎 = 𝑛𝑝𝑞 𝑧= Discrete Random Variable Continuous Random Variable 𝜎 𝐸(𝑥) = ∑𝑥𝑝 𝐸(𝑥) = 𝑥 𝑓(𝑥)𝑑𝑥 SAMPLE DISTRIBUTION Sample Sample Standard Z-Score Mean Deviation 𝜎 = ∑(𝑥 − 𝑥̅ ) 𝑝 𝜎 = (𝑥 − 𝑥̅ ) 𝑓(𝑥)𝑑𝑥 𝜎 𝑥̅ − 𝜇 ̅ 𝜇 ̅=𝜇