Scarica Engineering Entrance Exam Cheat Sheet e più Formulari in PDF di Ingegneria Meccanica solo su Docsity! LOGARITHM If dis a positive number not equal to 1 andx > 0, ANGLES “ An acute angle measures less than 90° * A rîghi angle measures 90° * An obruse angle measures more than 90° but less than 180°. “ A sfraight angle measures 180°. If a pair of parallel lines îs cut by a transversal that is perpendicular to the parallel lines, all eight angles are right angles. For any positive base d # 1 and any positive numbers x,y, and rr: * logg(x1) = logg.x + loggy . (7) logi r—log y s logge" * log, d"= n (In particular, log, 1= log, 5° =0 and log, 5 = log, 5!=1) unlogga QUANDRIC EQUATIONS Ita, d, and care real numbers with a # 0 and if av 2 + dx + c= 0, then albi dae 2a Ifa, b, and care rational numbers with a # 0, if ax? dx + c=0, and if D= 8° — 4ac, then Value of Discriminant Nature of the Roots D=0 2 equal rational roots D=-0 2 unequal complex roots that are D>0 (© Disa perfect square 2 unequal rational roots (Éi) Dis not a perfect square 2 unequal rational roots If axî>bxv+c= 0, then the sumof the two roots is ci ‘and the product of the two roots ST SEQUENCES Arithmetic Sequences: each term is equal to the previous term plus d Sequence: t1, t1+ d, t1 + 2d, The n'! term istn =t1+(n-1)d Number of integers from în t0 im =iîm — in+1 Sum of n terms Sn = (n/2) - (t1+ tn) Geometric Sequences: each term is equal to the previous term times r Sequence: t1, t1 er, ta «n, ... The n°! term ist, = tr"! Sum of n terms S, = t1-(r°—1)/(r- 1) Penmitations and Combinations: The munber of permutations of n things is n = n! The mumber of permutations of n things take r at a time is 2, — ni/(n-r)! The number of combinations of n things taken r at a time îs ,0, = n!/{(n-r)! Lilmo aio Ifa pairof parallel lines is cut by a transversal that is not perpendicular to the parallel line : * Four of the angles are acute, and four are obtuse. * All four acute angles are congrue nt. + All four obtuse angles are congruent. * The sum of the measures of any acute angle and any obtuse angle is 180°. film no parso TRIANGLES The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. 8 warbv Inany triangle: * The longest side is opposite the largest angle. * The shortest side is opposite the smallest angle. * Sides with equal lengths are opposite angles with equal measures (the angles opposite congruent sides are congruent). * A triangle is called scalene ifthe three sides all have different lengths. * A triangle is called isosceles if two sides are congnient. - A triangle is called eguilaferal if all three sides are congruent. * Acute triangles are triangles în which all three angles are acute. An acute triangle could be scalene, isosceles, or equilateral. * Obtuse friangles are triangles in which one angle is obtuse and two are acute. An obtuse triangle could be scalene or isosceles. * Right triangles are triangles that have one right angle and two acute angles. A right triangle could de scalene or isosceles. The side opposite the 90° angle is called the Ayporenuse, and by KEY FACT HI, it is the longest side. The other two sides are called the legs dala POLYGONS a+b+c=180andd+2+f=180 a+(b+e)+(c+d)+f=360 PARALELLOGRAM * Opposite sides are parallel: 45 | CD and AD| FC. * Opposite sides are congruent: 483 D and AD= BC. - Opposite angles are congruent: ZA = 4 and 485 £D. * The sum of the measures of any pair of consecutive angles is 180°. For example, a + è = 180 andb+c=180. * A diagonal divides the parallelogram into two congruent triangles: A480= A4CD. * The two diagonals bisect each other: AE=EC and BE=FD. MARC MAGA A rectangle is a parallelogram in which all four angles are right angles. Two adjacent sides of a rectangle are usually called the lerrgt4 (() and the widil (w). Note that the length is not necessarily greater than the width p q b d b_d 7 7 - The measure of each angle in a rectangle is 90°. - The diagonals of a rectangle have the same lengih: AC= BD. A 8 nl Lu Since a rbombus is a parallelogram, all of the properties thombuses. In addition: “ The length of each side of a rhombus is the same. “The two diagonals of a rhombus are perpendicular “The diagonals of a rhombus are angle bisectors. è A B 40-80 b AG=BD ni o ti isosceles trapezoid AREAS « Fora parallelogram: 4= DI « Fora rectangle: d = fw trapezoid «Fora square: 4=s7 or 4 1 * Fora trapezoid: #5 3% + 224 CIRCLES Any triangle formed by connecting the endpoints of two radii is isosceles. If dis the diameter and r is the radius of a circle: d= 2r. circumierence , € C= nd and C=2nr NS diameter © d The formula for the area of a circle of radius r is A="wr2. Ina circle of radius r, if an are measures 1°; x o - The length of the arc is 10l”) Zia? * The area of the sector formed by che arc and two radii is va” } The measure of an inscribed angle is one-half the measure of its intercepted arc. B=55 & Line {is tangent to both ciscles. x a Yr ex and ey are each tangent to the circle. « From any poînt outside a circle, exactly two tangents can be drawn to the cirele. * If two tangents are drawn from a point P outside a circle, intersecting the circle at 4 and 8, then P4 = PB. * The measure of the angle formed by two tangents drawn from the same point is one-half the difference of the two intercepred arcs. * A line tangent to a circle is perpendicular to the radius (or diameter) drawn to the point of contact. 90° + 140° + 90° + m/ P= 360° > m/P=40° R * When a square is inscribed in circle, the diagonals of the square are diameters of the circle. AB isa diagonal and a diameter) * When a circle is inseribed in a square, the length of a diameter is equal to the length of a side of the square. (48 = PX) A Pel 8 TS, VW dI 7 n lE PARABOLAS For any real numbers a, b, c with e #0: i+ Dx +0 Î (he equation ofa paralola whose ax of yy immetry i a vertical line. * Conversely. the equation of any parabola with a vertical axis of symmetry has an equation of the formy — ani + dvi c. - Ts equationof the parabola" axis of symmenis i È * The vertex of the parabola is (he point on rhe parabola whose x-coordinare is «If 0>0, the parabola opens upward and the vertex.is the lowest point on the parabola. + 1f0 <0 the parabola opens downward and the vertex is the highest point on the parabola. vaaibrie Given any three points, £, Q, and £ that do not lie on a line, there is a parabola with a vertical axis of symmetry that passes through them. ax2+ bv + c are the (real) solutions of the ] The x-inrercepts of the graph of the parabola ; equarionar î+ Av +c=0. unit I Since the graph ofy = x °— 6x +8 crosses the x-axis at 2 and 4. the quedratic equation x 2 6x +8 = 0 has two seal solutions: x= 2 and =4. TL Since the grapli o0y = 21 2-81 + 8 ciosses de i-nis culy al 2, (he quadiatic equation 2x 2-84 > 8 = 0 has onlyone resi solution: x=2. IIL Since the graph cfy = 2 2- 8x +9 does not cross the x-axis, he quadratic equation 2x 2- 81 +9 = 0 has no real solutions. Similarly. from graphs iv. v. and vi. you can see that the equation x Î+ 4= 0 has two real solutions: the equation = + x -9 = 0 tas only one real solution: and the equation —x 2+ 6 Il = has no real solution. + xorientation:y? = dpr * The focus is at the point (p,0), and the directrix has the equation e = —p * The parabola opens right if p>0 and opens left if p<0 * y-orientation:e? = 4py *. The focus'is at the point (0,p), and the directrix hasthe equation y = —p * The parabola opens right if p>0 and opens left if p<0 FUNCTIONS RELATIONS Ri= {0.0 (1. 1), 8,2} Ra {(0. 1). (1. 1). 2, 1} = {(x.y)|x=0, 1. or2 andy =1} Ra = {(0. 0). (1. 1), 2, 4)} = {(x.p)|y=x%andx=0, 1, 0r2} Ra= {(x,y)|y=x?and x is an integer} Rs= {x y)|y=x°andx20} Re= {(.y)ly=x3} Ra= {(x.y)lx=y} Re= {(x,p)lx2+y?=25} Re= {(x,y)lx is a state in the United States and y is the capital of x} Rio= {(x,)lx is a word inthe English language and y is the mumber of letters inx} STATISTICS If A is the average (arithmetic mean) of a set of u numbers, then: 24 sum An *nA=sum COUNTING If two jobs need to be completed and there are m ways to do the first job and n ways to do the second job, then there are m x n ways to do one job followed by the other: This principle can be extended ro any number of jobs. PROBABILITY If E is any event, the probability that E will occur is given by the number of ways event E can occur the oral mimber of possible ontcomes assuming that all of the possible outcomes are equally likelv. PIE)= Let E be an event and P(£ ) the probability it will occur. «If E is impossible (such as getting a number greater than 15), P(E) = 0. *Ifit is certain that E will occur (such as getting a prime number), P(E) = 1. *Imall cases OSP(E) SI. * The probability that event E will not occur is 1 — P(E). * If two or more events are mutually exclusive and constitute all the outcomes, the sum of their probabilities is 1. (For example, P(even) + P(odd) =135=1.) * The more likely that an event will occur; the higher its probability (the closer to 1 it is); the less likely that an event will occur, the lower its probability (the closer to 0 it is). If an experiment is repeated two (or more) times, the probability that an event occurs and then a second event occurs is the product of the probabilities thar each event occurs. FUNCTIONS FUNCTIONS CONTINUED A function is a relation for which each value from the set the first components of the ordered pairs is associated with exactly one value from the set of second components of the ordered pair. ODD AND EVEN FUNCTIONS COMBINING FUNCTIONS MISCELLANEOUS FUNTIONS INVERSES POLYNOMIAL FUNCTIONS QUANDRIC FUNCTIONS EXPONENTIAL AND LOGARITHMIC FUNCTIONS COMPLEX NUMBERS Natural, relative, rational numbers; simple problems that can be solved with elementary methods {analysis direct, first degree equation, first degree system, ...) O Real numbers, radicals and calculation with powers and radicals 0 Basics of statistics and probability (elementary operations between events, elementary counting, arithmetic average and weighted average of statistical data, frequency tables) O Euclidean plane geometry (basic results on congruence, similarity, Pythagorean theorems and Euclid) and basic concepts of space geometry; areas and volumes of elementary figures 0 Algebra of monomials and polynomials (operations with monomials and polynomials, factorization of polynomials) Analytical geometry (Equation of the straight line and the circumference; ellipse and hyperbola with axes of symmetry parallel to the coordinated axes). Description and recognition of simple subsets of the plan described by inequalities in two variables. O Exponentials and logarithms and their properties (operations, base change) Goniometry and trigonometry (sine, cosine and tangent functions and their fundamental properties; resolution of right triangles) O Elementary graphs: first and second degree polynomial functions, absolute value, function exponential and logarithmic, trigonometric functions (basic graphs and graphs obtained with translations and symmetries with respect to the coordinated axes and to the straight lines y= x andy =) O Algebraic equations and inequalities (attributable to the | or Il degree), irrational, with absolute value. |D Exponential, logarithmic and trigonometric equations and inequalities.