University:
California State University, Northridge
Course:
ME 101/L | Introduction to Mechanical Engineering and Lab
Academic year:

2024
Views:

226

Pages:

13
Author:

Darnell Watts

Chapter 17 Mathematics Review 17.1 Algebra  Three basic laws    Commutative: a + b = b + a Distributive: a ( b + c ) = a b + a c Associative: a + ( b + c ) = ( a + b ) + c 17.1 Algebra  Exponents   Used for many manipulations Examples    xa xb=xa+b xab=(xa)b Logarithms  Related to exponents   bx = y then x = logby Table 17.1.5 17.1 Algebra  Quadratic Formula    Binomial Theorem    Used to expand (a+x)n Formula 17.1.7 Partial Fractions    Solves ax2 + bx + c = 0 Formula 17.1.6 Used for simplifying rational fractions Formulas 17.1.8, 17.1.9, 17.1.10, 17.1.11 Examples 17.2 Trigonometry    Involves the ratios between sides of a right triangle sine, cosine, tangent, cotangent, secant, and cosecant are the primary functions Trigonometry identities are often used    For all triangle we can also use the laws of sines and cosines Some other equations that can be found in your book are    17.2.3, 17.2.4, 17.2.5, 17.2.6, 17.2.7 Pythagorean Theorem 17.2.10 Hyperbolic Trig Functions 17.2.11 Examples Не удается отобразить рисунок. 17.3 Geometry   Used to analyze a variety of shapes and lines The equation for a straight line  Ax + By + C = 0    This equation can also be written in Pint-slope, Slopeintercept, and Two-intercept forms Distance between a line and a point is given in Formula 17.3.5 The general equation of the second degree is Ax 2 + 2 Bxy + Cy 2 + 2 Dx + 2 Ey + F = 0 17.3 Geometry  This equation is used to represent conic sections   Classified on page 473 Ellipse, Parabola, Hyperbola   More information on pages 474-475 Examples 17.4 Complex Numbers  Complex numbers consist of a real (x) and imaginary (y) part     x+iy where i= In electrical engineering j is used instead of i because i is used for current x + iy = re iθ Useful to express in polar form Euler’s equation is also commonly used eiθ = cos θ + i sin θ   Other useful equations can be found on page 477 Examples 17.5 Linear Algebra  Used to solve n linear equations for n unknowns    Determinants of matrices are often used in calculations    Uses m x n matrices Many manipulations of this basic equation are shown on page 479 Illustrated on page 480 Eigenvalues are used to solve first-order differential equations Examples [c ] = ∑ [a ][b ] n ij k =1 ik kj n aij = ∑ aij Aij j =1 ( A − λI ) x = 0 17.6 Calculus  We first write derivatives using limits    Some basic derivatives are shown on pages 484-485 Used to indicate points of inflection, maxima, and minima L’Hospial’s rule when f(x)/g(x) is 0 or infinity 17.6.6 17.6 Calculus  Inversely we have integration         Used for finding the area under a curve Equation 17.6.7 Can be used to find the length of a curve Used to find volumes Definite when there are limits When indefinite a constant is added to the solution Basic Integrals on page 486 Examples 17.7 Probability and Statistics   The probability of one events’ occurrence effects the probability of another event Probabilities n! P(n, r ) = (n − r )!  (n − 1)! P(n, r ) = (n − r )! C ( n, r ) = Many combinations can occur     n! r!(n − r )! P(A or B) = P(A)+P(B) P(A and B)=P(A)P(B) P(not A) = 1-P(A) P(either A or B)=P(A)+P(B)-P(A)P(B) 17.7 Probability and Statistics   Probability ranges from 0 to 1 Additional equations on page 490       Arithmetic Mean Median Mode Standard Deviation Variance Examples