By Alan Jeffrey
NUMBERS, TRIGONOMETRIC features AND COORDINATE GEOMETRYSets and numbersIntegers, rationals and mathematics legislation Absolute price of a true numberMathematical inductionReview of trigonometric propertiesCartesian geometryPolar coordinatesCompleting the squareLogarithmic functionsGreek symbols utilized in mathematicsVARIABLES, capabilities AND MAPPINGSVariables and functionsInverse functionsSome detailed functionsCurves and Read more...
summary: NUMBERS, TRIGONOMETRIC capabilities AND COORDINATE GEOMETRYSets and numbersIntegers, rationals and mathematics legislation Absolute price of a true numberMathematical inductionReview of trigonometric propertiesCartesian geometryPolar coordinatesCompleting the squareLogarithmic functionsGreek symbols utilized in mathematicsVARIABLES, capabilities AND MAPPINGSVariables and functionsInverse functionsSome certain functionsCurves and parametersFunctions of numerous genuine variablesSEQUENCES, LIMITS AND CONTINUITYSequencesLimits of sequencesThe quantity eLimits of capabilities -/ continuityFunctions of numerous variables -/ l
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Additional resources for Mathematics for Engineers and Scientists, Sixth Edition
Example text
0 B (cos2 u(sin2 u)'2 (C (A) sin u cos u; C? 0 A sin2 u(B sin u cos u'C cos2 u; (1:52) D? 0 D cos u'E sin u; E? 51) is simplified if the cross-product term XY vanishes, and this occurs when B? 0 0. Using the trigonometric identities / cos2 u(sin2 u 0 cos 2u; 2 sin u cos u 0 sin 2u; the condition B? 53). 49). , E? and where A; F? for this value of u. 54), e and C e are non-zero and of opposite signs. It is (c) a hyperbola if A possible for the hyperbola to degenerate into a pair of straight lines.
Recalling / / / / / / / / Fig. 22 (a) Cardioid; (b) lemniscate r2 0/9 cos 2u ; (c) archimedean spiral; (d) five petal rose. 45 Polar coordinates from Fig. 8 how negative angles are measured, this means that the curve above the reference line (the x-axis) in Fig. 20 will be repeated below the line as though the x-axis were a mirror, so to the right of the origin the curve must be symmetrical about the x-axis. As the cosine function has period 2p, it follows directly that for those values of u for which cos 2u ] 0, the curve to the left of the origin must be a repeat of the curve to the right as though the y-axis were a mirror.
However, if a ( x B 0, then by the definition of the absolute value of a ( x we must have ja ( xj0 ( (a ( x), showing that ( (a( x)] 2, or, x ] a'2. Taken together the results require that x may be equal to or greater than 2'a or equal to or less than a( 2. x may not lie in the intervening interval of length 4 between x0 a( 2 and x0 a'2. This is illustrated in Fig. 4(a), where a thick line is again used to indicate points in the interval satisfied by ja ( xj] 2, and the solid dots are to be included at the ends of the appropriate interval.