Examples of the Processes of the Differential and Integral Calculus |
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Page 74
... fraction both lines must be retained , except for some particular values of n which cause the factor of one or other series to vanish . find ( 7 ) To expand sin næ in ascending powers of sin . Proceeding in the same manner as in the ...
... fraction both lines must be retained , except for some particular values of n which cause the factor of one or other series to vanish . find ( 7 ) To expand sin næ in ascending powers of sin . Proceeding in the same manner as in the ...
Page 79
... fractions which present themselves in the operation above described , may be replaced by any equivalent fraction re- sulting from the multiplication or division of both its numerator VI Evaluation of Functions which for certain values ...
... fractions which present themselves in the operation above described , may be replaced by any equivalent fraction re- sulting from the multiplication or division of both its numerator VI Evaluation of Functions which for certain values ...
Page 80
... fraction coinciding with P the required value of for the assigned value of x . We ૨ may likewise substitute for any finite factor of all the terms of the numerator or denominator of any of the series of fractions , the value which it ...
... fraction coinciding with P the required value of for the assigned value of x . We ૨ may likewise substitute for any finite factor of all the terms of the numerator or denominator of any of the series of fractions , the value which it ...
Page 82
... fraction is = 2 = n Ρ a " - x " log a log a u = na " . ( 13 ) Let W = When x = a , ( 14 ) Let u = log a ( 1 − æ ) 1⁄2 * When x = 1 , u = 0 . ( 15 ) Find the true value of MX Ε ma 6 11 = " when a = a , X- a u = Mε " " . ( 16 ) Let u ...
... fraction is = 2 = n Ρ a " - x " log a log a u = na " . ( 13 ) Let W = When x = a , ( 14 ) Let u = log a ( 1 − æ ) 1⁄2 * When x = 1 , u = 0 . ( 15 ) Find the true value of MX Ε ma 6 11 = " when a = a , X- a u = Mε " " . ( 16 ) Let u ...
Page 83
... ( 21 ) Let u = sin no , when x = n . 1- sin x + cos x sin + cos ∞ - 1 π When ∞ = u = 1 . ( 22 ) Let u = ( ε * — € s - 7 ) 2 -- x2 cos x The value of this fraction , when a = 0 6-2 EVALUATION OF INDETERMINATE FUNCTIONS . 83.
... ( 21 ) Let u = sin no , when x = n . 1- sin x + cos x sin + cos ∞ - 1 π When ∞ = u = 1 . ( 22 ) Let u = ( ε * — € s - 7 ) 2 -- x2 cos x The value of this fraction , when a = 0 6-2 EVALUATION OF INDETERMINATE FUNCTIONS . 83.
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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³