Examples of the Processes of the Differential and Integral Calculus |
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Page 37
... particular conditions of any given transformation will usually give us its value more readily than a substitution in the general formula . * ( 1 ) Transform x d R dy - dR y dx having given x = r cos 0 , and therefore x2 + y2 = r2 , dR ...
... particular conditions of any given transformation will usually give us its value more readily than a substitution in the general formula . * ( 1 ) Transform x d R dy - dR y dx having given x = r cos 0 , and therefore x2 + y2 = r2 , dR ...
Page 56
... particular case of that of Taylor . Ex . ( 1 ) Let u = ƒ ( x ) = ( 1 + x ) 3 ; ƒ ( 0 ) = 1 , du 1 1 = ƒ ' ( 0 ) dx d2 u da 2 ( 1 + x ) 3 = 1 , = 22 ( 1 + x ) } ' f " ( 0 ) = = = ď3 u dx3 = + & c . 1 * 56 DEVELOPMENT OF FUNCTIONS .
... particular case of that of Taylor . Ex . ( 1 ) Let u = ƒ ( x ) = ( 1 + x ) 3 ; ƒ ( 0 ) = 1 , du 1 1 = ƒ ' ( 0 ) dx d2 u da 2 ( 1 + x ) 3 = 1 , = 22 ( 1 + x ) } ' f " ( 0 ) = = = ď3 u dx3 = + & c . 1 * 56 DEVELOPMENT OF FUNCTIONS .
Page 69
... particular methods . The preceding Theorems sometimes fail from the function which is to be expanded becoming infinite or indeterminate for particular values of the variable , and , more DEVELOPMENT OF FUNCTIONS . 69.
... particular methods . The preceding Theorems sometimes fail from the function which is to be expanded becoming infinite or indeterminate for particular values of the variable , and , more DEVELOPMENT OF FUNCTIONS . 69.
Page 70
... particular ar- tifices depending on the nature of the function which is given . One of the most useful methods is to assume a series with indeterminate coefficients , and then to compare the differential of the function with that of the ...
... particular ar- tifices depending on the nature of the function which is given . One of the most useful methods is to assume a series with indeterminate coefficients , and then to compare the differential of the function with that of the ...
Page 74
... 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 last example , we 712 n2 ( n2 - 2 " ) sin nx = sinnrπ { 1 ...
... 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 last example , we 712 n2 ( n2 - 2 " ) sin nx = sinnrπ { 1 ...
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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