Examples of the Processes of the Differential and Integral CalculusJ and J. J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 64
Сторінка 16
... factors of the first degree , as into ( x + a ) ( x + ẞ ) , and then differentiating the product ( x + a ) " ( x + B ) " by the Theorem of Leibnitz ; but instead of doing so we shall make use of two formulæ given by Lagrange * . Let u ...
... factors of the first degree , as into ( x + a ) ( x + ẞ ) , and then differentiating the product ( x + a ) " ( x + B ) " by the Theorem of Leibnitz ; but instead of doing so we shall make use of two formulæ given by Lagrange * . Let u ...
Сторінка 31
... 6 " x -x E dx dx d dx This may be put under the form ...... d -x d d ( n - 1 ) x ( n - 1 ) x ( n - 2 ) x 6 6 dx dx to n factors . d € * 6 dx - I d da Now by the theorem given in Ex . 18 , CHANGE OF THE INDEPENDENT VARIABLE . 31.
... 6 " x -x E dx dx d dx This may be put under the form ...... d -x d d ( n - 1 ) x ( n - 1 ) x ( n - 2 ) x 6 6 dx dx to n factors . d € * 6 dx - I d da Now by the theorem given in Ex . 18 , CHANGE OF THE INDEPENDENT VARIABLE . 31.
Сторінка 32
... factors , we find d " u y " d y = [ { d d - ( n ( n - dx dx ( 7 ) Change the independent variable in d'u du - ( 1 − y2 ) y + n2 u = 0 , dy dy d U. x dr from y to , having given y = cos æ . The result is d'u + n2u = 0 . dx ( 8 ) Change ...
... factors , we find d " u y " d y = [ { d d - ( n ( n - dx dx ( 7 ) Change the independent variable in d'u du - ( 1 − y2 ) y + n2 u = 0 , dy dy d U. x dr from y to , having given y = cos æ . The result is d'u + n2u = 0 . dx ( 8 ) Change ...
Сторінка 74
... factor of one or other series to vanish . find ( 7 ) To expand sin na in ascending powers of sin a . Proceeding in the same manner as in the last example , we sin næ = sin nrπ { 1 ( sin x ) * - & c . } n2 n2 ( n2 - 22 ) - ( sin x ) 2 + ...
... factor of one or other series to vanish . find ( 7 ) To expand sin na in ascending powers of sin a . Proceeding in the same manner as in the last example , we sin næ = sin nrπ { 1 ( sin x ) * - & c . } n2 n2 ( n2 - 22 ) - ( sin x ) 2 + ...
Сторінка 82
... factor alone becomes indeterminate , we need only differentiate that factor so as to find its real value when x = 1 , and then multiply it by the value of the first factor when x = 1. The real value of the fraction is ( 13 ) Let u ...
... factor alone becomes indeterminate , we need only differentiate that factor so as to find its real value when x = 1 , and then multiply it by the value of the first factor when x = 1. The real value of the fraction is ( 13 ) Let u ...
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a² b2 a²x² angle arbitrary constant asymptote axis becomes C₁ c²x² Cambridge circle co-ordinates condition curvature curve cycloid cylinder determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz eliminate ellipse equal Euler find the value formula function Geometry gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum minimum Multiply negative origin parabola perpendicular radius radius of curvature singular solution spiral Substituting subtangent surface tangent plane theorem tractory triangle vanish whence x²)³