A Treatise on Infinitesimal Calculus: Differential calculus. 1857University Press, 1857 - Calculus |
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Page 189
... sign of such a quantity changes only by the quantity passing through zero or infinity , according as the factor , to the change of sign of which the function's change of sign is due , is in the numerator or denominator , so , when x = a ...
... sign of such a quantity changes only by the quantity passing through zero or infinity , according as the factor , to the change of sign of which the function's change of sign is due , is in the numerator or denominator , so , when x = a ...
Page 191
... change sign within these limits ; then F ( x ) F ( xo ) = { x + ( x - xo ) } ƒ ( Xn ) — ƒ ( xo ) ƒ ' { xo + 0 ( xn− xo ) } where represents a fraction mean to 0 and 1 . 9 Let the difference x - xo be divided into n parts , and let 1 ...
... change sign within these limits ; then F ( x ) F ( xo ) = { x + ( x - xo ) } ƒ ( Xn ) — ƒ ( xo ) ƒ ' { xo + 0 ( xn− xo ) } where represents a fraction mean to 0 and 1 . 9 Let the difference x - xo be divided into n parts , and let 1 ...
Page 192
... change sign between ro and xo + h , is necessary , in order that the sum of the denominators of the right - hand members of ( 4 ) may not vanish , for thereby the first member of ( 5 ) might be equal to an infinite quantity , and the ...
... change sign between ro and xo + h , is necessary , in order that the sum of the denominators of the right - hand members of ( 4 ) may not vanish , for thereby the first member of ( 5 ) might be equal to an infinite quantity , and the ...
Page 193
... change sign within these limits ; then , by virtue of ( 9 ) , we have F ′ ( xo + h1 ) F " ( xo + 0h1 ) = f ' ( xo + h1 ) f ' ( xo + 0h ) ; and replacing th1 by h2 , and observing that he is less than h1 , we have = F ′ ( xo + h1 ) F ...
... change sign within these limits ; then , by virtue of ( 9 ) , we have F ′ ( xo + h1 ) F " ( xo + 0h1 ) = f ' ( xo + h1 ) f ' ( xo + 0h ) ; and replacing th1 by h2 , and observing that he is less than h1 , we have = F ′ ( xo + h1 ) F ...
Page 194
... change sign between the limits ; then F ( xo + h ) F ( x ) - F " ( xo + 0h ) = f ( xo + h ) − f ( x6 ) fr ( xo + 0h ) * COR . I. Hence , if r ( x ) = 0 , and f ( x ) = 0 , F ( xo + h ) F " ( xo + 0h ) = f ( xo + h ) ƒ " ( xo + 0h ) ...
... change sign between the limits ; then F ( xo + h ) F ( x ) - F " ( xo + 0h ) = f ( xo + h ) − f ( x6 ) fr ( xo + 0h ) * COR . I. Hence , if r ( x ) = 0 , and f ( x ) = 0 , F ( xo + h ) F " ( xo + 0h ) = f ( xo + h ) ƒ " ( xo + 0h ) ...
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Common terms and phrases
a₁ algebraical ascending powers b₁ becomes calculated change of sign changes sign coefficients constant curve d²u d²x d²y d2y dx2 d³u d³y denominator determine differential equation dr dr dr dx dx dr dx dx dx dy dx dz dx² dx2 dy2 dy d2u dy dx dy dy dy dz dy² dy³ dz dx dz dz equal equicrescent Evaluate explicit function expression F(xo F(xo+h factor finite quantity fraction given Hence homogeneous function increases increments indeterminate form infinite infinitesimal Infinitesimal Calculus infinity logarithm loge maxima and minima minimum value negative partial derived-functions particular values positive primitive equation proper fraction replaced result roots Similarly sin x singular value substituting suppose symbols Taylor's Series tion total differentials vanish variation whence x+▲x zero
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