Examples of the Processes of the Differential and Integral Calculus |
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Page 94
... maximum or minimum . 66 du Equate to zero or dx infinity let a be a possible value of a obtained from either : du of these equations ; then , if changes sign from + to or - dx - from to when , h being an indefinitely small quantity , a ...
... maximum or minimum . 66 du Equate to zero or dx infinity let a be a possible value of a obtained from either : du of these equations ; then , if changes sign from + to or - dx - from to when , h being an indefinitely small quantity , a ...
Page 95
... maximum or a minimum accordingly as this differential coefficient is negative or positive . " If du dx = † ( x ) .v , y ( x ) being an es- sentially positive function of x , the following modification of this principle in many cases ...
... maximum or a minimum accordingly as this differential coefficient is negative or positive . " If du dx = † ( x ) .v , y ( x ) being an es- sentially positive function of x , the following modification of this principle in many cases ...
Page 96
... maximum . u2n + 1 2n is so also ; If u be a positive maximum or minimum , au2 " is also a maximum or minimum . If u be a negative maximum or minimum , au2n will be a minimum or maximum . The same remarks apply to fractional powers of ...
... maximum . u2n + 1 2n is so also ; If u be a positive maximum or minimum , au2 " is also a maximum or minimum . If u be a negative maximum or minimum , au2n will be a minimum or maximum . The same remarks apply to fractional powers of ...
Page 97
... maximum nor a minimum ; ( 4 ) the root 1 makes u a maximum ; the root 3 makes u a minimum . - u = ( x − a ) " . du = n ( x − a ) " -1 ; - dx x = a makes u = 0 , which is a minimum when n is even , du because changes sign from to when ...
... maximum nor a minimum ; ( 4 ) the root 1 makes u a maximum ; the root 3 makes u a minimum . - u = ( x − a ) " . du = n ( x − a ) " -1 ; - dx x = a makes u = 0 , which is a minimum when n is even , du because changes sign from to when ...
Page 98
... maximum . This is the solution of the problem . To divide the number a into two parts , such that the product of the mth power of the one by the nth power of the other shall be a maximum . du 1 x2 - x ( 6 ) u = = = 0 . 1 + x2 dx ( 1 + ...
... maximum . This is the solution of the problem . To divide the number a into two parts , such that the product of the mth power of the one by the nth power of the other shall be a maximum . du 1 x2 - x ( 6 ) u = = = 0 . 1 + x2 dx ( 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