Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 61
Сторінка 20
... preceding formula ďu = c ( - ) * a2 { ( − ) 13 ( 2x ) ' + ( − ) ' = ' r ( r − 1 ) ( 2.x ) r - 2 dx = + ( - ) ' ï32 r ( r − 1 ) ... ( r − 3 ) 1.2 Now generally ( - ) = c ( − ) * p 7 , and c ( - ) * 2 ( - ) * p ! = COS ( 2x ) ' 1 + & c ...
... preceding formula ďu = c ( - ) * a2 { ( − ) 13 ( 2x ) ' + ( − ) ' = ' r ( r − 1 ) ( 2.x ) r - 2 dx = + ( - ) ' ï32 r ( r − 1 ) ... ( r − 3 ) 1.2 Now generally ( - ) = c ( − ) * p 7 , and c ( - ) * 2 ( - ) * p ! = COS ( 2x ) ' 1 + & c ...
Сторінка 40
... preceding example d2 V d2V + dy dx * = d2 V + 1 d2 V 1 dv + dp p2 dp2 p dp - In exactly the same way , the equations of condition being similar , we find d'v 2 + d2 V - d'V 1 d2 V + 2 dp dx2 dr r2 do 1 dv + r dr Also , as in the first ...
... preceding example d2 V d2V + dy dx * = d2 V + 1 d2 V 1 dv + dp p2 dp2 p dp - In exactly the same way , the equations of condition being similar , we find d'v 2 + d2 V - d'V 1 d2 V + 2 dp dx2 dr r2 do 1 dv + r dr Also , as in the first ...
Сторінка 55
... preceding example let h = -x , then tan - 1 ( x + h ) = tan - 10 = 0 ; therefore tan - 1 x = sin y . sin y . + ( sin y ) sin 2y X3 x1 + ( sin y ) 3 sin 3 y + ( sin y ) sin 4y - + & c . 3 4 Now tan - x = 2 114 π COS sy y , and cot y ...
... preceding example let h = -x , then tan - 1 ( x + h ) = tan - 10 = 0 ; therefore tan - 1 x = sin y . sin y . + ( sin y ) sin 2y X3 x1 + ( sin y ) 3 sin 3 y + ( sin y ) sin 4y - + & c . 3 4 Now tan - x = 2 114 π COS sy y , and cot y ...
Сторінка 63
... preceding , made by assuming the given equation in y to be d y = F { x + x + ( y ) } . Then if u = f ( y ) , and if we put ƒ F ( x ) = ƒ1 ( * ) , and ~ _ƒF ( x ) = ƒ ' ( x ) , and μF ( ~ ) = $ . ( ~ ) , dx 20 d x2 u = f ( y ) = ƒ1 ( x ) ...
... preceding , made by assuming the given equation in y to be d y = F { x + x + ( y ) } . Then if u = f ( y ) , and if we put ƒ F ( x ) = ƒ1 ( * ) , and ~ _ƒF ( x ) = ƒ ' ( x ) , and μF ( ~ ) = $ . ( ~ ) , dx 20 d x2 u = f ( y ) = ƒ1 ( x ) ...
Сторінка 64
... a2b + 32 ( log a ) 2 a3b + & c . 1 1.2 1.2.3 If b = 1 , or y = 1 + ca3 , C a2c2 a3 c3 1 y = 1 + a + 2 log a See Ex . 15 of the preceding Section . + 32 ( log a ) 2 + & c . 1. 2 1.2.3 ( 4 ) Let y = a + x log 64 DEVELOPMENT OF FUNCTIONS .
... a2b + 32 ( log a ) 2 a3b + & c . 1 1.2 1.2.3 If b = 1 , or y = 1 + ca3 , C a2c2 a3 c3 1 y = 1 + a + 2 log a See Ex . 15 of the preceding Section . + 32 ( log a ) 2 + & c . 1. 2 1.2.3 ( 4 ) Let y = a + x log 64 DEVELOPMENT OF FUNCTIONS .
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a² b2 a²x² angle arbitrary constant asymptote becomes C₁ c²x² Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy 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 plane of reference radius SECT singular solution spiral Substituting subtangent surface tangent plane theorem triangle vanish whence x²)³