A Treatise on the Differential Calculus and the Elements of the Integral Calculus: With Numerous ExamplesMacmillan, 1864 - 404 σελίδες |
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Σελίδα 6
... preceding articles . The following may be given as a defini- tion : " The limit of a function for an assigned value of the independent variable , is that value from which the function can be made to differ as little as we please , by ...
... preceding articles . The following may be given as a defini- tion : " The limit of a function for an assigned value of the independent variable , is that value from which the function can be made to differ as little as we please , by ...
Σελίδα 25
... preceding , we will determine ( ax ) , and point out its y + Ay = √√ { a2 — ( x + h ) 2 } ; - △ y = √ { a3 − ( x + h ) 3 } − √√ ( a2 − x3 ) , Ay therefore Δα = x2 - ( x + h ) 2 √ { a2 − ( x + h ) 2 } + √ ( a2 — x2 ) ...
... preceding , we will determine ( ax ) , and point out its y + Ay = √√ { a2 — ( x + h ) 2 } ; - △ y = √ { a3 − ( x + h ) 3 } − √√ ( a2 − x3 ) , Ay therefore Δα = x2 - ( x + h ) 2 √ { a2 − ( x + h ) 2 } + √ ( a2 — x2 ) ...
Σελίδα 29
... preceding fraction by v - 1 . Let now v approach the limit 1 , then the limit of the last fraction is P - q therefore r dy_p - q x - 1 = n ” - dx r 48. Differential coefficient of x " by another method . Let y = x " , therefore ...
... preceding fraction by v - 1 . Let now v approach the limit 1 , then the limit of the last fraction is P - q therefore r dy_p - q x - 1 = n ” - dx r 48. Differential coefficient of x " by another method . Let y = x " , therefore ...
Σελίδα 40
... preceding article . ) 67. Differential coefficient of tanx and cot1x . Let ytan x , therefore x = tan y , da 1 Art . 53 , = dy cos2y ' therefore dy therefore Similarly , if dx = = = cos2y , Art . 60 , 1 1 + tan2y 1 1 + x2 y = cot1x , dy ...
... preceding article . ) 67. Differential coefficient of tanx and cot1x . Let ytan x , therefore x = tan y , da 1 Art . 53 , = dy cos2y ' therefore dy therefore Similarly , if dx = = = cos2y , Art . 60 , 1 1 + tan2y 1 1 + x2 y = cot1x , dy ...
Σελίδα 41
... preceding articles the differential coefficients of the inverse trigonometrical functions are usually determined . They may however be found without using Art . 60 . For example , suppose y = tan 1x , -1 therefore y + Ay = tan1 ( x + h ) ...
... preceding articles the differential coefficients of the inverse trigonometrical functions are usually determined . They may however be found without using Art . 60 . For example , suppose y = tan 1x , -1 therefore y + Ay = tan1 ( x + h ) ...
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A Treatise On the Differential Calculus and the Elements of the Integral ... Isaac Todhunter Δεν υπάρχει διαθέσιμη προεπισκόπηση - 2022 |
Συχνά εμφανιζόμενοι όροι και φράσεις
angle asymptote axis centre chord circle of curvature co-ordinates constant cycloid d'u d'u d²u d³u d³y deduce denote determine dF dF Differential Calculus differential coefficient diminish without limit du du dx dx dx dy dx dx dz dx² dx³ dy dx dy dy dy dy dz dy² dz dx dz dy dz dz dz² eliminate ellipse example expansion expression finite fraction given curve given equation Hence indefinitely independent variable infinite locus maxima and minima maximum or minimum minimum value negative numerator obtain parabola perpendicular point of inflexion polar positive proper fraction quantities respect Result shew subtangent Suppose tangent Taylor's Theorem tion triangle ultimate intersections unity vanish x₁ zero аф
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Σελίδα 227 - Determine a point within a triangle, such that the sum of the squares of the distances from the three sides is a minimum.
Σελίδα 332 - Shew that in general a conic section may be found which has a contact of the fourth order with a given curve at a proposed point, and shew how to find it when the length of the curve is given in terms of the angle which the normal makes with a fixed line. If the curve be an equiangular spiral, and a be the angle between the radius vector and the tangent at any point, shew that the conic section is an ellipse, the major axis of which makes with the normal to the curve an angle o> given by the equation...
Σελίδα 357 - A given curve rolls on a straight line, explain the method of finding the locus of the centre of curvature at the point of contact of the curve and straight line. If the rolling curve be an equiangular spiral the required locus will be a straight line ; if a cycloid a circle ; and if a catenary a parabola.
Σελίδα 264 - In the curve x*y* = a* (x + y], the tangent at the origin is inclined at an angle of 135° to the axis of a:. 4. In the curve x" (x + y) = a' (x—y), the equation to the tangent at the origin is y = x.
Σελίδα 193 - A person being in a boat 3 miles from the nearest point of the beach, wishes to reach in the shortest time a place 5 miles from that point along the shore ; supposing he can walk 5 miles an hour, but row only at the rate of 4 miles an hour, required the place he must land.
Σελίδα 319 - The chord of curvature passing through the origin will be obtained by multiplying 2p by the cosine of the angle between the radius vector and the normal to the curve at the point considered. (Art. 320.) Hence the chord of curvature through the origin 324. If i/r be the angle which the tangent at the point (x, y) of a curve makes with the axis of x, we have , dx therefore -=*- = dy dfy dx3 dx dx* ds (dy\* ds + (dx) Г"Чй.
Σελίδα 196 - Of all the lines drawn from the vertex of a given ellipse to the circumference of the circumscribing circle, determine that for which the portion intercepted between the two curves is a maximum. If...