Elements of Differential Calculus for Mathematics, Physics and Engineering Students |
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Page 121
... concave upwards if f ' ( x ) is an increasing function at x = p , i.e. ƒ " ( † ) > 0 . f " When the curve is concave downwards at p , a portion of the curve lies below the tangent at p . If the curve is concave upwards at p , a portion ...
... concave upwards if f ' ( x ) is an increasing function at x = p , i.e. ƒ " ( † ) > 0 . f " When the curve is concave downwards at p , a portion of the curve lies below the tangent at p . If the curve is concave upwards at p , a portion ...
Page 122
... concave upwards and vice versa . If the nature of concavity changes from concavity upwards to concavity downwards , we must have f " ( p - h ) + ve ƒ " ( p ) 0 f " ( p + h ) -ve But if the change of concavity is from concavity downwards ...
... concave upwards and vice versa . If the nature of concavity changes from concavity upwards to concavity downwards , we must have f " ( p - h ) + ve ƒ " ( p ) 0 f " ( p + h ) -ve But if the change of concavity is from concavity downwards ...
Page 125
... concave upwards . In the rtn of x 0 , dy / dx is -ve and the curve there is concave downwards and x = 0 must be a point of inflexion even though the second derivative does not exist there . Illustrative Examples 1. Let f ( x ) = X x1 ...
... concave upwards . In the rtn of x 0 , dy / dx is -ve and the curve there is concave downwards and x = 0 must be a point of inflexion even though the second derivative does not exist there . Illustrative Examples 1. Let f ( x ) = X x1 ...
Contents
INTRODUCTION | 1 |
SETS FUNCTIONS RELATIONS AND GRAPHS | 3 |
THEIR LIMITS AND CONVERGENCE | 26 |
Copyright | |
13 other sections not shown
Common terms and phrases
angle approximation asymptote axis cartesian circle cone Consider constant continuous coordinates corresponding cos² cos³ cosec cosh decreasing denote derivative differentiable divergent dx dy dy dx dy/dx elements ellipse f(p+h following curves ft./sec function given hyperbola independent variables intersect interval limit loge mean value theorem minimum monotonic monotonic decreasing neighbourhood origin parabola particle pedal equation plane point of inflexion polar equation positive pedal prove radius of curvature real numbers relative maximum respect Rolle's theorem satisfied sec² sequence set of points Similarly simple harmonic motion sin² sin³ sinh straight line subtangent surface surface of revolution tangent tanh u-series is convergent values velocity x²+y² y₁ zero ди ди дх ду