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Let y =f (x) be a twice differentiable, non- negative function defined on [a.b]. The area int _(a) ^(b) f(x) dx, b gt a bounded by y = f(x), the x-axis and the ordinates at x = a and x = b can be approximated as int _(a ) ^(b) f (x) dx ~~((b-a))/( 2) {f(a) + f (b)}. Since int _(a)^(b) f (x) dx = int _(a) ^(c ) f (x) dx + int _(c ) ^(b) f (x) dx, x hatI (a,b) , abetter approximation to int _(a) ^(b) f(x) dx can be written as int _(a) ^(b)f (x) dx "@" ((c-a))/(2) {f(a)+f(c )}_ ((b-c))/(2) {f(c )+ f (b)} ""^(@)F(c ), If c = (b-a)/(2), then this gives: int _(a) ^(b )f (x) dx "@" (b-a)/(4) {f(a) + 2f (a) + 2f (c )+ f(b) },.......(1) If Lim _(t to a ) {(int _(0)^(t) f (x)dx -((t-a))/(2)(f(t)+ f(a)))/((t-a )^(3))}=0, for each fixed a, then f(x) is a polynomial of degree utmost

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