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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 f ''(x) lt 0, x in (a,b), then at the point C(c, f (c))on y = f (x)for which F(c ) is a maximum, f '(c )is given by |
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Answer» `F'(c ) (f (B) -f (a))/(b-a)` |
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