By simply drawing a graph of the function, I can find that whenever the function is concave up, the trapezoidal rule gives me a definite integral approximation larger than the actual integral of the function;whenever the function is concave down, the trapezoidal rule gives me a definite integral approximation smaller than the actual integral of the function. Is there any way to prove it true other than drawing out the function graph?
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So, what you're trying to prove is: If a curve is concave up, then any line segment connecting two points of the curve goes below the curve, and vice versa. – ThePortakal Feb 22 '18 at 10:24
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Do you know the error involved in trapezoidal rule? The error expression gives the conclusion directly. – Paramanand Singh Feb 23 '18 at 11:56