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I'm having trouble determining when a left-hand approximation is an over/under estimate for a given function.

enter image description here

For example, for the graph above $f(x)$=$ln(x)$ and the function is increasing for all $x$ > 0.Since the function continues to increase, doe that mean the left-hand approximation would be an underestimate? $$\\$$ Update:

enter image description here

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  • $\begingroup$ Yes. Try drawing the rectangles for a Riemann sum approximation. Since they are 'anchored' at the left side, the curve lies above the rectangles, and therefore its area exceeds the areas of the rectangles. $\endgroup$ – Nick Peterson Jul 11 '17 at 5:35
  • $\begingroup$ @NickPeterson so would that mean the right-hand approximation would be an overestimate? $\endgroup$ – AmR Jul 11 '17 at 5:35
  • $\begingroup$ Yep. Again: draw the rectangles. $\endgroup$ – Nick Peterson Jul 11 '17 at 5:36
  • $\begingroup$ @NickPeterson is the updated image in my question correct for the left-hand? $\endgroup$ – AmR Jul 11 '17 at 5:42
  • $\begingroup$ Yep! That's the image I was suggesting. $\endgroup$ – Nick Peterson Jul 11 '17 at 5:53
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When the function is always increasing, that means the left-hand sum will be an underestimate and the right-hand sum will be an overestimate.

When the function is always decreasing, that means the right-hand sum will be an underestimate and the left-hand sum will be an overestimate. $$\\$$

For the function $f$($x$)=$ln$($x$), it is always increasing.

This site may help you understand better: http://www.shmoop.com/definite-integrals/compare-left-right-sum.html

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