# If the area under the curve of f(x) = 25 – x2 from x = –4 to x = 0 is estimated using four approximating rectangles and left endpoints, will the estimate be an underestimate or overestimate?

underestimate

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The 4-bar Riemann sum will be an underestimate of the true area.

The rate of change of f(x) is given by its first derivative:

#f(x) = 25 - x^2#

#f'(x) = -2x#

Since our interval is from

*When the function is increasing on the entire interval, a left sum will ALWAYS produce an UNDERESTIMATE of the area*.

This is because the bar "levels out" at the leftmost point, while the curve continues to rise above it, covering more area than the bar itself.

If that doesn't quite make sense, here's a picture of the graph with the four left-sum bars. I find that pictures have a way of intuitively explaining things that words just can't convey sometimes. Hopefully this graph helps your understanding of why a left-sum will always underestimate an increasing function.

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The estimate using four approximating rectangles and left endpoints will be an underestimate.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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