Question

True or False For a given number of rectangles, MRAM always gives a more accurate approximation to the true area under the curve than \(\operatorname{RRAM}\) or LRAM. Justify your answer.

Short answer

False. While MRAM often provides a more accurate approximation compared to LRAM and RRAM, it is not always the case. The accuracy of these methods depends on the function's behavior on the considered interval.

Step-by-step solution

Understanding the Methods

First, understand how the three methods work. LRAM, RRAM, and MRAM are methods used in calculus to approximate the definite integral of a function. They divide the area under the curve into a number of rectangles and add up the area of these rectangles to estimate the integral.

Left and Right Rectangle Approximation Method (LRAM & RRAM)

In LRAM, the height of each rectangle is determined by the value of the function at the left endpoint of the respective interval, while in RRAM, the height is determined by the value at the right endpoint. These methods can overestimate or underestimate the true area under the curve depending on the nature of the function on the given interval.

Middle Rectangle Approximation Method (MRAM)

MRAM, on the other hand, uses the value of the function at the midpoint of the interval to determine the height of each rectangle. This often provides a more accurate approximation compared to LRAM and RRAM.

Comparison and Conclusion

However, it's not always correct to assert that MRAM always gives a more accurate result than LRAM or RRAM. The accuracy of these methods depends on the behavior of the function on the considered interval. For instance, if a function is increasing or decreasing continuously on the interval, then LRAM and RRAM can provide a more accurate result than MRAM.