Question

Multiple Choice Suppose \(f, f^{\prime},\) and \(f^{\prime \prime}\) are all positive on the interval \([a, b],\) and suppose we compute LRAM, RRAM, and trapezoidal approximations of \(I=\int_{a}^{b} f(x) d x\) using the same number of equal subdivisions of \([a, b] .\) If we denote the three approximations of \(I\) as \(L, R,\) and \(T\) respectively, which of the following is true? ( A ) R < T < I < L (B) R < I < T< L (C) L < I < T < R (D) L < T< I < R (E) L < I < R < T

Short answer

(B) R < I < T < L

Step-by-step solution

Understand the functions

The question tells us that the function \( f \), as well as its first and second derivatives, are positive over the interval \( [a, b] \). This means that the function \( f \) is increasing and concave up over the interval \( [a, b] \). This understanding of the nature of the function is fundamental to analyze the approximations.

Analyze the Left Rectangle Approximation Method (LRAM)

In LRAM, the height of the rectangle for each sub-interval is the value of the function at the left endpoint of the sub-interval. Since \( f \) is increasing on \( [a, b] \), the value of the function at the beginning of each sub-interval will be less than that at any other point in the sub-interval. Thus, LRAM underestimates the area under the curve of \( f \) on \( [a, b] \). In LRAM, our approximation \( L \) would be less than the exact integral \( I \).

Analyze the Right Rectangle Approximation Method (RRAM)

In RRAM, the height of the rectangle for each sub-interval is the value of the function at the right endpoint of the sub-interval. Since \( f \) is increasing on \( [a, b] \), the value of the function at the end of each sub-interval will be more than that at any other point in the sub-interval. Thus, RRAM overestimates the area under the curve of \( f \) on \( [a, b] \). In RRAM, our approximation \( R \) would be more than the exact integral \( I \).

Analyze the Trapezoidal approximation

The Trapezoidal approximation can be thought of as the average of LRAM and RRAM. Since \( f \) is increasing, the trapezoid's approximation will be more than \( L \) and less than \( R \). Thus, \( T \) lies between \( L \) and \( R \). Since \( L < I \) and \( I < R \), it means that \( T \) could either be less than \( I \) or more than \( I \). But given that \( f \) is concave up, the trapezoidal rule will overestimate the area under \( f \) in this case, meaning \( T > I \).

Compare the approximations

Based on steps 2 to 4, we found that \( L < I \), \( I < R \), and \( T > I \). Therefore, \( L < I < R \) and \( T \) is somewhere greater than \( I \) but less than \( R \). The only option that fits this is (B) which states: \( R < I < T < L \).