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 Let 
A=int_(0)^(1)cos xdx. We estimate 
A using the 
L,R, and 
T approximations with 
n=100 subintervals. Which is true?
(A) 
L < A < T < R
(B) 
L < T < A < R
(C) 
R < A < T < L
(D) 
R < T < A < L

Let A=01cosxdxA=\int_{0}^{1}\cos x\,dx. We estimate AA using the LL, RR, and \newlineTT approximations with n=100n=100 subintervals. Which is true?\newline(A) L<A<T<RL < A < T < R\newline(B) L<T<A<RL < T < A < R\newline(C) R<A<T<LR < A < T < L\newline(D) R<T<A<LR < T < A < L

Full solution

Q. Let A=01cosxdxA=\int_{0}^{1}\cos x\,dx. We estimate AA using the LL, RR, and \newlineTT approximations with n=100n=100 subintervals. Which is true?\newline(A) L<A<T<RL < A < T < R\newline(B) L<T<A<RL < T < A < R\newline(C) R<A<T<LR < A < T < L\newline(D) R<T<A<LR < T < A < L
  1. Understand Approximations: Let's first understand what LL, RR, and TT approximations are. The Left (LL) approximation sums up the values of the function at the left endpoints of the subintervals, the Right (RR) approximation sums up the values of the function at the right endpoints, and the Trapezoidal (TT) approximation averages the left and right approximations. Since cos(x)\cos(x) is decreasing on the interval [0,1][0, 1], we know that the Left approximation will overestimate and the Right approximation will underestimate the actual integral value. The Trapezoidal approximation, being an average of LL and RR, should lie between them.
  2. Calculate Left Approximation: We can calculate the Left approximation LL by summing the values of cos(x)\cos(x) at the left endpoints of the subintervals. Since cos(x)\cos(x) is decreasing on [0,1][0, 1], LL will be an overestimate.
  3. Calculate Right Approximation: Similarly, we calculate the Right approximation (R)(R) by summing the values of cos(x)\cos(x) at the right endpoints of the subintervals. Since cos(x)\cos(x) is decreasing on [0,1][0, 1], RR will be an underestimate.
  4. Calculate Trapezoidal Approximation: The Trapezoidal approximation TT is the average of LL and RR. Since LL is an overestimate and RR is an underestimate, TT should be closer to the actual value AA than either LL or RR.
  5. Compare Approximations: Now, we can compare the approximations. Since cos(x)\cos(x) is decreasing on [0,1][0, 1], we have R<A<LR < A < L. The Trapezoidal approximation TT should be between LL and RR, so we have R<T<LR < T < L. Combining these, we get R<T<A<LR < T < A < L.

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