#50 Use Trapezoidal, Mid-point, and Simpsons Rule ## Section 7.8: Improper Integrals### Problems and Exercises46. **Graph Analysis Task:** Sketch the graph of a continuous function on the interval [0, 2] for which the right endpoint approximation with $ n = 2 $ is more accurate than Simpson’s Rule.47. **Function Properties:** If $ f $ is a positive function and $ f''(x) < 0 $ for $ a \leq x \leq b $, show that \[ T_n < \int_a^b f(x) \, dx < M_n \] where $ T_n $ and $ M_n $ are the trapezoidal and midpoint rule approximations, respectively.48. **Polynomial Integration:** Show that if $ f $ is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of \[ \int_a^b f(x) \, dx. \]49. **Trapezoidal and Midpoint Rule Relation:** Show that \[ \frac{1}{2}(T_n + M_n) = T_{2n}. \]50. **Simpson’s Rule Relation:** Show that \[ \frac{1}{3}T_n + \frac{2}{3}M_n = S_{2n}. \]These exercises challenge the understanding of numerical integration and approximations, focusing on developing a deeper grasp of error analysis and accuracy in various rules used for estimating definite integrals.

The given relation is 13Tn+23Mn=S2n Here, Tn is a trapezoidal approximation, Mn is the midpoint…

Answered: Show that T,+M, = S2n- Mn=S2n-

Math

Calculus

Show that T,+M, = S2n- Mn=S2n-

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Numerical Integration

In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differential equations. The most important aspect of this theory is error analysis.

Question

#50 Use Trapezoidal, Mid-point, and Simpsons Rule

## Section 7.8: Improper Integrals

### Problems and Exercises

46. **Graph Analysis Task:** Sketch the graph of a continuous function on the interval [0, 2] for which the right endpoint approximation with $ n = 2 $ is more accurate than Simpson’s Rule.

47. **Function Properties:** If $ f $ is a positive function and $ f''(x) < 0 $ for $ a \leq x \leq b $, show that
\[
T_n < \int_a^b f(x) \, dx < M_n
\]
where $ T_n $ and $ M_n $ are the trapezoidal and midpoint rule approximations, respectively.

48. **Polynomial Integration:** Show that if $ f $ is a polynomial of degree 3 or lower, then Simpson’s Rule gives the exact value of
\[
\int_a^b f(x) \, dx.
\]

49. **Trapezoidal and Midpoint Rule Relation:** Show that
\[
\frac{1}{2}(T_n + M_n) = T_{2n}.
\]

50. **Simpson’s Rule Relation:** Show that
\[
\frac{1}{3}T_n + \frac{2}{3}M_n = S_{2n}.
\]

These exercises challenge the understanding of numerical integration and approximations, focusing on developing a deeper grasp of error analysis and accuracy in various rules used for estimating definite integrals.

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