Consider ∫1^2 f(x) d x, where f(x)=3 lnx. (a) Make a rough...

Consider $\int_{1}^{2} f(x) d x,$ where $f(x)=3 \ln x.$ (a) Make a rough sketch of the graph of the fourth derivative of $f(x)$ for $1 \leq x \leq 2.$ (b) Find a number $A$ such that $\left|f^{\prime \prime \prime \prime}(x)\right| \leq A$ for all $x$ satisfying $1 \leq x \leq 2.$ (c) Obtain a bound on the error of using Simpson's rule with $n=2$ to approximate the definite integral. (d) The exact value of the definite integral (to four decimal places) is 1.1589, and Simpson's rule with $n=2$ gives 1.1588. What is the error for the approximation by Simpson's rule? Does this error satisfy the bound obtained in part (c)? (e) Redo part (c) with the number of intervals tripled to $n=6 .$ Is the bound on the error divided by $3 ?$

Consider $\int_{1}^{2} f(x) d x,$ where $f(x)=3 \ln x.$
(a) Make a rough sketch of the graph of the fourth derivative of $f(x)$ for $1 \leq x \leq 2.$
(b) Find a number $A$ such that $\left|f^{\prime \prime \prime \prime}(x)\right| \leq A$ for all $x$ satisfying $1 \leq x \leq 2.$
(c) Obtain a bound on the error of using Simpson's rule with $n=2$ to approximate the definite integral.
(d) The exact value of the definite integral (to four decimal places) is 1.1589, and Simpson's rule with $n=2$ gives 1.1588. What is the error for the approximation by Simpson's rule? Does this error satisfy the bound obtained in part (c)?
(e) Redo part (c) with the number of intervals tripled to $n=6 .$ Is the bound on the error divided by $3 ?$

Key Concepts

Simpson's Rule

Simpson's rule is a numerical integration technique that approximates the value of a definite integral by replacing the integrand with a piecewise quadratic function. It works particularly well for smooth functions and provides an error estimate that depends on the fourth derivative of the function. This method is widely used because it combines accuracy with efficiency, especially when the function to be integrated has continuous and bounded derivatives over the integration interval.

Error Estimation in Numerical Integration

Error estimation in numerical integration involves determining the maximum possible error that may arise when approximating an integral using a numerical method. For Simpson's rule, the error bound is typically expressed in terms of the fourth derivative of the function, where an upper bound on the absolute value of this derivative over the integration interval is used. Understanding and applying this error bound is crucial for ensuring that the numerical approximation meets a desired level of accuracy.

Higher-Order Derivatives

Higher-order derivatives, such as the fourth derivative used in Simpson's rule error analysis, provide insights into the behavior of a function's curvature. In the context of numerical integration, these derivatives measure how rapidly the curvature of the function changes, and they are integral in forming error bounds. Accurate analysis and estimation of higher-order derivatives allow for more reliable error estimates, leading to better informed decisions about the number of subintervals needed for an approximation.

Graphical Analysis of Derivatives

Graphical analysis of derivatives involves sketching and interpreting the graphs of a function's derivatives to understand its behavior over an interval. In numerical integration contexts, such graphs can help visualize how the derivatives behave, indicating where the function might be most curved. This is particularly useful when determining the maximum absolute value of a specific derivative, which is a key component in establishing error bounds for numerical methods like Simpson's rule.

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