Consider ∫0^2 f(x) d x, where f(x)=(1)/(12) x^4+3 x^2. (a) Make a rough sketch of the graph of f

step 1 For part A, we have a function F of X. F of X is equal to 1 over 12 times X to the fourth plus 3

Consider ∫0^2 f(x) d x, where f(x)=(1)/(12) x^4+3 x^2. (a)...

Consider $\int_{0}^{2} f(x) d x,$ where $f(x)=\frac{1}{12} x^{4}+3 x^{2}.$ (a) Make a rough sketch of the graph of $f^{\prime \prime}(x)$ for $0 \leq x \leq 2$ (b) Find a number $A$ such that $\left|f^{\prime \prime}(x)\right| \leq A$ for all $x$ satisfying $0 \leq x \leq 2.$ (c) Obtain a bound on the error of using the midpoint rule with $n=10$ to approximate the definite integral. (d) The exact value of the definite integral (to four decimal places) is 8.5333, and the midpoint rule with $n=10$ gives 8.5089 . What is the error for the midpoint approximation? Does this error satisfy the bound obtained in part (c)? (e) Redo part (c) with the number of intervals doubled to $n=20 .$ Is the bound on the error halved? Quartered?

Consider $\int_{0}^{2} f(x) d x,$ where $f(x)=\frac{1}{12} x^{4}+3 x^{2}.$
(a) Make a rough sketch of the graph of $f^{\prime \prime}(x)$ for $0 \leq x \leq 2$
(b) Find a number $A$ such that $\left|f^{\prime \prime}(x)\right| \leq A$ for all $x$ satisfying $0 \leq x \leq 2.$
(c) Obtain a bound on the error of using the midpoint rule with $n=10$ to approximate the definite integral.
(d) The exact value of the definite integral (to four decimal places) is 8.5333, and the midpoint rule with $n=10$ gives 8.5089 . What is the error for the midpoint approximation? Does this error satisfy the bound obtained in part (c)?
(e) Redo part (c) with the number of intervals doubled to $n=20 .$ Is the bound on the error halved? Quartered?

Key Concepts

Impact of Subinterval Quantity on Error

The number of subintervals (n) in numerical integration directly affects the error bound. As the number of intervals increases, the error decreases, and the relationship is typically quadratic (error is proportional to 1/n^2 for the midpoint rule). This concept explains how doubling or further increasing n influences the approximation error, aiding in decisions regarding computational trade-offs between precision and efficiency.

Error Bound for the Midpoint Rule

Error analysis for the midpoint rule involves establishing an upper limit on the absolute error of the approximation. This bound is typically derived using the second derivative of the function and is mathematically expressed in terms of the maximum absolute value of the second derivative over the integration interval. The error bound indicates how far the numerical approximation could deviate from the exact value of the integral.

Midpoint Rule for Numerical Integration

The midpoint rule is a numerical integration technique where the function is approximated by evaluating it at the midpoint of each subinterval. The rule works by partitioning the interval into equal segments, computing the function’s values at these midpoints, and then summing the products of these values with the subinterval width. This method provides an approximate value for the definite integral and is particularly useful when the function is reasonably smooth.

Second Derivative Analysis

This concept involves computing the second derivative of a function to understand its concavity and curvature on an interval. Analyzing the second derivative helps in determining maximum and minimum values, which are crucial in bounding the error terms in numerical integration formulas. Studying its behavior, often through a sketch, provides insight into how the function’s curvature might influence approximation errors.

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