Error Estimates for the Trapezoidal RuleAs with the Trapezoidal Rule, we have a formula that suggests how we can choose to ensure that the errors are within acceptable boundaries. The following method illustrates how we can choose a sufficiently large Suppose for Then the error estimate is given by Example 3:a. Use Simpson’s Rule to approximate with .

. b. Find so that the Simpson Rule Estimate for is accurate to

:

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44Hence we must take to achieve the desired accuracy. Technology Note: Estimating a Definite Integral with a TI-83/84 CalculatorWe will estimate the value of . 1.Graph the function with the [WINDOW] setting shown below. 2.The graph is shown in the second screen. 3.Press 2nd[CALC]and choose option 7(see menu below) 4.When the fourth screen appears, press [1][ENTER]then [4][ENTER]to enter the lower and upper limits. 5.The final screen gives the estimate, which is accurate to 7 decimal places. Lesson Summary 1.We used the Trapezoidal Rule to solve problems. 2.We estimated errors for the Trapezoidal Rule. 3.We used Simpson’s Rule to solve problems. 4.We estimated Errors for Simpson’s Rule. Multimedia Links For video presentations of Simpson's Rule (21.0), see Simpson's Rule, Approximate Integration (7:21)

45and Math Video Tutorials by James Sousa, Simpson's Rule of Numerical Integration (8:48). For a video presentation of Newton's Method (21.0), see Newton's Method (7:29). Review Question 1.Use the Trapezoidal Rule to approximate with 2.Use the Trapezoidal Rule to approximate with 3.Use the Trapezoidal Rule to approximate with 4.Use the Trapezoidal Rule to approximate with 5.How large should you take n so that the Trapezoidal Estimate for is accurate to within .001? 6.Use Simpson’s Rule to approximate with 7.Use Simpson’s Rule to approximate with 8.Use Simpson’s Rule to approximate with 9.Use Simpson’s Rule to approximate with 10.How large should you take n so that the Simpson Estimate for is accurate to within .00001? Review Answers 1.2.3.4.5.Take 6.7.8.9.10.Take

46Day 1 Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. 4393dxCxx323113dxxxCxIntegrate. 1)6dx2) 23t dt3)35xdx4) du5) 32xdx6) 3xdx7) 1dxxx8) 312dxx9) 32xdx10) 43231xxdx11)23x dx12) 31dxx*Continue on next page

4713) 214dxx14) 222tdtt15)231uudu16) 165xxdx17) 2yydy

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