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AP Calculus BC Monday , 09 December 2013

AP Calculus BC Monday , 09 December 2013. OBJECTIVE TSW use numerical integration to solve integrals of non-standard equations. ASSIGNMENTS DUE WEDNESDAY/THURSDAY Sec. 4.4 Sec. 4.5 Sec. 4.6 SCHEDULE CHANGE Wednesday/Thursday: TEST Sec. 4.2 – 4.6 Friday: TEST DPM

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AP Calculus BC Monday , 09 December 2013

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  1. AP Calculus BCMonday, 09 December 2013 • OBJECTIVETSW use numerical integration to solve integrals of non-standard equations. • ASSIGNMENTS DUE WEDNESDAY/THURSDAY • Sec. 4.4 • Sec. 4.5 • Sec. 4.6 • SCHEDULE CHANGE • Wednesday/Thursday: TEST Sec. 4.2 – 4.6 • Friday: TEST DPM • Monday: FINAL EXAM(Class held in DOWNSTAIRS LGI)

  2. Sec. 4.6: Numerical Integration

  3. Sec. 4.6: Numerical Integration Some integral do not possess antiderivatives that are elementary functions. Ex:

  4. Sec. 4.6: Numerical Integration One way to approximate a definite integral is to use n trapezoids.

  5. Sec. 4.6: Numerical Integration Theorem 4.16 The Trapezoidal Rule Let f be continuous on [a, b]. The Trapezoidal Rule for approximating is given by Moreover, as n → , the right-hand side approaches

  6. Sec. 4.6: Numerical Integration Ex: Use the Trapezoidal Rule to approximate Compare the results for n = 4 and n = 8. For n = 4:

  7. Sec. 4.6: Numerical Integration Ex: Use the Trapezoidal Rule to approximate Compare the results for n = 4 and n = 8. For n = 8:

  8. Sec. 4.6: Numerical Integration Ex: Use the Trapezoidal Rule to approximate Compare the results for n = 4 and n = 8. For n = 4: For n = 8: "n = 8 trapezoids gives an area that is ≈ 0.01 greater than the area given by n = 4." The exact area is:

  9. Sec. 4.6: Numerical Integration Two important points concerning the Trapezoidal Rule: 1. The approximation tends to become more accurate as n increases. 2. The Trapezoidal Rule is easily applied if the function cannot be integrated or if the function is not known.

  10. Sec. 4.6: Numerical Integration If you must approximate, it's important to know how accurate you can expect your approximation to be. The next theorem gives the formula for estimating the error in the Trapezoidal Rule.

  11. Sec. 4.6: Numerical Integration Theorem 4.19 Error in the Trapezoidal Rule If f has a continuous second derivative on [a, b], then the error E in approximating by the Trapezoidal Rule is This formula must be memorized for the test.

  12. Sec. 4.6: Numerical Integration Ex:(a) Determine the smallest value of n (fewest number of trapezoids) so that the Trapezoidal Rule will approximate the following integral with an error of less than 0.01. (b) Then, approximate the integral. The maximum value of |f''(x)| on the interval[0, 1] is |f''(0)| = 1.

  13. Sec. 4.6: Numerical Integration By the Error theorem,

  14. Sec. 4.6: Numerical Integration The error is to be less than 0.01, so The fewest number of trapezoids to give an approximation with an error less than 0.01 is 3. ALWAYS round up!!!

  15. Sec. 4.6: Numerical Integration For n = 3: Actual area  1.148 Difference ≈ 0.006556…

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