Math191_Week12_section

# Math191_Week12_section - dx x 2(1 e x is discontinuous at x...

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Math 191 Problem Set for Week 11 Section 1. You are planning to use Simpson’s Rule to estimate the value of the integral R 2 1 f ( x ) dx with an error magnitude less than 10 - 5 . You have determined that | f (4) ( x ) | ≤ 3 throughout the interval of integration. How many subintervals should you use to assure the required accuracy? (Remember that for Simpson’s Rule the number has to be even.) 2. Evaluate the improper integrals. a. R 0 -∞ xe 3 x dx b. R -∞ 4 dx x 2 +16 c. R - 2 ( θ +1) 3 / 5 3. Which of the improper integrals converge and which diverge? a. R 0 e - u cos udu b. R 1 e - t t dt c. R -∞ dx x 2 (1+ e x ) sol) R -∞ dx x 2 (1+ e x ) = R - 1 -∞ dx x 2 (1+ e x ) + R 0 - 1 dx x 2 (1+ e x ) + R 1 0 dx x 2 (1+ e x ) + R 1 dx x 2 (1+ e x ) ; The reason that we need four subintervals of the integration is 1) the function
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Unformatted text preview: dx x 2 (1+ e x ) is discontinuous at x = 0 and so two limits are required for 0-and 0 + ; and 2) the interval of the integrations includes ∞ and-∞ , and so two limits are required for ∞ and-∞ . Then lim x → 1 x 2 1 x 2 (1+ e x ) = lim x → x 2 (1+ e x ) x 2 = lim x → (1 + e x ) = 1 2 and R 1 dx x 2 diverges ⇒ R 1 dx x 2 (1+ e x ) diverges by another version of Limit Com-parison Test that I mentioned in section ⇒ R ∞-∞ dx x 2 (1+ e x ) diverges. 1...
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