Math 1B Section 112 Quiz 4 Thursday 20 September 2007 Theo Johnson Freyd theojf math berkeley edu Name 1 3 pts Let s say I believe that y is a function of x and I do an experiment and get the following values of x and y y x y 00 x 0 0 1 1 1 3 1 4 x 0 1 2 3 4 For the particular measurement I m making I want to estimate the trapezoid approximation with n 4 I get Z 4 y x dx 6 R4 0 y x dx If I use 0 How good of an estimate is this I e what is the expected error You can t evaluate the second derivative y 00 x exactly but you can estimate it if y x 1 a y x b and y x 1 c then y 00 x a 2b c So estimate y 00 1 y 00 2 and y 00 3 and use these to estimate the error E4T K b a 3 12n2 1 pt K max y 00 x 1 1 pt EnT E4T 1 43 12 42 1 3 1 1 pt Determine whether the following definite integrals are convergent or divergent Evaluate each convergent integral Z 1 dx 2 3 pts x x 0 Z 1 dx 3 2 0 x p 3 2 1 1 pt So integral diverges by the p Test 2 pt x 4 dx 3x2 2x 1 x 4 x 1 2 which converges by p Test x3 3x2 2x x3 x Z 3 4 pts x3 So integral converges by Comparison Test x 4 3x2 2x x 4 dx 3x2 2x x3 Z 1 x3 2 5 3 x x 1 x 2 Z t x 4 dx lim 3 t 1 x 3x2 2x lim 2 ln x 5 ln x 1 3 ln x 2 t1 t t x 1 5 lim ln t x2 x 2 3 1 t 1 5 1 1 5 lim ln 2 ln t t t 2 3 12 1 2 3 5 2 ln 1 ln ln 27 32 33 2 pt 1 pt 5 pt 2 5 pt
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