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11Breview4sol - ucsc econ/ams 11b Review Questions 4...

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ucsc econ/ams 11 b Review Questions 4 Solutions 1. Compute the differentials of the functions below. a. y = x 2 - 3 x + 1 , dy = (2 x - 3) dx b. u = e x 2 - 3 x +1 , du = e x 2 - 3 x +1 (2 x - 3) du 2. Use differentials to estimate 3 28. Express your answer as a simple fraction, a/b , not in decimal form. We use the approximation formula f ( x 0 + dx ) f ( x 0 ) + dy . First, we identify the function, which is straightforward: f ( x ) = x 1 / 3 . Next, we identify x 0 . We want to set 28 = x 0 + dx , and we want dx to be relatively small, and we also want x 0 to be a point for which it is easy to evaluate f ( x ). In other words we are looking for a point, x 0 , that is close to 28 and for which the cube root is known. x 0 = 27 fills the bill. So, we have f ( x ) = x 1 / 3 , x 0 = 27 and dx = 28 - 27 = 1. Next we compute dy : dy = f 0 ( x 0 ) dx = 1 3 x - 2 / 3 0 dx = 1 3 27 - 2 / 3 · 1 = 1 27 . Finally, we plug everything back into the approximation formula, above. 28 1 / 3 27 1 / 3 + dy = 3 + 1 27 = 82 27 . Note: estimate is within 0 . 00045 of the true value of 3 28.
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