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1 151 WebCalc Fall 2002-copyright Joe Kahlig Quiz #20 MATH 151 Section Name: November 14, 2002 1. Find the intervals where the function f is concave up and where it is concave down. 2 f (x) = e−x The domain is all real numbers. 2 f 0 (x) = −2xe−x now use the product rule to find f 00 (x) 2 2 2 f 00 (x) = −2e−x + 4x2 e−x = (−2 + 4x2 )e−x There is no place where f 00 (x) is undefined so we only have to check where f 00 (x) = 0 2 Since e−x is always positive, we need 0 = −2 + 4x2 . Solving this gives x = ± f 00 −5 − 0.5 q concave down − 0.5 0 concave up: −∞, − 1 2 + − + q q q 1 2, 1 2 , q 5 1 2, ∞ 1 2 2. If the domain of f (x) is all real numbers and 3 2 f 00 (x) = (x − 9)2 (x + 3)(x − 2)e2x −3x +6 , find the intervals where f (x) is concave up/down and label the inflection values. The possible inflection values are x = −3, 2, and 9. f 00 + −5 − −3 0 + + 2 5 9 10 Concave up: (−∞, −3), (2, 9), (9, ∞) concave down: (−3, 2) inflection values at x = −3 and x = 2 3. Give the asymptotes( vertical and horizontal) for these functions. (x − 3)(40x + 3) x2 − 9 Horizontal asym: y = 40 Vertical asym: x = −3 (a) y = (b) y = ln(70 − 3x) Horizontal asym: none Vertical asym: x = 70 3