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Unformatted text preview: Version 079 – EXAM 1 – sachse – (56620) 1 This printout should have 18 questions. Multiplechoice questions may continue on the next column or page – find all choices before answering. 001 10.0 points If the graph of f is which one of the following contains only graphs of antiderivatives of f ? 1. 2. 3. 4. 5. 6. cor rect Explanation: If F 1 and F 2 are antiderivatives of f then F 1 ( x ) F 2 ( x ) = constant independently of x ; this means that for any two antiderivatives of f the graph of one is just a vertical translation of the graph of the other. But no horizontal translation of the graph of an antiderivative of f will be Version 079 – EXAM 1 – sachse – (56620) 2 the graph of an antiderivative of f , nor can a horizontal and vertical translation be the graph of an antiderivative. This rules out two sets of graphs. Now in each of the the remaining four fig ures the dotted and dashed graphs consist of vertical translations of the graph whose line style is a continuous line. To decide which of these figures consists of antiderivatives of f , therefore, we have to look more carefully at the actual graphs. But calculus ensures that (i) an antiderivative of f will have a local extremum at the xintercepts of f . This eliminates two more figures since they contains graphs whose local extrema occur at points other than the xintercepts of f . (ii) An antiderivative of f is increasing on interval where the graph of f lies above the xaxis, and decreasing where the graph of f lies below the xaxis. Consequently, of the two remaining figures only consists entirely of graphs of antiderivatives of f . keywords: antiderivative, graphical, graph, geometric interpretation /* If you use any of these, fix the comment symbols. 002 10.0 points For each n the interval [2 , 8] is divided into n subintervals [ x i − 1 , x i ] of equal length Δ x , and a point x ∗ i is chosen in [ x i − 1 , x i ]. Express the limit lim n →∞ n summationdisplay i = 1 (3 x ∗ i sin x ∗ i ) Δ x as a definite integral. 1. limit = integraldisplay 2 8 3 x sin x dx 2. limit = integraldisplay 8 2 3 x dx 3. limit = integraldisplay 8 2 3 x sin x dx correct 4. limit = integraldisplay 8 2 3 sin x dx 5. limit = integraldisplay 2 8 3 x dx 6. limit = integraldisplay 2 8 3 sin x dx Explanation: By definition, the definite integral I = integraldisplay b a f ( x ) dx of a continuous function f on an interval [ a, b ] is the limit I = lim n →∞ n summationdisplay i =1 f ( x ∗ i ) Δ x of the Riemann sum n summationdisplay i =1 f ( x ∗ i ) Δ x formed when [ a, b ] is divided into n subinter vals [ x i − 1 , x i ] of equal length Δ x and x ∗ i is some point in [ x i − 1 , x i ]....
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 Fall '09
 GOGOLEV
 Calculus, lim g, dx

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