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Unformatted text preview: Version 079 – EXAM 1 – sachse – (56620) 1 This print-out should have 18 questions. Multiple-choice 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 anti-derivatives of f ? 1. 2. 3. 4. 5. 6. cor- rect Explanation: If F 1 and F 2 are anti-derivatives of f then F 1 ( x )- F 2 ( x ) = constant independently of x ; this means that for any two anti-derivatives 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 anti-derivative of f will be Version 079 – EXAM 1 – sachse – (56620) 2 the graph of an anti-derivative of f , nor can a horizontal and vertical translation be the graph of an anti-derivative. 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 anti-derivatives of f , therefore, we have to look more carefully at the actual graphs. But calculus ensures that (i) an anti-derivative of f will have a local extremum at the x-intercepts of f . This eliminates two more figures since they contains graphs whose local extrema occur at points other than the x-intercepts of f . (ii) An anti-derivative of f is increasing on interval where the graph of f lies above the x-axis, and decreasing where the graph of f lies below the x-axis. Consequently, of the two remaining figures only consists entirely of graphs of anti-derivatives 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
- Calculus, lim g, dx