exam1solution_pdf

exam1solution_pdf - Version 045 EXAM 1 Radin(56635 This...

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Version 045 – EXAM 1 – Radin – (56635) 1 This print-out should have 18 questions. Multiple-choice questions may continue on the next column or page – fnd 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. cor- rect 6. 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
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Version 045 – EXAM 1 – Radin – (56635) 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 Fg- 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 Fgures 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 Fgures 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 Fgures only consists entirely of graphs of anti-derivatives of f . keywords: antiderivative, graphical, graph, geometric interpretation /* If you use any of these, Fx the comment symbols. 002 10.0 points ±or each n the interval [5 , 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 s i = 1 (4 x i sin x i ) Δ x as a deFnite integral. 1. limit = i 8 5 4 x dx 2. limit = i 5 8 4 sin x dx 3. limit = i 5 8 4 x dx 4. limit = i 5 8 4 x sin x dx 5. limit = i 8 5 4 sin x dx 6. limit = i 8 5 4 x sin x dx correct Explanation: By deFnition, the deFnite integral I = i b a f ( x ) dx of a continuous function f on an interval [ a, b ] is the limit I = lim n n s i = 1 f ( x i ) Δ x of the Riemann sum n s 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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exam1solution_pdf - Version 045 EXAM 1 Radin(56635 This...

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